Simon Masnou

Benedikt Wirth

Jean-François Babadjian

Jean-Michel Morel

Séverine Rigot D'

Christèle Nina Tony

Alpàr Yueyan Anthony

Clémentine, Samer, Hugo, Magda Antonin Fatima

Fabien Perla

Un

singulier à Pierre pour les découvertes mathématiques et extra-mathématiques des plus sérieuses aux plus amusantes, pour les bouquins et les discussions, pour une collaboration rondement menée ! Et un autre à Antonin, qui est associé à la plupart de mes souvenirs de voyages mathématico-touristiques et culinaires ces dernières années ! Je dois également remercier particulièrement Clémentine et Fatima sans qui je ne m'en serais jamais sorti avec les papiers administratifs, et qui m'ont apporté une aide bienvenue dans une période dicile. Enn, merci à Hugo et Magda de me permettre de discuter politique au quotidien ! Je remercie également mes amis, Didier, Clément, Camille, Raphaël et Guillaume pour les moments de détente qui m'ont permis de souer et de déconnecter (sans doute pas assez, qu'ils m'en excusent) du travail quand il le fallait. Enn, c'est évidemment à ma famille que je dois le plus, pour leur soutien constant, de près comme de loin ; à ma mère, qui ne m'aura malheureusement pas vu achever cette thèse, à Marc et à ma douce, pour tout.

À ma mère,

Introduction

L'objet de cette thèse est l'étude du transport branché, de problèmes variationnels qui y sont liés et des structures fractales qui peuvent y apparaître. Avant d'introduire le transport branché proprement dit et d'en faire le panorama, commençons par observer que c'est une branche du transport optimal, dont on donne un très bref aperçu.

Transport optimal

Le problème du transport optimal remonte à 1781, lorsque Gaspard Monge publie son Mémoire sur la théorie des déblais et des remblais [START_REF] Monge | Mémoire sur la théorie des déblais et des remblais[END_REF]. Une répartition initiale de masse (il s'agissait de terre dans la formulation originale) étant donnée, et une nouvelle répartition étant souhaitée, il s'agit pour chaque particule se trouvant en x, de la déplacer vers une destination T (x) de façon à obtenir cette nouvelle répartition, et ce de la manière la plus économique possible. Dans le modèle de Monge, le coût de transport d'une particule est proportionnel à sa masse multipliée par la distance parcourue, et le but est de minimiser la somme de toutes ces quantités. Dans un langage moderne, le problème se formalise de la manière suivante : étant données deux mesures de même masse µ et ν, disons de probabilité, sur un compact K ⊆ R d , trouver une fonction T : K → K solution du problème min T ˆK|T (x) -x| dµ(x) : T # µ = ν , (M) où T # µ est la mesure image de µ par T , c'est-à-dire T # µ(B) = µ(T -1 (B)) pour tout borélien B, ou encore ´φ • T dµ = ´φ dν pour tout φ ∈ C (K). Dans ce cadre, on suppose que toute la masse µ(x) se trouvant en x est envoyée en un même point T (x), sans qu'elle ne se sépare. Posé sous cette forme, c'est un problème dont le coût est certes convexe, mais dont la contrainte est fortement non convexe. En eet, si µ et ν ont pour densité f et g, par changement de variable la contrainte T # µ = ν se reformule f = |det DT | • g • T, (0.0.1) en admettant que T soit régulière et injective. Ainsi, la contrainte ne passe pas à la limite pour des topologies faibles, et l'existence de solutions 1 est restée inconnue jusque très récemment. 1 Question que les mathématiciens ne se posaient pas à l'époque de Monge.

CONTENTS

Une généralisation au problème de Monge a été posée par Kantorovich en 1942 [Kan04b;Kan04a], où l'inconnue n'est plus une fonction mais un plan de transport π ∈ Π(µ, ν), soit une mesure sur l'espace produit π ∈ Prob(R d ×R d ) telle que (p 1 ) # π = µ, (p 2 ) # π = ν. Dans cette formulation, on autorise désormais les particules qui se trouvaient au départ en x à rejoindre diérentes destinations, le mouvement étant décrit par π: dπ(x, y) représente simplement la quantité de particules transportée de x vers y. Le problème étendu, appelé problème de Monge-Kantorovich, s'exprime ainsi :

min π ˆRd ×R d
|y -x| dπ(x, y) : π ∈ Π(µ, ν) .

(MK)

Il est possible de voir ce nouveau problème comme une extension (par relaxation) du problème de Monge. En particulier, à chaque fonction T est associé canoniquement un plan de transport π T ∈ Π(µ, ν) de sorte que les coûts de π T et de T coïncident : π T est la mesure concentrée sur le graphe de T , telle que la mesure d'une portion du graphe corresponde à la mesure, selon µ, de sa projection sur l'axe des abscisses : en d'autres termes π T = (Id, T ) # µ. L'intérêt du problème de Monge-Kantorovich est qu'il s'agit d'un problème de programmation linéaire : c'est un problème linéaire sous contraintes convexes. On obtient de manière quasi-immédiate la compacité et la continuité susantes pour conclure à l'existence d'un minimiseur, et toute la théorie de dualité convexe s'applique, fournissant de nombreux outils pour étudier le problème.

Néanmoins, cela ne prouve pas l'existence d'une solution au problème de Monge Une vaste théorie a été élaborée à la suite des travaux pionniers de Brenier [START_REF] Brenier | Polar factorization and monotone rearrangement of vectorvalued functions[END_REF],

reliant le transport optimal à de nombreux domaines : géométrie dans les espaces métrique, EDPs d'évolution, traitement d'images, inégalités fonctionnelles et géométriques etc. Un vaste panorama de la théorie et ses principales applications pourra être trouvé dans les ouvrages [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF], [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Villani | Grundlehren der Mathematischen Wissenschaften[END_REF], [START_REF] Ambrosio | Gradient ows in metric spaces and in the space of probability measures[END_REF].

CONTENTS

Du transport optimal au transport branché Dans le problème du transport optimal classique, le coût pour chaque particule ne dépend que de sa position de départ et de sa position d'arrivée, et non de sa trajectoire.

C'est un implicite qui vient du problème posé au départ par Monge : le coût de transport d'une particule de masse m est proportionnel à m × où est la distance parcourue, or cette distance est minimale le long d'une géodésique, c'est-à-dire d'un segment dans R d , de sorte que si le transport est optimal, les particules p doivent se déplacer en ligne droite et le coût total s'exprime bien comme p m p |y p -x p |. En réalité, dès lors que le coût de transport c d'une particule ne dépend que de sa masse m et de sa propre trajectoire γ, le problème peut se réduire à un coût ne dépendant que de m et du couple (x, y).

Évidemment, cet axiome n'est pas satisfaisant pour modéliser nombre de situations où le coût de transport de chaque particule peut dépendre du mouvement des autres particules, par exemple en fonction de la quantité de particules empruntant le même chemin. Dans le cas d'une foule voulant évacuer une pièce, chaque personne désire rejoindre une issue tout en évitant de se rentrer dedans : les fortes concentrations sont coûteuses voire proscrites, ce qui se traduit par une pénalisation ou une contrainte sur la densité de transport. C'est le domaine de ce qu'on appelle le transport congestionné Dans le cas du transport de Monge, si l'on veut transporter une masse située en un point vers une distribution uniformément répartie sur un segment, le transport optimal consiste à faire voyager toutes les particules en ligne droite, le long d'une innité (non dénombrable) de rayons de transport (voire Figure 1a); si l'on considère le transport routier, ce n'est évidemment par la manière la plus économique de procéder ! Dans un modèle de transport branché où le déplacement conjoint est moins coûteux, il est préférable de faire voyager les particules ensemble le plus longtemps possible puis de les faire se séparer an qu'elles rejoignent leurs diérentes destinations, d'où l'apparition de points de branchements et d'un réseau de transport comme l'illustre la Figure 1b.

De nombreuses structures naturelles, comme les réseaux racinaires, les branches des arbres, les réseaux uviaux, le système de vaisseaux sanguins dans le poumon etc.

partagent des traits similaires, de même que de nombreuses structures construites par l'homme : réseaux routiers, gazoducs, oléoducs, systèmes d'irrigation, réseaux électriques, réseaux d'assainissement etc. Une explication peut tenir au fait que toutes ses structures partagent un principe commun : elles doivent irriguer une certaine distribution de masse à partir d'une source (ponctuelle ou étendue) tout en optimisant un coût favorisant les fortes concentrations (par économie d'échelle dans les réseaux articiels, par soucis de préservation dans les réseaux naturels ; par exemple en biologie, les gros vaisseaux sanguins sont préférables car moins fragiles). Un objectif de la théorie du transport branché est notamment de mettre en évidence des propriétés qualitatives générales de ces structures à l'aide d'un modèle commun.

Le transport branché et ses modèles

Le principal axiome que l'on supposera est le suivant : déplacer une masse m sur une distance a un coût proportionnel à m α où α est un exposant dit concave : α ∈ [0, 1]. Le cas plus général de la H-masse (introduit dans [START_REF] Pauw | Size minimization and approximating problems[END_REF] suite aux travaux [START_REF] White | Rectiability of at chains[END_REF]), où H est croissante, sous-additive et H(0) = 0 a commencé a être étudié récemment [START_REF] Brancolini | Equivalent formulations for the branched transport and urban planning problems[END_REF][START_REF] Colombo | On the lower semicontinuous envelope of functionals dened on polyhedral chains[END_REF], mais nous nous concentrerons dans cette thèse sur le cas H(m) = m α . Le premier chapitre de la thèse dénit précisément les principaux modèles du transport branché en fournissant certaines preuves on l'espère simpliées. Commençons par le modèle le plus simple : le modèle discret, qui date des travaux de Gilbert [START_REF] Gilbert | Minimum Cost Communication Networks[END_REF] en 1967, dont l'objet était l'optimisation des réseaux de communications.

On se donne deux ensembles nis de points (x i ) i≤N , (y j ) j≤M dans R d , pondérés par des masses (a i ) i , (b j ) j et on suppose qu'ils ont même masse total, disons unitaire : i a i = j b j = 1. On peut les voir comme deux mesures de probabilité niment atomiques µ = i≤N a i δ x i et ν = j≤M a j δ y j . Le but est de connecter ces deux mesures via un graphe pondéré orienté G qui consiste en un triplet G = (V, E, Θ = (θ e ) e∈E ) où V est l'ensemble des sommets, E celui des arêtes (orientées) et θ e représente pour chaque arête e un poids associé. Un tel graphe connecte µ et ν s'il vérie les lois de Kirchho : en chaque sommet v ∈ V , masse rentrant en v = masse sortant en v, les poids a i étant considérés comme rentrant en x i , les b j comme sortant en y j . À chaque graphe est associé un coût qu'on appelle énergie de Gilbert : La principale propriété qualitative des minimiseurs est que ce sont tous des arbres (lorsque α < 1). Plus précisément, le fait que H(x) = x α satisfasse H(0) = 0 et H croissante empêche la présence de circuits (boucles qui suivent l'orientation des arêtes), et sa stricte concavité empêche la présence de cycles (boucles non orientées). Prenons l'exemple d'une source s et de deux puits p 1 , p 2 . Dans ce cas, trois topologies sont a priori possible : en V, en Y ou en L. Lorsque les deux puits sont à même distance de CONTENTS la source, la forme en L n'est jamais optimale, et lorsque l'angle (p 1 -s, p 2 -s) n'est pas trop grand, la forme en Y est préférable, comme l'illustre la Figure 3. Ceci motive l'introduction du problème d'irrigation dans sa forme eulérienne, où l'inconnue est mesure vectorielle v qu'on appelle ot d'irrigation :

E α (G) =
min {M α (v) : ∇ • v = µ -ν} , (EI α )
où la α-masse M α est dénie comme

M α (v) =    ˆK dv d H 1 (x) α d H 1 (x) si v est rectiable, +∞ sinon.
2 En théorie géométrique de la mesure, on le formule plutôt en termes de courants.

CONTENTS

On dit que v est rectiable si c'est une mesure vectorielle de la forme v = θτ H 1 E où θ est une fonction positive appelée multiplicité, E est un ensemble 1-rectiable 3 , τ (x) un vecteur unitaire tangent à E en x. Des propriétés de coercivité et de semicontinuité peuvent être démontrées pour établir l'existence de minimiseurs. Notons toutefois que la topologie considérée doit être plus forte que la convergence faible des mesures pour v, puisqu'on peut approcher faiblement un champs de vecteur lisse par une mesure rectiable de α-masse bornée, de sorte que M α n'est pas semicontinue inférieurement pour cette topologie.

Passons maintenant au modèle lagrangien, plus riche au sens où on suit, dans ce modèle, la trajectoire de chaque particule plutôt que de considérer uniquement leur champ de vitesse de ces particules, comme c'était le cas dans le modèle eulérien. Sans entrer dans les détails, un mouvement de particules est modélisé par une mesure η sur un ensemble Γ de courbes paramétrées (pour des raisons techniques, elles seront dénies sur [0, +∞] et 1-Lipschitziennes), par exemple une mesure de probabilité η ∈ Prob(Γ), de sorte que dη(γ) représente la quantité de particules voyageant suivant la trajectoire de γ. On appelle η un plan d'irrigation. La contrainte imposant à η d'envoyer µ sur ν se formalise par les conditions (e 0 ) # η = µ, (e ∞ ) # η = ν où e 0 (γ) = γ(0) et e ∞ (γ) = lim t→∞ γ(t), ce qu'on note µ η -→ ν. En chaque point x, la quantité de masse θ η (x) visitant x, appelée multiplicité, est dénie comme θ η (x) = η(γ ∈ Γ : x ∈ γ).

On note ´γ θ η (x) α-1 dx := ´+∞ 0 θ η (γ(t)) α-1 | γ(t)| dt l'intégrale de θ α-1 η le long de γ, et on dénit le α-coût de η comme la somme pour toutes les courbes γ de cette quantité, soit :

I α (η) =

ˆΓ

ˆγ θ η (x) α-1 dx dη(γ).

Cette expression, a priori barbare, admet toutefois une expression très simple sous certaines hypothèses raisonnables sur η : (LI α )

I α (η) =    ˆK θ η (x) α d H 1 (x) si η est rectiable,
Les modèles eulérien et lagrangien sont équivalents, au sens que les minima des deux problèmes d'irrigation correspondants sont les mêmes, et qu'il est possible de 3 Une réunion dénombrables de courbes lipschitziennes et d'un ensemble de mesure H 1 nulle.

CONTENTS construire un minimiseur d'un problème à partir d'un minimiseur de l'autre. Ce fait était connu depuis un certain temps mais non démontré en toute rigueur et en totalité.

Le Chapitre 2 est consacré à la preuve de cette équivalence. Un ingrédient essentiel est la preuve de la formule d'énergie (0.0.2), qui résulte d'une simple application du théorème de Fubini-Tonelli, dont l'utilisation licite n'a pas toujours été clairement justiée, puisque les mesures doivent être σ-nies pour appliquer le théorème. C'est la raison pour laquelle on démontre au préalable la rectiabilité des plans d'irrigation de coût ni. Écrivons formellement le calcul:

I α (η) . = ˆΓ ˆγ θ η (x) α-1 dx dη(γ) = ˆΓ ˆK 1 x∈γ θ η (x) α-1 d H 1 (x) dη(γ) = ˆK θ η (x) α-1 η({γ : x ∈ γ}) d H 1 (x) = ˆK θ η (x) α d H 1 (x).
I α (η) = min v:µ→ν M α (v),
(ii) si v est optimal, il peut être représenté par un plan d'irrigation v = v η où η : µ → ν est optimal, (iii) if η est optimal, alors v η est optimal.

Un bref aperçu de la théorie Donnons un aperçu rapide de la théorie, en restant assez vague sur le formalisme, certains résultats s'exprimant plus simplement dans un modèle ou dans un autre.

Angles minimaux Reprenons le cas d'une source x 0 de masse m 0 irriguant deux puits x 1 , x 2 à même distance de la source, de masses m 1 , m 2 telles que m 0 = m 1 + m 2 . Considérons un graphe G x de structure en Y, de point de branchement situé en x. Si G x est d'énergie minimale, x doit être un point critique de la fonction

x → E α (G x ) = m α 0 |x 0 -x| + m α 1 |x 1 -x| + m α 2 |x 2 -x|, ce qui s'écrit m α 0 n 0 + m α 1 n 1 + m α 2 n 2 = 0 où n i = x i -x |x i -x| . En posant k 1 = m 1 m 0 , k 2 = m 2 m 0
, on obtient les relations sur les angles:

cos θ 1 = k 2α 1 + 1 -k 2α 2 2k α 1 (0.0.3) cos θ 2 = k 2α 2 + 1 -k 2α 1 2k α 2 (0.0.4) cos(θ 1 + θ 2 ) = 1 -k 2α 2 -k 2α 1 2k α 1 k α 2 . (0.0.5) Si x 1 et x 2 sont équidistants de x 0 et m 1 = m 2 , alors θ 1 = θ 2 =: θ et x doit être équidistant de x 1 et x 2 . De plus 2θ = arccos(2 2α-1 -1) =: θ α et θ = arccos(2 α-1 ). Ainsi si θ ≥ θ α , l'optimum ne peut être une structure en Y. En notant = |x 1 -x 0 | = |x 2 -x 0 |, le coût de la structure en V est c V = m α 2 1-α ,
tandis que lorsque θ < θ α la structure en Y (avec l'angle θ α au branchement) coûte, après un calcul élémentaire : De la même manière, lorsque x 1 , x 2 , m 1 , m 2 sont quelconques, on peut vérier que la structure en Y est préférable dès lors que cos(θ) <

c Y = m α 2 1-α cos((θ α -θ)/2) < c V , et c'est le minimiseur. b 120 • (a) α = 0 b 90 • (b) α = 1/2 b (c) α = 3/4 (d) α = 1
1-k 2α 2 -k 2α 1 2k α 1 k α 2 , c'est à dire θ > arccos 1 -k 2α 2 -k 2α 1 2k α 1 k α 2 ≥ arccos(2 2α-1 -1) . = θ α .
Ainsi, si x est un point de branchement, c'est-à-dire qu'il y a au moins 2 arêtes rentrantes ou 2 arêtes sortantes en x, notées e 1 , e Theorem 0.0.5 (Irrigabilité [START_REF] Xia | Optimal paths related to transport problems[END_REF]). Si α > 1 -1 d , alors toute mesure µ supportée sur un compact de R d est α-irrigable.

N η = {x : θ η (x) > 0}.
Cette condition est nécessaire au sens où la mesure uniforme sur un cube n'est pas On constante que l'exposant β varie de 0 à 1 lorsque α va de 1 -1/d à 1, or on s'attend à ce que le réseau devienne de plus en plus dius lorsque α → 1 pour retomber sur le problème de Monge. Or on sait que l'ensemble le plus proche de 0 pour la distance de Wasserstein W 1 est évidemment la boule, dont le bord est lisse et de dimension d -1. On fait donc la conjecture suivante : Conjecture 0.0.7. Si E est solution de (0.0.6), alors

α-irrigable si α = 1 -1 d et d > 1. Si α > 1-1 d , alors pour toute paire de mesures µ, ν ∈ Prob(K) où K est un compact xé de R d , la quantité d α (µ, ν) = min{I α (η) : µ η - →
d α (µ, ν) ≤ Cm α diam(supp µ ∪ supp ν), où m = µ = ν et C = C(α, d).
X α (ν) = d α (δ 0 , ν) au sens où X α (ν ) ≤ X α (ν) + α ˆzη d(ν -ν)
min {X α (ν) : 0 ≤ ν ≤ 1, ν ∈ Prob(R d )}.
dim H (∂E) = dim M (∂E) = d -β.
Démontrer ce résultat revient à obtenir la minoration dim H ∂E ≥ d -β, mais nos stratégies n'ont pas encore permis de l'établir, laissant pour l'heure la question de la fractalité ouverte. Une résolution numérique du problème a été entreprise, basée sur le modèle de type Modica-Mortola proposé par Oudet et Santambrogio [START_REF] Oudet | A Modica-Mortola approximation for branched transport and applications[END_REF]. La fonctionnelle du transport branché, dans sa version eulérienne, est approchée au sens de la Γ-convergence par D'autres problèmes variationnels faisant intervenir X α sont proposés dans [START_REF] Maddalena | A variational model of irrigation patterns[END_REF].

M α ε (v) = ε -γ 1 ˆ|v(x)| σ dx + ε γ 2 ˆ|∇v(x)| 2 dx,
F 2 (ν) =    ˆRd f (x) 2 2 dx si ν = f L et ´Rd f = 1, +∞
Le problème d'optimisation de forme déni précédemment entre dans ce cadre, avec

F (ν) = 0 si ν ∈ Prob(R d ) et 0 ≤ ν ≤ 1, +∞ sinon,
de même que le problème d'évolution de forme envisagé dans le Chapitre 5, où on considère une suite de fonctionnelles

F = F τ k avec F k (ν) = ˆRd d s (x, E k ) τ dν -λ ˆRd dν, où λ > 0 est xé et d s (•, E k ) représente la distance signée à l'ensemble E k .
Un outil essentiel pour l'étude de tels problèmes est la fonction paysage z. Par exemple, dans le cas F = F 2 , des résultats descriptifs et de régularité en fonction de z devraient pouvoir se démontrer. En particulier, l'équation d'Euler-Lagrange devrait être de la forme αz + f = cst sur {f > 0}, de sorte que f s'exprime en fonction de s. De plus, comme z(x) ≥ |x| on obtient une borne sur le support de f . Enn, si z est β-Hölder, ce qui est le cas lorsque ν est assez régulière qu'il faudrait donc démontrer au préalable alors f l'est également.

Néanmoins, ce type d'étude ne peut être eectué que pour la fonctionnelle X α = d α (δ 0 , ν), pour laquelle on sait dénir la fonction paysage, or on aimerait pouvoir remplacer δ 0 par plusieurs sources, éventuelle une mesure diuse µ. Le Chapitre 4 s'intéresse donc à dénir la fonction paysage dans le cas de sources multiples, et montrer des résultats de régularité Hölder tels que ceux connus dans le cas d'une simple source.

Pour l'instant, on arrive à généraliser la dénition et les résultats de régularités pour un plan d'irrigation optimal η : µ → ν qui possède un système ni de racines : il existe N points x 1 , . . . , x N tels que η-presque toute courbe passe par l'un de ces points. Cette condition est en particulier garantie lorsque d(supp µ, supp ν) > 0. Ceci est un premier pas vers une dénition générale de la fonction paysage pour µ et ν quelconques. La régularité de la fonction paysage devrait nécessiter en revanche certaines hypothèses sur le réseau ou les mesures à connecter (séparation des mesures, réseau admettant un nombre ni de racine, par exemple) qui ne sont pas réalisées en général.

La thèse s'achève par un dernier chapitre dédié à des perspectives pouvant prolonger les travaux présentés dans les précédents chapitres.

THE DISCRETE MODEL

The aim of this rst chapter is threefold. First of all, we want to give a brief but precise description of what we consider to be the main models of branched transport.

We are going to dene the discrete model, which is the oldest one as it was introduced in the 60's by Gilbert in [START_REF] Gilbert | Minimum Cost Communication Networks[END_REF], though it was not termed branched transport at that time. After that, we will go on with the two main continuous and static models: the Lagrangian model, which uses the measures on path formalism, and the Eulerian model, which uses the language of 1-dimensional currents or vector measures. Our second purpose is to make an emphasis on the Lagrangian model, which is the most used in the next chapters, and which we want to rephrase using measures on paths, which we call irrigation plans, rather than parameterized trac plans used in the works of Bernot, Caselles, Morel [START_REF] Bernot | Optimal transportation networks[END_REF], or patterns dened by Morel, Maddalena, Solimini [START_REF] Maddalena | A variational model of irrigation patterns[END_REF]. With this formalism we are able to give simple proofs of the elementary properties of minimizers, and to provide a rigorous basis to establish the equivalence between the two continuous models, which is the object of Chapter 2. The nal purpose of this chapter is to set all the denitions and state all the results to be used in the rest of the thesis, which will consequently be almost self-contained. Most of the results of this chapter were already proved or could be assembled from the existing literature, but we give proofs of some of them when we feel we were able to provide a simpler proof or when the adaptation of the existing proofs to our irrigation plans framework is not straightforward.

The discrete model

Let us start with the discrete model, which is the most concrete and easiest to grasp.

We consider two atomic probability measures µ, ν on R d :

µ = n i=1 a i δ x i , ν = m j=1 b j δ y j ,
and we want to connect them in the cheapest possible way. In this discrete case, the connection will be achieved through a graph, and the cost associated to each graph will be the so-called Gilbert energy [START_REF] Gilbert | Minimum Cost Communication Networks[END_REF].

Irrigation graph An irrigation graph G is a weighted oriented graph, that is a triple G = (V, E, Θ = (θ e ) e∈E ), where V is a set of vertices, E a set of oriented edges between vertices and θ e is a nonnegative weight associated to each edge e. We say that G sends µ to ν if (i) it satises Kirchho 's Law, that is at each vertex v ∈ V , incoming mass at v = outgoing mass at v, (ii) µ is considered as incoming and ν as outgoing: if v = x i then a i counts as incoming and if v = y j then b i counts as outgoing.

We denote the set of irrigation graphs by IG(R d ) and by IG(µ, ν) the set of those sending µ to ν. For short, we will write G : µ → ν or µ G -→ ν to say that G sends µ to ν.

Gilbert energy For α ∈ [0, 1], we associate to every irrigation graph G its Gilbert energy (see [START_REF] Gilbert | Minimum Cost Communication Networks[END_REF][START_REF] Gilbert | Steiner Minimal Trees[END_REF]):

E α (G) = e∈E θ α e |e|,
where |e| denotes the length of e. Notice that the cost is proportional to m α × : if one wants to send a mass m at distance along the graph then one should pay m α .

The discrete irrigation problem The goal is to minimize Gilbert energy among all graphs which send µ to ν, which reads:

min E α (G) . = e∈E θ α e |e| : µ G -→ ν .
(DI α )

Since the map x → x α is subadditive, i.e. (x + y) α ≤ x α + y α , ∀x, y ≥ 0, it is less expensive for the mass to travel together as much as possible. This is responsible for the observed branching structures in optimizers in the strictly subadditive case (α < 1). Let us look at the two extreme cases.

• α = 0

The mass does not appear in the cost, leaving only the total length of the graph:

it is called Steiner problem (see [START_REF] Gilbert | Steiner Minimal Trees[END_REF]),

• α = 1

The cost is linear in the mass, which corresponds the classical Monge problem in optimal transport (see [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF] for a general reference on optimal transport).

Consequently, one may view the irrigation problem as an interpolation between two well-known problems: Steiner and Monge's problems.

Before sketching a proof of existence, we will need some terminology. If e is an oriented edge, ê ∈ S d-1 denotes its orientation, ē the same edge with opposite orientation, |e| its length, and e -, e + its initial and nal extremity respectively. Denition 1.1.1. Given p = (e 1 , . . . , e k ) a sequence of oriented edges, we say that p is

• a circuit if e + i = e - i+1 for every i = 1, . . . , k, • a cycle if p = (ẽ 1 , . . . , ẽk ) is a circuit, where ẽi ∈ {e i , ēi },
with the convention e k+1 = e 1 and ẽk+1 = ẽ1 . Theorem 1.1.2. The discrete irrigation problem (DI α ) admits solutions. Moreover, if G is a solution, then (i) it has no (non-trivial) inner vertex of degree 2, (ii) it lies in the convex envelope of the sources and targets, (iii) it has no circuit, (iv) it has no cycle if α < 1.

Proof. Let us make some remarks concerning general graphs G : µ → ν.

Support. The measures µ, ν are concentrated on the convex envelope C of the atoms, which is a compact convex set, and one may project any graph G : µ → ν on C getting another graph G still sending µ to ν, but with reduced cost E α ( G) ≤ E α (G), since the projection is 1-Lipschitz. Notice that the inequality is strict if G does not lie entirely in C.

Circuits and cycles. If G has a circuit p = (e 1 , . . . , e k ), then one may remove it from the graph by removing a mass = min i θ e i to the edges e i (and then removing edges with 0 multiplicity), thus getting a new graph G. Since p was a circuit of G, G still satises Kirchho 's Law and sends µ to ν. The fact that x → x α is increasing and that we have for sure removed at least one edge guarantees that E α ( G) < E α (G).

Hence one may remove circuits iteratively while strictly reducing the cost. Now, if G has a cycle p = (e 1 , . . . , e k ), choose an orientation τ . For |t| < = min i θ e i , add t to the weights of the edges of orientation τ (we write e i ∈ τ ) and remove t to those of opposite orientation, resulting in a graph G t . At each vertex v, we have added and removed the same amount of mass owing in, thus Kirchho 's Law is still satised so that G t : µ → ν. Now let us look at the cost:

E α (G t ) -E α (G) = i:e i ∈τ |e i |((θ e i + t) α -θ α e i ) + i:e i / ∈τ |e i |((θ e i -t) α -θ α e i ) =: f (t).
If α < 1 then x → x α is strictly concave, and so is f , hence f (t) < f (0)+tf (0) = tf (0). Now choose t with a sign opposite to f (0) so that E α (G t ) < E α (G). One may repeat the operation until there is no more cycle while strictly reducing the cost.

Degree 2 vertices. One may remove vertices v which are neither a source nor a target and which have degree 2 (i.e. they have exactly two adjacent edges), replacing the two adjacent edges by a single one. Here again we reduce the cost E α ( G) ≤ E α (G) and preserve Kirchho 's Law with G : µ → ν. Notice that the inequality is strict if there are degree 2 vertex which are not trivial (i.e. such that the adjacent edges are not colinear).

Existence of minimizers In the case α = 1, it is easy to build a minimizer: construct G as a superposition of straight paths going from each source x i to each destination y j with mass a i b j . Now we place ourselves in the case α < 1 and we consider a minimizing sequence (G n ) n∈N . Thanks to the previous paragraphs, we may assume that the graphs G n enjoy all the properties (i)(iv). We may get a uniform bound on the number of edges and vertices of the graphs G n . Denote by f (L) the supremal number of vertices of graphs connecting L sources and targets, termed leaves, and satisfying (i) and (iv). Let us show that f (L) ≤ 2(L -1) by induction. For L = 2 it is clear. Now suppose that it is true for k < L and take such a graph G with L leaves. Consider a simple non-oriented path p joining two leaves x, y with a maximal number of vertices. Set x the next vertex after x along this path and ê the corresponding edge. Since x has degree at least 3, it is connected to a point which does not belong to p. We claim that the connected component C of x in G \ p consists only of edges joining x and a leaf. If it was not the case, one could either nd a simple path p ∈ G \ p connecting x to another vertex u which has at least two edges, or an edge e / ∈ p between x and a point u which is not a leaf. In all cases, if u was not a leaf then p or e may be extended to a simple path, connecting x to a leaf because G is a cycle-free1 . This extended path is still denoted by p and u the leaf which is its other extremity. This path cannot intersect p since G has no cycle. Thus the concatenation of p \ ê and p is a simple path joining a pair of leaves and which has strictly more vertices than p because p has at least 3 vertices: a contradiction. We denote by A the leaves in C and we remove C and ê from G, thus obtaining a new graph Ĝ. By construction, Ĝ : μ → ν where μ and ν are obtained from µ, ν by removing the atoms in A and possibly adding an atom at x. Moreover this graph still enjoys properties (i) and (iv). Consequently we have removed card A + 1 leaves (those in A, and x) while adding a possible one at x, hence there are at most Lleaves where = card A ≥ 1, which implies by the induction hypothesis:

card V (G) = card V ( Ĝ) + + 1 ≤ f (L -) + + 1 ≤ 2(L --1) + + 1 ≤ 2(L -1), thus f (L) ≤ 2(L -1)
for all L ≥ 2. Consequently, the number or edges and vertices of the graphs G n is uniformly bounded, and they are all contained in a common compact set, so that one may extract a subsequence making all vertices and edges converge, leading to a limit graph G : µ → ν (even though some edges may disappear in the limit). The continuity of the cost in the weights and vertices allows us to say that E α (G) = min (DI α ).

Suppose G optimal. The three rst paragraphs imply that G is supported in the convex envelope of the sources and targets, that G has no circuit and no (non-trivial) inner vertex of degree 2. Moreover if α > 1, G has no cycle.

Remark 1.1.3. In the case α = 1, it is possible to have optimizers with cycles: consider for example a square with two sources of mass 1/2 at vertices of a diagonal, and two targets of mass 1/2 at vertices of the other diagonal. An optimizer consists in 4 segments connecting each source to the two targets, with mass 1/4. However, it is possible to remove the cycles with unchanged cost (destroying the symmetry).

Examples In the case when there is only one source s and two targets t 1 , t 2 , an optimal graph is made of at most three edges. If the targets are equidistant from s with same mass 1/2, then there are two possible topologies: Y-shaped or V-shaped. When t 1 , t 2 are far enough from s, compared to their mutual distance |t 1 -t 2 |, the Y -shaped graph is the cheapest for α < 1, i.e. there is branching. The optimal angle can be explicitly computed in terms of α.

Notations and general framework

Let X be subset of R d . We denote by Γ(X) the space of 1-Lipschitz curves in X parameterized on R + , embedded with the topology of uniform convergence on compact sets. We write for short Γ = Γ(R d ), but we will often take X to be some compact subset

K of R d . If X is closed (for example a compact set or R d ) then Γ(X) is a Polish space, which can be metrized by d(γ 1 , γ 2 ) = sup n∈N 1 n min(1, γ 1 -γ 2 L ∞ ([0,n]) ). If φ : X → R and ψ : X → R d are Borel functions on X, γ ∈ Γ(X), we set ˆγ φ(x) dx := ˆ∞ 0 φ(γ(t))| γ(t)| dt and ˆγ ψ(x) • dx := ˆ∞ 0 ψ(γ(t)) • γ(t) dt
provided these integrals exist.

Length and stopping time If γ ∈ Γ(X), we dene its stopping time and its length respectively by

T (γ) = inf{t ≥ 0 : γ is constant on [t, +∞[}, L(γ) = ˆ∞ 0 | γ(t)
(π 0 ) # η = µ, (π ∞ ) # η = ν,
where π 0 (γ) = γ(0), π ∞ (γ) = γ(∞) := lim t→+∞ γ(t) and f # η denotes the push-forward of η by f whenever f is a Borel map 2 . We denote by IP(µ, ν) the set of irrigation plans irrigating ν from µ:

IP(µ, ν) = {η ∈ IP(R d ) : (π 0 ) # η = µ, (π ∞ ) # η = ν}. 2 Notice that lim t→∞ γ(t) exists if γ ∈ Γ 1 (R d ),
and this is all we need since any irrigation plan is concentrated on Γ 1 (K).

Arc length parameterization

We say that a curve γ ∈ Γ is parameterized by arc length if it has unit speed until it stops, i.e. | γ(t)| = 1 for a.e. t ∈ [0, T (γ)[. If an irrigation plan η ∈ IP(R d ) is such that η-a.e. curve γ is parameterized by arc length, we say that η is itself parameterized by arc length. Set φ : γ → γ the map which associates to each curve γ ∈ Γ(K) its arc-length parameterization 3 . One may dene the arc length parameterization of η as η := φ # η and check that (π 0

) # η = (π 0 ) # η and (π ∞ ) # η = (π ∞ ) # η.
Multiplicity Given an irrigation plan η ∈ IP(R d ), let us dene the multiplicity

θ η : R d → [0, ∞] as θ η (x) = η(γ ∈ Γ(R d ) : x ∈ γ).
It represents the amount of mass which ows by x through curves of η. We call network associated to η the set N η of points with positive multiplicity, i.e. points that are really visited by η:

N η := {x ∈ R d : θ η (x) > 0}. Simplicity If γ ∈ Γ, we denote by m(x, γ) = #{t ∈ [0, T (γ)] ∩ R + : γ(t) = x} ∈ N ∪ {∞}
the multiplicity of x on γ, that is the number of times the curve γ visits x. We call simple points of γ ∈ Γ those which are visited only once, i.e. such that m(x, γ) = 1 and denote by S γ the set of such points. We say that γ is simple if γ \ S γ = ∅ and essentially simple if H 1 (γ \ S γ ) = 0. As usual we extend these denitions to irrigation plans, saying that η is simple (resp. essentially simple ) if η-a.e. curve is simple (resp.

essentially simple). Finally we set m η (x) := ˆΓ m(x, γ) dη(γ)

which represents the mean number of times curves visit x. Notice that

θ η (x) = ˆΓ 1 x∈γ dη(γ) ≤ ˆΓ m(x, γ) dη(γ) . = m η (x)
so that m η (x) is in a way the full multiplicity at x.

The Lagrangian irrigation problem

For α ∈ [0, 1] we consider the irrigation cost

I α : IP(R d ) → [0, ∞] dened by I α (η) := ˆΓ ˆγ θ η (x) α-1 dx dη(γ),
with the conventions 0 α-1 = ∞ if α < 1, 0 α-1 = 1 otherwise, and ∞ × 0 = 0. If µ, ν are two probability measures on R d , the Lagrangian irrigation problems consists in minimizing the cost I α on the set of irrigation plans which send µ to ν, which reads min η∈IP(µ,ν) ˆΓ ˆγ θ η (x) α-1 dx dη(γ).

(LI α )

3 One may check that it is Borel.

CHAPTER 1. GENERAL THEORY Notice that I α is invariant by arc length reparameterization, thus we will often assume without loss of generality that irrigation plans are parameterized by arc length.

First variation The cost I α may be written as

I α (η) = ˆΓ Z α η (γ) dη(γ) where Z α η (γ) := ˆγ θ η (x) α-1 dx
is the α-cost the curve γ ∈ Γ w.r.t. η. This map Z η is a rst variation of I α in the following sense:

Proposition 1.2.1 (First variation inequality for I α ). If η is an irrigation plan with nite α-cost, then for all irrigation plan η the following holds:

I α (η) ≤ I α (η) + α ˆZη (γ) d(η -η).
(1.2.2)

Notice that the integral ´Zη d(η -η) is well-dened since ´Zη dη < ∞ and Z η is nonnegative, though it may be innite.

This result may be obtained by adapting [San07, Theorem 3.1].

Existence of minimizers

In this section we prove the lower semicontinuity and compactness results leading to the proof of existence of minimizers by the direct method in the calculus of variations [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]. We recall here that, unless stated otherwise, continuity properties on Γ relate to the topology of uniform convergence on compact sets and on IP(R d ) to the weaktopology in the duality with C b (Γ(R d )).

Closedness and compactness results For M > 0, we set

Γ M (X) := {γ ∈ Γ(X) : T (γ) ≤ M }.
Notice that if K ⊆ R d is a compact set then Γ M (K) is a compact set for the uniform topology by Ascoli's Theorem. For C > 0, we dene IP C (X) as the set of irrigation plans η on X which satisfy:

T(η) := ˆΓ(X) T (γ) dη(γ) ≤ C,
: η → (π 0 ) # η and π ∞ : η → (π ∞ ) # η on IP(R d ). Proposition 1.2.2. The maps π 0 , π ∞ : IP C (R d ) → Prob(R d ) are continuous 4 . In particular IP C (µ, ν) is a closed subset of Prob(Γ(R d ))
. 4 Recall that we always endow spaces of measures with their weak-topology in the duality with C b .

Proof. Clearly the map π 0 : γ → γ(0) is continuous on Γ(R d ), hence π 0 is continuous on IP(R d ). However π ∞ : γ → γ(∞) dened on Γ 1 is not necessarily continuous thus π ∞ needs not be continuous on IP(R d ). Nevertheless, given any η ∈ IP C (µ, ν), by Markov's inequality:

η(γ : T (γ) > M ) ≤ C M , (1.2.3) which rewrites η(Γ(R d ) \ Γ M (R d )) ≤ C M . (1.2.4)
Since π ∞ is continuous on all the sets Γ M (R d ) for M > 0, inequality (1.2.4) allows us to conclude. Indeed take η n η in IP C (R d ). Take > 0 and M large enough so that

C M ≤ . For any φ ∈ C b (R d ) one has ˆΓ φ(γ(∞)) dη n (γ) = ˆΓ φ(γ(M )) dη n (γ) + ˆΓ\Γ M (φ(γ(∞)) -φ(γ(M ))) dη n (γ) and ˆΓ φ(γ(∞)) dη(γ) = ˆΓ φ(γ(M )) dη(γ) + ˆΓ\Γ M (φ(γ(∞)) -φ(γ(M ))) dη(γ) thus ˆΓ φ(γ(∞))η n (dγ) -ˆΓ φ(γ(∞)) dη(γ) ≤ ˆΓ φ(γ(M ))η n (dγ) -ˆΓ φ(γ(M )) dη(γ) + 4 φ ∞ .
We pass to the lim sup n using the continuity of π M : γ → γ(M ) on Γ(R d ), then take the limit as

→ 0 to get ˆRd φ(x)(π ∞ ) # η n (dx) → ˆRd φ(x)(π ∞ ) # dη(x), which means that π ∞ is continuous on IP C (R d ). Since IP C (R d ) is a closed subset of Prob(R d ), then so is IP C (µ, ν).
Now we just have to investigate its tightness to show that it is compact, as done in

the next lemma. Lemma 1.2.3 (Compactness). Given µ, ν ∈ Prob(R d ) and C > 0, the set IP C (µ, ν) is a compact subset of Prob(Γ(R d )).
Proof. Thus it remains to prove that it is tight. We know that there exists a compact

set X ⊆ R d such that µ(R d \ X) ≤ ε. (1.2.5) Notice that any curve γ ∈ Γ(R d ) which starts in X and such that T (γ) ≤ M stays inside the compact set K = X M := {x : d(x, X) ≤ M } because it is 1-Lipschitz. η(Γ(R d ) \ Γ M (K)) ≤ η({γ : γ(0) / ∈ X}) + η({γ : T (γ) > M }) = µ(R d \ X) + η({γ : T (γ) > M }) ≤ ε + C M ≤ 2ε if M ≥ C/ε. This holds for all η ∈ IP C (µ, ν) hence IP C (µ, ν) is a tight subset of Prob(R d ) since Γ M (K) is compact (it is actually compact for the uniform topology).
Continuity results When A is a closed subset of K, we set

[A] = {γ ∈ Γ 1 : γ ∩ A = ∅}, |A| η = η([A]), so that θ η (x) = |{x}| η =: |x| η . One may show that [A] is Borel. Our rst continuity result is that of |•| η along decreasing sequences of closed sets. Lemma 1.2.4. If (A n ) n∈N is a decreasing sequence of closed sets and A = ↓ n A n then |A| η = lim n→∞ |A n | η . Proof. Let us prove that [A] = ↓ n [A n ]. The inclusion [A] ⊆ n [A n ] is clear since [A] ⊆ [A n ] for all n. Let us take γ ∈ n [A n ].
Since belonging to some [B] only depends on the trajectory of γ, we may assume that it is parameterized by arc length. Because γ has nite length L, there is a sequence (A n ) n and a sequence (t n ) n ∈ [0, L] such that γ(t n ) ∈ A n for all n. One may extract a converging subsequence, still denoted

(t n ) n , such that t n n→∞ ---→ t ∈ [0, L]. Since the (A n )'s are decreasing closed sets, γ(t) belongs to their intersection A, hence γ ∈ [A]. By the monotone convergence theorem lim n |A n | η . = lim n η([A n ]) = η([A]) . = |A| η .
Proposition 1.2.5. For C > 0, the map

θ : R d × IP C (R d ) -→ [0, 1] (x, η) -→ θ η (x) is upper semicontinuous. Proof. Given x n → x and η n η, take > 0. If n is large enough, x n ∈ B(x, ), hence lim sup n |x n | ηn ≤ lim sup n B(x, ) ηn . Besides, using (1.2.3) one gets B(x, ) ηn ≤ η n ({T > M }) + η n ({T ≤ M } ∩ [ B(x, )]) ≤ C/M + η n (A)
where we have set

A := Γ M (R d ) ∩ [ B(x, )].
It is easy to check A is closed since T is lower semicontinuous and the ball is closed. Hence passing to the lim sup in n yields lim sup

n B(x, ) ηn ≤ C/M + η(A) ≤ C/M + B(x, ) η . (1.2.6) 1.2. THE LAGRANGIAN MODEL Taking M → ∞, one gets lim sup n |x n | ηn ≤ lim sup n B(x, ) ηn ≤ B(x, ) η ,
then we pass to the limit → 0 using Lemma 1.2.4:

lim sup n |x n | ηn ≤ lim →0 B(x, ) η = θ η (x). Recall that Z α η (γ) := ˆγ θ η (x) α-1 dx is the α-cost the curve γ ∈ Γ w.r.t. η. Thus we set Z α : Γ(X) × IP(X) -→ R (γ, η) -→ Z α η (γ)
.

Proposition 1.2.6. For any C > 0, the function Z α is lower semicontinuous on

Γ(R d ) × IP C (R d ). Proof. The case α = 1 is clear since Z 1 (γ, η) = L(γ) hence we assume α < 1. We know that the map f : (x, η) → θ η (x) α-1 is lower semicontinuous on R d × IP C (R d ) since θ : (x, η) → θ η (x)
is upper semicontinuous by Proposition 1.2.5 and because α -1 < 0. Since T is lower semicontinuous, for > 0 and n large enough we have

T (γ) ≤ T (γ n ) + , which implies that ˆT(γn) 0 f (γ n (t), η n )| γn (t)| dt ≥ ˆT(γ)- 0 f (γ n (t), η n )| γn (t)| dt. (1.2.7) Recall that IP C (R d ) is metrizable 5 for the narrow convergence as a subset of Prob(Γ(R d )) where Γ(R d ) is a Polish space, hence R d × IP C (R d ) is metrizable by some distance d. Suppose for a moment that f is Lipschitz continuous on R d × IP C (R d ) equipped with this distance. Since γ n converges uniformly to γ on [0, T (γ) -)], the func- tions g n : t → f (γ n (t), η n ) converge uniformly to g : t → f (γ(t), η) on [0, T (γ) -]. Now we have to take care of the | γn (t)| factor. Since the sequence ( γn ) n is bounded in L ∞ ([0, T (γ) -]) one may extract a subsequence ( γn k ) k such that γn k L ∞ --γ and | γn k | L ∞ --u. It is a classical result that | γ(t)| ≤ u(t) almost everywhere on [0, T (γ) -]. Denoting by •, • the duality bracket L 1 -L ∞ on [0, T (γ) -], we have ˆT(γ)- 0 f (γ n k (t), η n k )| γn k (t)| dt . = g n k , | γn k | k→∞ ---→ g, u ≥ ˆT(γ)- 0 f (γ(t), η)| γ(t)| dt
since g n k → g uniformly hence strongly in L 1 . Prior to extracting the subsequence ( γn k ) k we could have taken rst a subsequence of γn such that the left hand-side con-

verged to lim inf n ´T(γ)- 0 f (γ n (t), η n )| γn (t)| dt. Thus we actually have lim inf n ˆT(γ)- 0 f (γ n (t), η n )| γn (t)| dt ≥ ˆT(γ)- 0 f (γ(t), η)| γ(t)| dt.
5 E.g. by the Lévy-Prokhorov metric.

Finally, we use this inequality together with (1.2.7) and pass to the limit → 0 thanks to the monotone convergence theorem, which yields

lim inf n ˆT(γn) 0 f (γ n (t), η n )| γn (t)| dt ≥ ˆT(γ) 0 f (γ(t), η)| γ(t)| dt.
(1.2.8)

In general f is not Lipschitz continuous but only lower semicontinus. Nevertheless (1.2.8) still holds for our function f because it may be written as an increasing sequence of Lipschitz continuous functions f k ↑ f , hence writing the inequality with f k and using the monotone convergence theorem as k → ∞ proves that,

lim inf n Z α (γ n , η n ) ≥ Z α (γ, η) thus Z α is lower semicontinuous on Γ(R d ) × IP C (R d ).
Recall that

I α (η) = ˆΓ Z α (γ, η) dη(γ),
hence the lower semicontinuity of I α on IP C (R d ) will be obtained as a corollary to the following lemma.

Lemma 1.2.7. Let X be a subset of Prob(Γ(R d )).

(i) If f : Γ(R d ) × X → R is uniformly continuous and bounded, then the functional F : η → ´Γ f (γ, η) dη(γ) is continuous on X. (ii) If f : Γ(R d )×X → [0, +∞] is lower semicontinuous, then F : η → ´Γ f (γ, η) dη(γ)
is lower semicontinuous on X.

Proof. Let us prove the rst item. Take η n η. We set g n

(γ) = f (γ, η n ) and g(γ) = f (γ, η). The fact that f is uniformly continuous on Γ × X implies that g n → g strongly in C b (Γ). Since η n η, we have ´Γ f (γ, η n ) η n (dγ) = η n , g n → η, g = ´Γ f (γ, η) η(dγ) where •, • denotes the C b (Γ) -Prob(Γ) duality bracket.
The second item is a straightforward consequence of the fact that f can be written as the increasing limit of Lipschitz bounded functions f k , and of the monotone convergence theorem.

Theorem 1.2.8. For α ∈ [0, 1], the irrigation functional I α is lower semicontinuous on IP C (R d ).

Proof. We just put together Proposition 1.2.6 which states that Z α is lower semicontinuous on Γ(R d ) × IP C (R d ) and the previous lemma with X = IP C (R d ) and f = Z α , recalling that

I α (η) = ˆΓ Z α (γ, η) dη(γ).
The existence theorem We are now able to prove the existence theorem for the Lagrangian irrigation problem (LI α ).

Theorem 1.2.9. If µ, ν are probability measures on R d , there exists a minimizer η for the problem min η∈IP(µ,ν)

I α (η) . = ˆΓ ˆγ θ η (x) α-1 dx dη(γ) . (LI α )
Proof. We assume I α ≡ +∞, otherwise there is nothing to prove. Let us take a minimizing sequence η n , which we may assume to be parameterized by arc length. In particular

I α (η n ) ≤ C for some C > 0. Consequently T(η n ) = L(η n ) . = ˆΓ ˆγ dx dη n (γ) ≤ ˆΓ ˆγ θ η (x) α-1 dx dη n (γ) . = I α (η n ) ≤ C, which implies that η n ∈ IP C (µ, ν). Thanks to Proposition 1.2.2, IP C (µ, ν) is a compact (and metrizable) subset of Prob(Γ(R d )) hence we can extract a converging sequence η n η ∈ IP C (µ, ν) up to some renaming. By Theorem 1.2.8, I α is lower semicontinu- ous on IP C (R d ), thus I α (η) ≤ lim inf n I α (η n ) = inf (LI α ),
which shows that η is a minimizer for the problem (LI α ).

Notice that we do not claim that for given µ, ν ∈ Prob(R d ) and α ∈ [0, 1], the minimum is nite, i.e. that there is an irrigation plan in IP(µ, ν) with nite α-cost. Hence in the sequel we will say that η ∈ IP(µ, ν) is optimal for I α if it is a minimizer of (LI α ) and if it has nite α-cost, in which case we write η ∈ OIP α (µ, ν).

Existence for the H-cost Let us just remark that all the computations and the proofs done from Section 1.2.2 hold if we replace the map x → x α by H : R + → R + and x α-1 with H(x)/x, where H satises:

(i) H(0) = 0, (ii) H is continuous and nondecreasing, (iii) x → H(x)/x is nonincreasing, (iv) lim x→0 H(x) x = +∞.
Notice that items (ii)-(iii) hold if H is concave and increasing, and that (iii) implies that H is subadditive.

Operations on irrigation plans

In this section we dene restrictions, cuts and concatenations of irrigation plans. These standard operations were dened in [START_REF] Bernot | Optimal transportation networks[END_REF], although there was a minor glitch in the construction of a concatenation. In [START_REF] Pegon | Master's Thesis on Branched Transport[END_REF], we dened it via approximation by nite graphs, while here we propose to prove a gluing lemma (analogous to the one in optimal transport, see [San15, Lemma 5.5]) by disintegration of measures. In this section X is a closed subset of R d .

Restriction. Given an irrigation plan η ∈ IP(X) and a Borel set E ⊆ Γ(X), the restriction of η to E is merely the measure-theoretic restriction: for all Borel set A ⊆ Gluing. We are given two irrigation plans η 1 , η 2 such that η + 1 = η - 2 and we want to glue them together. Lemma 1.2.10 (Gluing lemma). There exists an irrigation plan η 12 and a Borel map τ : Γ(X) → Prob(R + ), such that

Γ(X), η E(A) = η(E ∩ A). Cut. If χ is a Borel map which associates to each curve γ ∈ Γ(X) a probability measure χ γ ∈ Prob(R 2 + ) concentrated on H = {(a, b) : a ≤ b}, then we dene the cut η[χ] of η with respect to χ by: φ, η[χ] = ˆΓ(X) ˆH φ(γ |[a,b] (• -a) dχ γ (a, b) dη(γ), for all φ ∈ C 0 (Γ).
η 1 = η 12 [0, τ ] and η 2 = η 12 [τ, +].
The plan η 12 is called a concatenation of η 1 and η 2 and τ a recovery cutting time.

Proof. Let us disintegrate η 1 and η 2 with respect to π ∞ and π 0 respectively, recalling that (π ∞ ) # η 1 = (π 0 ) # η 2 =: σ, so that:

dη 1 (x, y) = dη y 1 (x) dσ(y) and dη 2 (y, z) = dη y 2 (z) dσ(y),
where η y 1 is concentrated on curves which stop at y and η y 2 on curves that start at y. We set 6 π y = η y 1 ⊗ η y 2 and dene for all pair (γ 1 , γ 2 ) of curves which start and stop at 6 Any coupling π in Π(η y 1.2. THE LAGRANGIAN MODEL a common point:

γ 1 : γ 2 (t) = γ 1 (t) if t ≤ T (γ 1 ), γ 2 (t -T (γ 1 )) if t > T (γ 1 ).
We dene η 12 in the following way:

η 12 , φ = ˆRd ˆΓ(X)×Γ(X) φ(γ 1 : γ 2 ) dπ y (γ 1 , γ 2 ) dσ(y),
for all φ ∈ C 0 (Γ(X)). Now let us build τ . We start by dening a measure ρ ∈

M + (R + × Γ(X)): ρ, ψ = ˆRd ˆΓ(X)×Γ(X) ψ(T (γ 1 ), γ 1 : γ 2 ) dπ y (γ 1 , γ 2 ) dσ(y),
for all ψ ∈ C 0 (R + ×Γ(X)). Then we disintegrate ρ with respect to the second projection π 2 , noticing that (π 2 ) # ρ = η 12 , thus getting:

dρ(t, γ) = dτ γ (t) dη 12 (γ).
Now let us check that the desired properties hold. Take a test function φ ∈ C 0 (Γ(X)):

η 12 [0, τ ], φ = ˆΓ(X) ˆ∞ 0 φ(γ |[0,t] ) dτ γ (t) dη 12 (γ) = ˆR+ ×Γ(X) φ(γ |[0,t] ) dρ(t, γ) = ˆRd ˆΓ(X)×Γ(X) φ (γ 1 : γ 2 ) |[0,T (γ 1 )] dπ y (γ 1 , γ 2 ) dσ(y) = ˆRd ˆΓ(X)×Γ(X) φ(γ 1 ) dπ y (γ 1 , γ 2 ) dσ(y) = ˆRd ˆΓ(X) φ(γ 1 ) dη y 1 (γ 1 ) dσ(y) = ˆΓ(X) φ(γ) dη 1 (γ),
and

η 1 = η 12 [0, τ ]. A similar computation shows that η 2 = η 12 [τ, +], which ends the proof.
Remark 1.2.11. The result generalizes easily with n irrigation plans η 1 , . . . , η n such that η + i = η - i+1 : one may nd an irrigation plan η and recovery cutting times τ

i : Γ(X) → Prob(R + ) for 0 < i < n such that η i = η[τ i-1 , τ i ],
with τ 0 = δ 0 and τ n = δ T (•) . We write η ∈ η 1 : η 2 : . . . : η n to say that η is such a gluing. Remark 1.2.12. From two irrigation plans η ∈ IP(µ, ν) and η ∈ IP(ν, ξ) one may build an irrigation plan η ∈ IP(µ, ξ) with cost

I α (η ) ≤ I α (η) + I α (η ).
Indeed, just take η to be a gluing of η and η , with τ a recovery cutting time. Then one has:

I α (η ) = ˆΓ(X) ˆγ θ η (x) α-1 dx dη (γ) = ˆΓ(X) ˆR+ ˆγ|[0,b] θ η (x) α-1 dx + ˆγ|[b,∞] θ η (x) α-1 dx dτ γ (b) dη (γ) = ˆΓ(X) ˆR+ ˆγ|[0,b] θ η (x) α-1 dx dτ γ (b) dη (γ) + ˆΓ(X) ˆR+ ˆγ|[b,∞] θ η (x) α-1 dx dτ γ (b) dη (γ). Notice that θ η (x) ≥ θ η (x) and θ η (x) ≥ θ η (x), hence ˆΓ(X) ˆR+ ˆγ|[0,b] θ η (x) α-1 dx dτ γ (b) dη (γ) ≤ ˆΓ(X) ˆR+ Z η (γ |[0,b] ) dτ γ (b) dη (γ) = ˆZη (γ) dη(γ) = I α (η),
and the same goes for the other term, leading to

I α (η ) ≤ I α (η) + I α (η ).
Moreover it is clear that η ∈ IP(µ, ξ).

Irrigability and irrigation distance

We have already noticed that a pair of probability measures cannot necessarily be connected with nite α-cost. However, according to the previous subsection, the relation µ and ν can be connected with nite α-cost is transitive, thus every pair of measures can be connected with nite α-cost if and only if every measure µ be connected with nite α-cost to δ 0 , in which case we say that µ is α-irrigable. The next result from [START_REF] Xia | Optimal paths related to transport problems[END_REF] provides a sucient condition on α and d for irrigability of compactly supported measures.

Proposition 1.2.13 (Irrigability). If α > 1 -1 d then for every pair of compactly

supported measures (µ, ν) ∈ Prob(R d ), there is an irrigation plan η ∈ IP(µ, ν) of nite α-cost, i.e. such that I α (η) < ∞.
Remark 1.2.14. One can actually show that the uniform measure on a unit cube can be irrigated from a unit Dirac mass if and only if α > 1 -1 d , hence this condition is necessary for an arbitrary pair (µ, ν) to be irrigable with nite α-cost.

For µ, ν ∈ Prob(R d ), we set d α (µ, ν) to be the optimal α-cost connecting the two, namely:

d α (µ, ν) = min {I α (η) : η ∈ IP(µ, ν)} ∈ R ∪ {+∞}.
The lower semicontinuity of I α and the continuity of π 0 , π ∞ on IP C (R d ) readily imply that d α is lower semicontinuous. From Remark 1.2.12 we get the triangle inequality, thus d α is a distance on compactly supported measures, and we may state the following (proven in [START_REF] Xia | Optimal paths related to transport problems[END_REF]).

Proposition 1.2.15. Fix a compact subset X ⊆ R d and α > 1-1 d . Then d α (µ, ν) < ∞ for every pair of measures µ, ν ∈ Prob(X) and d α is a distance on Prob(X) which metrizes the weak-convergence of measures in the duality with C (X).

Actually, sharp inequalities comparing d α and Wasserstein distances W p have been established in [START_REF] Brasco | A Benamou-Brenier approach to branched transport[END_REF] and [START_REF] Morel | Comparison of distances between measures[END_REF]:

cW 1/α (µ, ν) ≤ d α (µ, ν) ≤ W 1 (µ, ν) β , where β = 1 + αd -d.

Basic structure properties of optimizers

From now on, we assume that we are in the branched case α < 1. We show that optimizers of the Lagrangian irrigation problem (LI α ) enjoy some tree-like properties: the single-path property and the cycle-free property. The rst one is actually a rst step to prove the second, which is a stronger property. In order to do so, let us anticipate a bit of content from Chapter 2, that is:

(i) any η ∈ OIP α (µ, ν) is simple, meaning that it is concentrated on curves γ which are injective on [0, T (γ)], (ii) any η ∈ OIP α (µ, ν) is rectiable, i.e. there is a 1-rectiable set R such that η-a.e. curve γ stays essentially in R: H 1 (γ \ R) = 0,
(iii) the following energy formula (see Theorem 2.1.9) is true for all simple and rectiable irrigation plans (hence for optimal ones):

I α (η) = ˆRd θ η (x) α d H 1 (x), (EF) (iv) if η ∈ OIP α (µ, ν), and η ∈ IP(µ, ν) is a candidate irrigation plan which is rectiable, then I α (η) ≤ ˆRd θ η (x) α d H 1 (x) ≤ I α (η ),
even if η is not simple and last inequality is strict, thanks to the simple replacement lemma (Lemma 2.1.12).

Let us start with a few notations. Given n points (x 1 , . . . , x n ) ∈ R d , we denote by Γ(x 1 , . . . , x n ) the set of curves γ ∈ Γ(R d ) which visit points (x 1 , . . . , x n ) in that order, i.e. there are times t 1 ≤ . . . ≤ t n such that γ(t i ) = x i for all i. Also, we denote by Γ[x 1 , . . . , x n ] the set of curves which visit (x 1 , . . . , x n ) no matter the order of visit.

Proposition 1.2.16 (Single path property). Assume that η ∈ OIP α (µ, ν) with α < 1. For all pair of points x, y such that η(Γ(x, y)) > 0, there is a curve parameterized by arc length on a segment [0, ] and denoted by γ η (x, y) such that η-a.e. curve γ ∈ Γ(x, y) coincide in its trajectory with γ η (x, y).

Remark 1.2.17. Let us clarify a bit. We say that γ 1 and γ 2 coincide in their trajectory if they can be reparameterized in γ1 , γ2 so that γ1 = γ2 , or equivalently if this equality holds with γ1 , γ2 being the arc-length parameterization of γ 1 , γ 2 .

Proof. Let A and B be two subsets of Γ(x, y) with positive mass η(A), η(B) > 0.

For ε > 0 small, we dene another irrigation plan η ε obtained from η by replacing a proportion ε of curves of B by curves where the path between x and y has been replaced by the paths of those in A. We shall be more precise. We set

η B = η B η(B)
and

η A = η A η(A) ,
and t x (γ), t y (γ) to be the unique times 7 for which γ(t x (γ)) = x, γ(t y γ)) = y.

η ε = η -εη B + εη A . Obviously η ε ∈ IP(µ, ν) because by construction (π 0 ) # ηA = (π 0 ) # η B and (π ∞ ) # ηA = (π ∞ ) # η B . Given u ∈ R d , notice that θ ηA (u) = ˆΓ(R d ) (1 u∈γ([0,tx]) + 1 u∈γ(]tx,ty[) + 1 u∈γ([ty,T ]) ) dη A (γ) = θ η B (Γ(u, x, y)) + θ η A (Γ(x, u, y)) + θ η B (Γ(x, y, u)),
so that the multiplicity θ ηε (u) can be expressed as

θ ηε (u) = θ η (u) -εθ B η (u) + εθ A η (u) where θ A η (u) = η(A ∩ Γ(x, u, y)) η(A)
and η(B ∩ Γ(x, u, y)) η(B) .

We know that η ε is rectiable, hence by optimality of η (recall (iv) at the beginning of this section), one has

ˆRd θ ηε (u) α d H 1 (u) ≥ I α (η), or equivalently ˆRd θ η (u) + ε θ A η (u) -θ B η (u) α d H 1 (u) ≥ ˆRd θ η (u) α d H 1 (u).
By interchanging the roles of A and B, one gets the same inequality but for ε of arbitrary sign. Notice that the map ε

→ (θ η (u) + ερ) α is strictly concave if ρ = 0 hence the map ε → ˆRd θ η (u) + ε θ A η (u) -θ B η (u) α d H 1 (u) is strictly concave if θ A η (u) -θ B η (u) is not identically 0. Since it is minimal at ε = 0, it cannot be strictly concave, therefore θ A η (u) = θ B η (u) 7 It is well dened for simple curves, which is true η-a.e. for H 1 -a.e. u ∈ R d . For any nonnegative function f dened on R d , one has ˆRd f (u)θ A η (u) d H 1 (u) = ˆRd f (u)θ B η (u) d H 1 (u).
(1.2.9) By Fubini's Theorem, we can give another expression for these:

ˆRd f (u)θ A η (u) d H 1 (u) = 1 η(A) ˆRd ˆA 1 γ∈Γ(x,u,y) f (u) dη(γ) d H 1 (u) = 1 η(A) ˆA ˆγ f (u)1 γ∈Γ(x,u,y) du dη(γ) = A ˆγ[x,y] f (u) du dη(γ),
where we have set γ[x, y] = γ |[tx(γ),ty(γ)] , thus (1.2.9) rewrites:

A ˆγ[x,y] f (u) du dη(γ) = B ˆγ[x,y] f (u) du dη(γ).
(1.2.10)

Now we consider a countable basis of open sets (O n ) n∈N in R d and we take f = 1 On . For all n ∈ N and all A, B ∈ Γ(x, y), one has:

A H 1 (γ[x, y] ∩ O n ) dη(γ) = B H 1 (γ[x, y] ∩ O n ).
(

(γ[x, y] ∩ O n ) = 0 = H 1 (γ[x, y] ∩ O n ): a contradiction.
As a consequence all curves γ ∈ Γ(x, y) \ E coincide in their trajectory between x and y. We denote this curve, parameterized by arc length on an interval [0, ], by γ η (x, y).

We are now able to prove the stronger cycle-free property, which is a Lagrangian counterpart to the cycle-free property in the discrete case ((iv) of Theorem 1.1.2).

Theorem 1.2.18 (Cycle-free property). If η ∈ OIP α (µ, ν) with α < 1 then there is no loop sequence x 1 , x 2 , . . . , x n ,

x n+1 = x 1 such that η(Γ[x i , x i+1 ]) > 0 for 1 ≤ i ≤ n.
Remark 1.2.19. Recall that Γ[x, y] is the set of curves which visit x and y, no matter in which order. Thus η(Γ[

x i , x i+1 ]) > 0 means that either η(Γ(x i , x i+1 )) > 0 or η(Γ(x i+1 , x i )) > 0.
Proof. The idea is similar to the proof of the previous proposition, thus we are going to give fewer details. For all i, we set

s i = 1, A i = Γ(x i , x i+1 ) if if η(Γ(x i , x i+1 )) > 0,
and

s i = -1, A i = Γ(x i+1 , x i ) otherwise, in which case η(Γ(x i+1 , x i )) > 0.
We know by the single-path property that all curves in A i follow a common trajectory γ i between x i and x i+1 (from x i to x i+1 if s i = 1, from x i+1 to x i if s i = -1). We are going to remove a small mass ε > 0 from the γ i 's such that s i > 0 and add ε to the other ones. Here again, the construction is based on the gluing lemma. We set

η A i = η A i η(A i )
. We consider a maximal (i.e. longest) sequence of points σ = x k , x k+1 , . . . , x l such that s i = 1 for all i = k, . . . , (l -1), and we take

η σ = η A k-1 [0, t x k ] : δ γσ : η A l [t x l , +] (or just η σ = δ γσ if σ
is the whole sequence), where γ σ = γ k : . . . : γ l is the concatenation of the consecutive curves associated to σ. We denote by U the set of all such maximal sequences σ.

Now for each consecutive triple τ = x k-1 , x k , x k+1 such that s k-1 = s k = -1 we take η τ ∈ η A k-1 [0, t x k ] : η A k [t x k
, +]: these triples form a set V . We dene the competitor

η ε = η -ε s i =-1 η A i + ε τ ∈V η τ + ε σ∈U η σ .
The rst sum corresponds to the mass we remove, the second is a reconnection term which we must add due to this removal, while the third sum corresponds to the mass that we want to add. We leave it to the reader to check that η ε ∈ IP(µ, ν) and that

θ ηε (u) = θ η (u) + ε∆θ(u), where ∆θ(u) = s i θ(u) if u ∈ γ i . By optimality of η, we have ˆRd θ η (u) α d H 1 (u) ≤ ˆRd (θ η (u) + ε∆θ(u)) α d H 1 (u),
which holds for ε ≥ 0, but also for ε ≤ 0, had we reversed the order of the loop sequence. Here again we are faced with a concave function of ε which is minimal at the interior point 0. This can only happen if θ(u) = 0 for H 1 -a.e. u ∈ i γ i , which is a contradiction since on each γ i , θ η (u) ≥ η(A i ) > 0.

The Eulerian model

We keep assuming till the end of this chapter, that α ∈ [0, 1[.

Irrigation ows We call irrigation ow on R d any vector measure v ∈ M d (R d ) such that ∇ • v ∈ M 1 (R d ),
where ∇ • v is the divergence of v in the sense of distribution. We denote by IF(R d ) the set of irrigation ows, and we equip it with the M div topology, that is v n

M div ---v if v n and ∇ • v n converge to v and ∇ • v weakly as measures, in the duality with C 0 (R d , R d ) and C 0 (R d , R) respectively.
Remark 1.3.1. These objects have several names. They are called 1-dimensional normal currents in the terminology of Geometric Measure Theory, and are called trac paths by Xia in [START_REF] Xia | Optimal paths related to transport problems[END_REF].

If E ⊆ R d is an H 1 -measurable set, τ : E → S d-1 is H 1 -measurable and θ : E → R + is H 1 -integrable, we dene the vector measure E, τ, θ ∈ M d (K) by < E, τ, θ , ψ >= ˆE θ(x)ψ(x) • τ (x) d H 1 (x), for all ψ ∈ C 0 (R d , R d ). In other terms E, τ, θ . = θτ H 1 E.
∇ • v G : Proposition 1.3.2. If G is a graph, G ∈ IG(µ, ν) if and only if ∇ • v G = µ -ν.
Consequently, we say that v sends µ to ν if ∇ • v = µ -ν and denote by IF(µ, ν)

the set of such irrigation ows. Now on the generalization of Gilbert Energy, if we identify v G with its H 1 -density, one has

E α (G) . = e∈E(G) θ α e |e| = ˆRd |v G (x)| α d H 1 (x).
This leads to dening the following cost on IF(R d ):

M α (v) =    ˆ|v(x)| α d H 1 (x) if v is rectiable, +∞ otherwise,
which is called the α-mass of v.

On the denition and semicontinuity of the α-mass. Several denitions of αmass have been given depending on the setting (currents or vector measures, euclidean or metric setting). In [START_REF] Pauw | Size minimization and approximating problems[END_REF] it was dened in a euclidean setting for integer rectiable currents as the integral of the multiplicity to the α (actually they consider more general costs than x → x α but concave integrands H, and dene a so-called H-mass). Lower semicontinuity is proven in this space of integer rectiable currents for a α-at norm F α .

A similar denition is used in a metric setting in [Ste10]. Though lower semicontinuity is not explicitly proven, the same denition for real rectiable currents (in a Euclidean setting) is used by Stepanov and Paolini in [START_REF] Paolini | Optimal transportation networks as at chains[END_REF] and extended to at chains. In the works [START_REF] Xia | Optimal paths related to transport problems[END_REF] and [START_REF] Xia | Interior regularity of optimal transport paths[END_REF], two dierent denitions of M α are used, both by relaxation: either in the space M div of vector measures v whose divergence are measures (for the weak convergence of v and ∇ • v), or in the space of at chains (for the usual at norm). Things are claried in [START_REF] Colombo | On the lower semicontinuous envelope of functionals dened on polyhedral chains[END_REF] where it is proven that the lower semicontinuous envelope of the α-mass dened on real-valued polyhedral chains is precisely

M α (v) =    ˆ|v(x)| α H 1 (dx) if v is rectiable, +∞ otherwise,
on the space of real at chains. Since this space is larger than IF(R d ), and the topology weaker than the weak convergence in M div , all these denitions coincide and in particular M α is lower semicontinuous on IF(R d ).

Eulerian irrigation problem We are now able to formulate an Eulerian irrigation problem in a continuous setting. Given two probability measures µ, ν ∈ Prob(R d ), we want to nd an irrigation ow v sending µ to ν which has minimal α-mass. This reads

min {M α (v) : v ∈ IF(µ, ν)} . (EI α )
Xia proved the following existence theorem in [START_REF] Xia | Optimal paths related to transport problems[END_REF].

Theorem 1.3.3 (Existence of minimizers). For any compactly supported probability measures µ, ν ∈ Prob(R d ), there is a minimizer v to the Eulerian irrigation problem (EI α ). Moreover if 1 -1 d < α < 1 the minimum is always nite.

As for lagrangian irrigation problem, we call optimal irrigation ow for M α with prescribed source and target µ, ν a minimizer of (EI α ) which has a nite cost, and we denote by OIF α (µ, ν) the set of such optimal irrigation ows.

The single source case

In this section, we place ourselves in the case of a single source by assuming that the initial measure is a unit Dirac: µ = δ 0 , and we assume α < 1. This is the most studied case in branched transport, either from the point of view of Maddalena, Morel, and Solimini's patterns dened in [START_REF] Maddalena | A variational model of irrigation patterns[END_REF], studied also by Devilanova, Solimini in [START_REF] Devillanova | Elementary properties of optimal irrigation patterns[END_REF][START_REF] Devillanova | On the dimension of an irrigable measure[END_REF], or in the trac plan framework (the closest to our irrigation plans framework) by Bernot, Caselles, Morel [BCM05; BCM08; BCM09]. In this case, Santambrogio was able to dene a landscape function in [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF], a terminology borrowed from geophysics ([RR01]), which can be thought of as a supergradient of the cost I α and has become an important tool to study optimal networks or variational problems related to branched transport, as done in Chapter 3.

Tree property

Optimal networks in the single source case have a tree structure stemming from a root located at 0, as stated in the next proposition.

Proposition 1.4.1 (Tree property). Let η ∈ IP(δ 0 , ν) be an optimal irrigation plan with nite α-cost. For all x ∈ N η , η-a.e. curve γ passing through x coincides in its trajectory with a common curve γ η (x) : [0, L(γ η (x))] → R d which goes from 0 to x.

Proof. We simply use Proposition 1.2.16, noticing that θ η (x) > 0 if and only η(Γ(0, x)) > 0 since η-a.e. curve starts at 0, and dening γ η (x) = γ η (0, x).

For γ ∈ Γ(R d ) and t ≥ 0, we dene the joint multiplicity of γ at t w.r.t. the irrigation plan η by

θ η (γ[0, t]) = η(γ ∈ Γ(R d ) : γ|[0,t] = γ |[0,t] ).
The previous proposition yields the following immediate corollary.

Corollary 1.4.2. Let η as in the previous proposition and parameterized by arc length. If x ∈ N η and γ = γ η (x) then:

∀t ≤ L(γ), θ η (γ(t)) = θ η (γ[0, t]).

The landscape function

The landscape function associated to an optimal irrigation plan was dened by Santambrogio in [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF] with some inspiration from optimal channel networks in geophysics (see [START_REF] Rodriguez-Iturbe | Fractal river basins: chance and selforganization[END_REF]). The terminology and the results of this section are extracted from [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF] and given without proofs, for which we refer to the original paper, or to Chapter 4 which generalizes these results to the multiple sources case.

To dene the landscape function associated to an optimal network η, we need to dene a large class of curves which follow the network in a sense. Denition 1.4.3 (η-good curves). We say that a curve γ is η-good if ˆT(γ)

0 θ η (γ[0, t]) α-1 dt < ∞
which is equivalent to the two assertions:

(i) Z η (γ) < ∞, (ii) θ η (γ(t)) = θ η (γ[0, t]
) for all t < T (γ).

Proposition 1.4.4. If η is an optimal irrigation plan, the following assertions hold:

(i) η-good curves γ are injective 9 on [0, T (γ)[, (ii)
they are paremeterized by arc length if η is itself parameterized by arc length, (iii) η is concentrated on the set of η-good curves.

One may view the set of η-good curves as the right generalization of support for irrigation plans. The following proposition is key to dening the landscape function.

Theorem 1.4.5 (Well-denedness property). If η is optimal and γ 1 , γ 2 are two η-good curves with the same endpoint x, then Z η (γ 1 ) = Z η (γ 2 ). 9 Actually, it is injective on the whole [0, T (γ)] but this is a consequence of Theorem 1.4.5.

Denition 1.4.6 (Landscape function). If η is optimal, the previous proposition allows us to dene the landscape function z η by:

z η (x) = Z η (γ) if γ is an η-good curve such that γ(∞) = x, +∞ otherwise.
The following rst variation formula is fundamental. A slightly weaker version was established in [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF], the proof of this one is given in Chapter 3, Proposition 3.1.5.

Proposition 1.4.7 (First variation). Suppose that η is an optimal irrigation plan between δ 0 and ν, with landscape function z η . The following holds:

d α (δ 0 , ν) ≤ d α (δ 0 , ν) + α ˆRd z η d(ν -ν), for all ν ∈ Prob(R d ).
Therefore, z η may be thought as a supergradient of ν → d α (δ 0 , ν). Let us state the other main properties of landscape functions.

Proposition 1.4.8 (Lower semicontinuity). The landscape function associated to an optimal irrigation plan is lower semicontinuous.

Proposition 1.4.9 (Steepest descent). The network N η follows the direction of steepest descent of z η , in the sense that for all x 0 ∈ N η :

∀x ∈ N η , z η (x) ≥ z η (x 0 ) -θ η (x 0 ) α-1 |x -x 0 | + o(|x -x 0 |).
With some extra hypotheses on ν, one may get more regularity of z η .

Proposition 1.4.10 (Hölder continuity). We assume that α > 1 -1 d . Suppose that ν is of the form:

ν = f L E with f ≥ c > 0 on E,
and E is regular in the sense that

∀x ∈ E, ∀r ≤ diam(E), Θ E (x, r) := |B r (x) ∩ E| |B r (x)| ≥ c ,
for some c > 0. Then the landscape function z η associated to any optimal irrigation plan η ∈ OIP α (δ 0 , ν) is β-Hölder continuous on E where

β = d α -1 - 1 d = 1 + dα -d.
Chapter 2

Equivalence of the models The goal of this chapter is to give a complete and rigorous proof of the equivalence between the Lagrangian and Eulerian irrigation problems (LI α ) and (EI α ). This is a fact which has been known for some time and was stated in the book [START_REF] Bernot | Optimal transportation networks[END_REF]. The rst connection between the two models was perhaps being made in [START_REF] Paolini | Optimal transportation networks as at chains[END_REF], where the Lagrangian model is set through what they call transports which correspond to our irrigation plans with a preliminary quotient with respect to reparameterization, and where the Eulerian model is cast in the framework of at chains. An ad hoc Smirnov decomposition theorem was proved for at chains (Smirnov originally proved it for normal currents in [START_REF] Smirnov | Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional ows[END_REF]), as a way to build a transport from a at chain, while a at chain is clearly induced by a transport. Although it is not explicitly stated, the way to pass from Eulerian to Lagrangian (our Section 2.3) can be extracted from their work. For the converse way, one needs to show some simplicity of optimizers (done by some recursive loop-removal in [START_REF] Bernot | Optimal transportation networks[END_REF]), and use an energy formula which expresses the Lagrangian cost as a generalized Gilbert energy. This is done in [START_REF] Bernot | Optimal transportation networks[END_REF] via Fubini-Tonelli's theorem also the σ-niteness is not explicitly stated. To prove it, one should prove rst the rectiability of optimal irrigation plans, as done in Bernot's PhD thesis ( [START_REF] Bernot | Optimal transport and irrigation[END_REF]), which we prove dierently in Section 2.1.1. The same fact was noticed in the recent paper [START_REF] Brancolini | Equivalent formulations for the branched transport and urban planning problems[END_REF] where they prove the equivalence between the Lagrangian (patterns) and Eulerian (mass ux) models with more general costs, replacing the α-cost and α-mass with H-costs and H-masses for a large class of functions H.

In Section 2.4 the equivalence between the two models is established, using the energy formula and a Smirnov decomposition (see [START_REF] Smirnov | Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional ows[END_REF]) for vector measures obtained by Santambrogio in [San14] via a Dacorogna-Moser approach, as stated in Theorem 2.4.1. In short, both Eulerian and Lagrangian irrigation problems they have same minimal value, and one can build minimizers of one problem from minimizers of the other. In this chapter we assume α < 1, and that we are on a compact set K ⊆ R d for simplicity.

The energy formula

The aim of this section is to establish an energy formula stated in [BCM09, Section 4.3], expressing the Lagrangian irrigation cost as follows:

ˆΓ ˆγ θ η (x) α-1 dx dη(γ) = ˆK θ η (x) α d H 1 (x), (EF)
provided η satises some hypotheses (namely essential simplicity and rectiability).

The term on the right-hand side is the so-called Gilbert Energy denoted by E α (η).

The proof relies solely on the correct use of Fubini-Tonelli's theorem, which requires σ-niteness of measures. The next subsection is therefore devoted to the rectiability of irrigation plans.

Rectiable irrigation plans

Intensity and ow We dene the intensity i η ∈ M + (K) and ow v η ∈ M d (K) of an irrigation plan η ∈ IP(µ, ν) by the action on continuous maps:

i η , φ = ˆΓ ˆγ φ(x) dx dη(γ), v η , ψ = ˆΓ ˆγ ψ(x) • dx dη(γ), for all φ ∈ C (K), ψ ∈ C (K) d .
The quantity di η (x) represents the total circulation at x and dv η (x) the total ow at x.

Concentration and rectiability

Let A be a Borel subset of K. By denition of i η , the following assertions are equivalent:

(i) i η is concentrated on A, i.e. i η (A c ) = 0, (ii) for η-a.e. γ ∈ Γ, γ ⊆ A up to an H 1 -null set, i.e. H 1 (γ \ A) = 0.
In that case we say (with a slight abuse) that η is concentrated on A. An irrigation plan η ∈ IP(K) is termed σ-nite if it is concentrated on a σ-nite set w.r.t. H 1 and rectiable if it is concentrated on a 1-rectiable set 1 . The intensity i η has a simple expression when η is σ-nite, as shown below.

Proposition 2.1.1. If η is an irrigation plan concentrated on a σ-nite set A, then i η = m η H 1 A.

Proof. For any Borel set B, one has

i η (B) = i η (A ∩ B) = ˆΓ ˆγ 1 A∩B dx dη(γ) = ˆΓ ˆB m(x, γ) d H 1 A(x) dη(γ) = ˆB ˆΓ m(x, γ) dη(γ) d H 1 A(x) = ˆB m η (x) d H 1 A(x).
The equality on the second line follows from the coarea formula, the next from Fubini-Tonelli's theorem which holds because measures are σ-nite, and the last one from the denition of m η . Now we would like to prove that irrigation plans η are concentrated on their network N η . = {x ∈ K : θ η (x) > 0} and that the latter is a rectiable set. The rst assertion is true provided η has nite α-cost, as we shall see later, and the second item holds in all generality, as we shall prove know. Our proof follows an idea of [BCM09, Theorem 4.10],

1 A 1-rectiable set is the union of an H 1 -null set with a countable union of Lipschitz curves. though we cannot use the energy formula yet (as its proof requires said rectiability or σ-niteness) and take care of the fact that the rectiability criterion of Lemma 2.1.3 (as well as Lemma 2.1.4) applies to sets of nite H 1 measure and not of σ-nite H 1 measure. In [BW15, Lemma 3.18], another proof is provided adapting a proof of Bernot ([Ber05, Lemma 4.6.4] or [BCM09, Lemma 6.3]). With our approach, we will need a few notions and lemmas from geometric measure theory.

If A is a subset of R d we dene the upper and lower 1-density of A at x as

Θ(x, A) = lim sup r↓0 H 1 (A ∩ B(x, r)) 2r , Θ(x, A) = lim inf r↓0 H 1 (A ∩ B(x, r)) 2r
.

When these quantities are equal, we call their common value Θ(x, A) the 1-density of A at x.

The rst lemma we will need is proved in

[Mat95, Chapter 8]. Lemma 2.1.2. If B ⊆ R d , H 1 (B) = sup{H 1 (K) : K ⊆ B compact such that H 1 (K) < ∞}.
The second is due to Besicovitch and may be obtained as a particular case of [Mat95, Theorem 17.6].

Lemma 2.1.3. Let E be an H 1 -measurable set such that H 1 (E) < ∞. If its 1-density Θ(x, E) exists and is equal to 1 for H 1 -a.e.

x in E then it is 1-rectiable.

Finally, we will need the following result which is included in [Mat95, Theorem 6.2].

Lemma 2.1.4. If E is a set such that H 1 (E) < ∞, then the upper 1-density Θ(x, E) is less than 1 for H 1 -a.e. x in E.

Proposition 2.1.5 (Rectiability of the network). If η ∈ IP(K) is an irrigation plan, its network N η is 1-rectiable.

Proof. First of all, since the network does not change under normalization, we may assume that η is parameterized by arc length. We have N η = n>0 D n where

D n := x : θ η (x) > 1 n .
Let us show that that H 1 (D n ) < ∞. By contradiction assume that for some n, H 1 (D n ) = ∞, hence thanks to Lemma 2.1.2 one can nd for M > 0 as large as we want a compact subset

K ⊆ D n such that M ≤ H 1 (K ) < ∞. Since H 1 (K ) < ∞ one can use Fubini-Tonelli's theorem to get L(η) . = ˆΓ L(γ) dη(γ) ≥ ˆΓ H 1 (γ ∩ K ) dη(γ) = ˆΓ ˆK 1 x∈γ d H 1 (x) dη(γ) = ˆK ˆΓ 1 x∈γ dη(γ) d H 1 (x) = ˆK θ η (x) d H 1 (x) > M n .
The inequality L(η) > M n must be true for all M > 0, i.e. L(η) = ∞, which contradicts the denition of an irrigation plan, hence H 1 (D n ) < ∞.

Now we shall prove that Θ(x, D n ) = 1 a.e. on D n . Since D n has nite H 1 -measure, we already know Θ(x, D n ) ≤ 1 for H 1 -a.e. x ∈ D n by Lemma 2.1.4, thus it remains only to prove Θ(x, D n ) ≥ 1. If A is a Borel subset of R we denote Leb(A) the set of Lebesgue points of A, which are points t such that

lim r↓0 |A ∩ [t -r, t + r]| 2r = 1,
where |X| denotes the Lebesgue measure of X ⊆ R. Recall that by Lebesgue's theorem we have |A \ Leb(A)| = 0. For any γ ∈ Γ 1 , we set

A γ = t : t ∈ Leb s : |γ(s)| η > 1 n for all n s.t. |γ(t)| η > 1 n , B γ = {t ∈]0, T (γ)[ : γ(t) exists} , D γ = γ(A γ ∩ B γ ). Notice that |[0, T (γ)] \ (A γ ∪ B γ )| = 0 hence H 1 (γ \ D γ ) = 0 since γ is Lipschitz.
Finally we set

D = γ∈Γ 1 D γ . Let us check that H 1 (N η \ D ) = 0. We have ˆNη\D θ η (x) d H 1 (x) = ˆNη\D ˆΓ 1 x∈γ dη(γ) d H 1 (x) = ˆΓ ˆNη\D 1 x∈γ d H 1 (x) dη(γ) = ˆΓ1 H 1 (N η ∩ γ \ D ) dη(γ) = 0.
The use of Fubini-Tonelli's theorem is justied since N η = n D n is σ-nite and the last equality follows from

H 1 (N η ∩γ \D ) ≤ H 1 (γ \D γ ) = 0. This implies H 1 (N η \D ) = 0 since θ η > 0 on N η . Now take any x ∈ D n ∩ D . By construction of D there is a curve γ ∈ Γ 1 and a t ∈ A γ ∩ B γ such that x = γ(t), which implies that |s ∈ [t -r, t + r] : γ(s) ∈ D n | 2r r↓0 --→ 1 (2.1.1)
and

H 1 (γ([t -r, t + r]) \ D n ) 2r r↓0 --→ 0.
(2.1.2)

It follows from (2.1.2) and the fact that γ([t-r, t+r])

⊆ B(x, r) because γ is 1-Lipschitz that Θ(x, D n ) . = lim inf r↓0 H 1 (B(x, r) ∩ D n ) 2r ≥ lim inf r↓0 H 1 (γ([t -r, t + r])) 2r
.

But γ has a derivative e at t which has unit norm. Moreover the H 1 -measure of γ([t -r, t + r]), which is a compact connected set, is larger than the distance between γ(t -r) and γ(t + r), and since γ(t ± r) = x ± re + o(r) one has

H 1 (γ([t -r, t + r])) ≥ |γ(t + r) -γ(t -r)| = 2r + o(r),
which yields Θ(x, D n ) ≥ 1. This proves that Θ(x, D n ) exists and is equal to 1 for H 1a.e. x ∈ D n hence D n is 1-rectiable by Lemma 2.1.3 and N η = n D n as well.

At this stage we have shown that the network of any irrigation plan is rectiable, yet this does not mean that any irrigation plan is rectiable (this is obviously not the case) since η needs not be concentrated on N η . However, it is essentially the only candidate rectiable set (or even candidate σ-nite set) on which η could be concentrated, as stated below.

Corollary 2.1.6. Given η ∈ IP(K) an irrigation plan, the following assertions are equivalent:

(i) η is concentrated on N η , (ii) η is rectiable, (iii) η is σ-nite.
Proof. It is enough to prove (iii) ⇒ (i) by the previous proposition. If η is concentrated on a σ-nite set A, we know by Proposition 2.1.

1 that i η = m η H 1 A. Therefore i η is also concentrated on {x : m η (x) > 0} = N η .
Remark 2.1.7. From this and Proposition 2.1.1 we get that i

η = m η H 1 if η is recti- able.
The most important consequence of Proposition 2.1.5 is the following rectiability result.

Theorem 2.1.8 (Rectiability). If η has nite α-cost with α ∈ [0, 1[, it is rectiable.

Proof. Because of the previous statement, we need only show that η is concentrated on N η . We have

I α (η) . = ´Γ ´γ θ η (x) α-1 dx dη(γ) < ∞ hence for η-almost every curve γ, for H 1 -almost every x in γ, θ η (x) α-1 < ∞, which implies that θ η (x) > 0 i.e. x ∈ N η .
By denition, it means that η is concentrated on N η .

Proof of the energy formula

We dene Gilbert energy E α :

IP(K) → [0, ∞] as E α (η) =    ˆK θ η (x) α d H 1 (x) if η is rectiable, +∞ otherwise.
and a variant Ēα :

IP(K) → [0, ∞],
which is kind of a full Gilbert energy:

Ēα (η) =    ˆK θ α-1 η (x)m η (x) d H 1 (x) if η is rectiable, +∞ otherwise.
Assuming α ∈ [0, 1[, we would like to establish the energy formula

I α (η) = E α (η).
(EF)

This does not hold in general. Actually we are going to show that I α (η) = Ēα (η) for all irrigation plan η ∈ IP(K) and that Ēα (η) = E α (η) provided η is essentially simple.

Theorem 2.1.9 (Energy formula). Assuming α ∈ [0, 1[, the following formula holds:

I α (η) = Ēα (η). (EF')
Moreover, if η is essentially simple this rewrites

I α (η) = E α (η). (EF)
Proof. By Theorem 2.1.8, if η is not rectiable then I α (η) = E α (η) = Ēα (η) = ∞ and the result is clear. Now we assume that η is rectiable, which means that it is concentrated on the rectiable set N η , according to Theorem 2.1.8 and Corollary 2.1.6.

Notice that by the coarea formula we have

ˆΓ ˆγ θ η (x) α-1 dx dη(γ) = ˆΓ ˆK θ η (x) α-1 m(x, γ) d H 1 (x) dη(γ),
thus the goal is to reverse the order of integration. Here Fubini-Tonelli's theorem applies because η is concentrated on its network, which is rectiable, which yields

I α (η) = ˆΓ ˆNη θ η (x) α-1 m(x, γ) d H 1 (x) dη(γ) = ˆNη θ η (x) α-1 m η (x) d H 1 (x) = ˆK θ η (x) α-1 m η (x) d H 1 (x) = Ēα (η).
and (EF') holds. Now if η is essentially simple then in all the previous calculations m η (x) = θ η (x) so that

I α (η) = ˆNη ˆΓ θ η (x) α-1 θ η (x) dη(γ) d H 1 (x) = ˆK θ η (x) α d H 1 (x) = E α (η),
thus getting (EF).

Remark 2.1.10. Actually, the proof shows that the equality I α (η) = Ēα (η) (and I α (η) = E α (η) if η is essentially simple) holds also for α = 1 provided η is rectiable. However, one may nd η non-rectiable such that I 1 (η

) ∈]0, ∞[ while H 1 (N η ) = 0. In that case one has 0 = ´K m η (x) d H 1 (x) < I 1 (η) < Ē1 (η) = ∞. Notice also that for such η one has 0 = ´K θ η (x) α-1 m η (x) d H 1 (x) < I α (η) = ∞ for α ∈ [0, 1[, which explains why we imposed E α (η) = Ēα (η) = ∞ if η is not rectiable.

Simplicity of optimal irrigation plans

In this section we shall prove that optimal irrigation plans are necessary simple using the energy formula.

Reduced intensity We associate to any irrigation plan η ∈ IP(K) a reduced intensity j η by

j η , φ = ˆΓ ˆγ φ(x) d H 1 (x) dη(γ),
for all φ ∈ C (K). It is a positive nite measure, since the total mass is

j η = ˆΓ H 1 (γ) dη(γ) ≤ ˆΓ L(γ) dη(γ) = L(η) < ∞. Remark 2.1.11. Notice that if A is a Borel set, j η (A) = 0 ⇔ i η (A) = 0 hence
j ζn (O) = ˆΓ H 1 (γ ∩ O)ζ n (dγ).
By a generalization of Golab's Theorem (see [START_REF] Brancolini | Optimal networks for mass transportation problems[END_REF]), the following holds

H 1 (γ ∩ O) ≤ lim inf n H 1 (γ n ∩ O) if γ n → γ uniformly on compact sets, which means that γ → H 1 (γ ∩ O) is lower semicontinuous on Γ. Consequently ζ → ´Γ H 1 (γ ∩ O)ζ(dγ) is lower semicontinuous and one gets j ζ (O) . = ˆΓ H 1 (γ ∩ O)ζ(dγ) ≤ lim inf n ˆΓ H 1 (γ ∩ O)ζ n (dγ) ≤ j η (O)
for all open set O. This implies that j ζ ≤ j η by regularity of nite measures hence ζ is a minimizer of (LEN η ).

Let us check that any minimizer ζ is simple. By contradiction, if it was not simple there would be a set Γ ⊆ Γ such that ζ(Γ ) > 0 and every γ ∈ Γ has a loop. One may dene a Borel map r : γ → r(γ) which removes from γ ∈ Γ the loop with maximal length (the rst one in case there are several), and is identical on

Γ \ Γ . Then set ζ := r(ζ). Obviously one has L( ζ) < L(ζ), ζ ∈ IP(µ, ν) and jζ ≤ j ζ , which contradicts the optimality of ζ in (LEN η ).
Finally, suppose η is rectiable and take ζ a minimizer of our problem. According to Remark 2.1.11, the inequality j ζ ≤ j η implies that ζ is rectiable and

j η = θ η H 1 , j ζ = θ ζ H 1 , which yields (ii). Proposition 2.1.13. Given α ∈ [0, 1], if η ∈ IP(µ, ν) is optimal with nite α-cost, then it is simple.
Proof. The case α = 1 is straightforward from Lemma 2.1.12 since L = I 1 . Now we assume that α < 1 and take η optimal, in which case the niteness of the α-cost implies the rectiability of η by Theorem 2.1.8. We need only show that η is a minimizer of (LEN η ). Take η a simple replacement of η. Then since η, η are rectiable and θ η ≤ θ η H 1 -a.e., one has

I α (η) = ˆK θ α-1 η m η d H 1 ≥ ˆK θ α η d H 1 ≥ ˆK θ α η d H 1 = I α (η)
Since η is optimal we have equality everywhere, which means that

m η = θ η = θ η = m η H 1 -a.e. Consequently L(η) = ˆK m η (x) d H 1 (x) = ˆK m η(x) d H 1 (x) = L(η)
hence η minimizes (LEN η ) and is as such simple by Lemma 2.1.12.

From Lagrangian to Eulerian

Recall that we have associated to any irrigation plan η ∈ IP(K) an intensity i η ∈ M + (K) and a ow v η ∈ M d (K). We will show that v η is an irrigation ow sending µ to ν and satisfying M α (v η ) ≤ I α (η) under some hypotheses.

Proposition 2.2.1. If η ∈ IP(µ, ν) then v η ∈ IF(µ, ν).

FROM LAGRANGIAN TO EULERIAN

Proof. Let us calculate the distributional divergence of v η . For φ ∈ C (K), we have

∇ • v η , φ = -v η , ∇φ = - ˆΓ ˆ∞ 0 ∇φ(γ(t) • γ(t) dt dη(γ) = ˆΓ1 (φ(γ(0)) -φ(γ(∞))) dη(γ) = ˆK φ(x)µ(dx) -ˆK φ(x)ν(dx), thus ∇ • v η = µ -ν ∈ M 1 (K), which implies that v η ∈ IF(µ, ν). Proposition 2.2.2. If η is rectiable, then v η is a rectiable irrigation ow.
Proof. Without loss of generality we may assume that η is parameterized by arc length.

We know that η is concentrated on its network N η , which is rectiable. Given ψ ∈ C (K) d , by the area formula one has

ˆΓ ˆ∞ 0 ψ(γ(t)) • γ(t) dt dη(γ) = ˆΓ ˆγ ψ(x) • γ(t)=x γ(t) dx dη(γ) = ˆK ψ(x) • ˆΓ(x) γ(t)=x γ(t) dη(γ) d H 1 (x) = ˆK ψ(x) • v(x) d H 1 (x),
where we have set for H 1 -almost every x ∈ N η :

v(x) = ˆΓ(x) γ(t)=x γ(t) dη(γ).
Let us denote by L(x) the approximate tangent line at H 1 -almost every x. Since a.e. γ stays in N η up to a H 1 -null set, we know that for η-almost every γ, for H 1 -almost every x ∈ γ, if x = γ(t) then γ(t) (is well-dened and) belongs to L(x). By Fubini's theorem, one can reverse the order of almost everywhere 's and state that for H 1 -a.e.

x ∈ N η , for η-a.e. γ ∈ Γ(x),

γ(t)=x γ(t) ∈ L(x) hence v(x) ∈ L(x). Consequently v η is the rectiable irrigation ow v η = N η , τ, θ where τ = v(x)/|v(x)| (whatever value in S d-1 if v(x) = 0) and θ(x) = |v(x)|.
Proposition 2.2.3. If η is an essentially simple and rectiable irrigation plan, in particular if η is optimal, we have

M α (v η ) ≤ I α (η). Proof. By Remark 2.1.7 we know that i η = m η H 1 = θ η H 1 . Since |v η | ≤ i η = θ η H 1 , v η
has an H 1 -density which is less than θ η . Using the energy formula, we have

I α (η) = E α (η) = ˆK θ η (x) α H 1 (dx) ≥ ˆK|v η (x)| α d H 1 (x) = M α (v η ),
the last equality because v η rectiable as state in the previous proposition.

We have therefore proven that we have inf

IF(µ,ν) M α ≤ inf IP(µ,ν) I α ,
and that if η is optimal, v η is a good optimal candidate for the Eulerian problem (EI α ).

From Eulerian to Lagrangian

Given an irrigation ow v ∈ IF(µ, ν) of nite cost M α , we would like to build an irrigation plan η ∈ IP(µ, ν) such that v = v η (and whose cost is less than v). Theorem 2.3.1 (Irrigation ow decomposition). Given an irrigation ow v ∈ IF(µ, ν), there is an irrigation plan η ∈ IP(µ, ν) and a cycle w ∈ IF(K) satisfying

(i) v = v η + w, (ii) i η ≤ |v|.
From this we deduce:

Corollary 2.3.2. If v is an optimal irrigation ow in IF(µ, ν), there is an irrigation plan η ∈ IP(µ, ν) such that

(i) v = v η , (ii) |v η | = i η .
Proof. Let us take v η , w as in the previous theorem. Since M α (v) < ∞, v and v η are rectiable, and by optimality of v one has Proposition 2.3.3. If v is an optimal irrigation ow in IF(µ, ν), one can nd an irrigation plan η ∈ IP(µ, ν) such that

ˆK|v η (x)| α d H 1 (x) = M α (v η ) ≥ M α (v) = ˆK(|v η (x)| + |w(x)|) α d H 1 (x),
I α (η) ≤ M α (v).
Proof. Take η as in the previous corollary. Since M α (v) < ∞, v is rectiable and i η = |v| is concentrated on a rectiable set, which means by denition that η is rectiable. As a consequence |v| = i η = m η H 1 and we have

M α (v) = ˆK|v(x)| α d H 1 (x) = ˆK m η (x) α d H 1 (x), while I α (η) = Ēα (η) = ˆK θ η (x) α-1 m η (x) d H 1 (x).
We would like I α (η) ≤ M α (v), which is a priori not necessarily the case for the η we have constructed, since it is not necessarily essentially simple. Instead, take a simple replacement η ∈ IP(µ, ν) satisfying m η = θ η ≤ θ η ≤ m η . Then we get

I α (η) = ˆK θ η(x) α d H 1 (x) ≤ ˆK m η (x) α d H 1 (x) = M α (v)
which yields the result.

Remark 2.3.4. Since the minima in the Eulerian and Lagrangian problems are actually the same as we shall see in Theorem 2.4.1, the previous inequality is an equality, which implies that θ η = θ η = m η H 1 -a.e. thus η was actually optimal hence simple.

The equivalence theorem

We are now able to formulate the equivalence between the Lagrangian and Eulerian models. M α (v),

(ii) if v is optimal in IF(µ, ν), it can be represented by an optimal irrigation plan, i.e. v = v η for some optimal η ∈ IP(µ, ν),

(iii) if η is optimal in IP(µ, ν), then v η is optimal in IF(µ, ν) and i η = |v η |.
Proof.

I α (η) = ˆK i α η (x) d H 1 (x) ≥ ˆK|v η (x)| α d H 1 (x) = M α (v η ),
since we have equality everywhere by optimality of v η and η. The chapter is divided into ve parts. In a preliminary section we briey recall the Lagrangian framework of branched transport set in Chapter 1, and we formulate our question as a shape optimization problem involving the irrigation distance. Section 3.2 is devoted to the proof of existence of minimizers and to elementary properties of minimizers. In Section 3.3 we prove the β-Hölder regularity of the landscape function, which appears in the description of optimizers, and use it to derive an upper bound on the Minkowski dimension of the boundary of the optimizers in Section 3.4. The nal section is an attempt at computing optimizers numerically by adapting the Modica-Mortola approach introduced by [OS11], where we provide some computer visualizations.

M H (v) = ˆRd H(|v(x)|) d H 1 (x) + H (0) v s , where v = v

Preliminaries

Let us make a very quick recall of the Lagrangian framework we have introduced in Chapter 1 and its main features.

The irrigation problem

We denote by Γ(R d ) (or Γ for short) the set of 1-Lipschitz curves in R d parameterized on [0, ∞], endowed with the topology of uniform convergence on compact sets. where L(γ) = ´∞ 0 | γ(t)| dt. Notice that any irrigation plan is concentrated on Γ 1 (R d ) := {γ : L(γ) < ∞}. We denote by IP(R d ) the set of all irrigation plans η ∈ Prob(Γ). If µ and ν are two probability measures on R d , one says that η ∈ IP(R d ) irrigates ν from µ if one recovers the measures µ and ν by sending the mass of each curve respectively to its initial point and to its nal point, which means that

Irrigation plans

(π 0 ) # η = µ and (π ∞ ) # η = ν,
where π 0 (γ) = γ(0), π ∞ (γ) = γ(∞) := lim t→+∞ γ(t) and f # η denotes the push-forward of η by f whenever f is a Borel map 1 . We denote by IP(µ, ν) the set of irrigation plans irrigating ν from µ:

IP(µ, ν) = {η ∈ IP(R d ) : (π 0 ) # η = µ, (π ∞ ) # η = ν}.
If η is a given irrigation plan, we dene the multiplicity at x, that is the total mass passing by x, as θ η (x) = η({γ ∈ Γ : x ∈ γ}),

where x ∈ γ means that x belongs to the image of the curve γ. Finally, for any nonnegative function f , we denote by ´γ f (x) dx the line integral of f along γ ∈ Γ:

ˆγ f (x) dx := ˆ+∞ 0 f (γ(t))| γ(t)| dt.
Irrigation costs For α ∈ [0, 1] we consider the irrigation cost I α : ˆΓ ˆγ θ η (x) α-1 dx dη(γ).

IP(R d ) → [0, ∞]
(LI α )

We set Z η (γ) = ´γ θ η (x) α-1 dx so that the cost may expressed as

I α (η) = ˆΓ Z η (γ) dη(γ).
Proposition 3.1.1 (First variation inequality for I α ). If η is an irrigation plan with I α (η) nite, then for all irrigation plan η the following holds:

I α (η) ≤ I α (η) + α ˆZη (γ) d(η -η).
(3.1.2) Theorem 3.1.2 (Existence of minimizers,). For any pair of probability measures µ, ν ∈ Prob(R d ) with compact support, the problem (LI α ) admits a minimizer.

Theorem 3.1.3 (Irrigability). If 1 -1 d < α < 1, for any µ, ν ∈ Prob(R d ) with compact support there exists some η ∈ IP(µ, ν) such that I α (η) is nite.

From now on we assume that α ∈ 1 -1 d , 1 . 1 Notice that lim t→∞ γ(t) exists if γ ∈ Γ 1 (K), and this is all we need since any irrigation plan is concentrated on Γ 1 (K).

d α (µ, ν) ≤ Cm α diam(supp µ ∪ supp ν ).
Landscape function Given an optimal irrigation plan η ∈ IP(δ 0 , ν), we say that a curve γ is η-good if ˆT(γ)

0 θ η (γ[0, t]) α-1 dt < ∞
which is equivalent to the two assertions:

(i) Z η (γ) < ∞, (ii) θ η (γ(t)) = θ η (γ[0, t]
) for all t < T (γ).

One may prove by optimality that η is concentrated on the set of η-good curves. Moreover it is proven in [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF] that for all η-good curves γ, the quantity Z η (γ) depends only on the nal point γ(∞) of the curve, thus we may dene the landscape function z η as follows:

z η (x) = Z η (γ) if γ is an η-good curve s.t. x = γ(∞), +∞ otherwise.
Notice that for an optimal η the cost may be expressed in terms of z η :

I α (η) = ˆΓ Z η (γ) dη(γ) = ˆRd z η (x) dν(x).
Finally, one may show that z η is lower semicontinuous and that the inequality z η (x) ≥ |x| holds.

2 Proving this is just an adaptation of the proof on compact sets. If µ is xed (for example) and ν n → ν with η n ∈ IP(µ, ν n ) optimal and parameterized by arc length, assuming that the cost is bounded, the irrigation plans η n are tight and one may extract a subsequence converging to some η which irrigates ν and whose cost is less than lim inf d α (µ, ν n ) by lower semicontinuity of I α .

The shape optimization problem

We ask ourselves the following question: what is the set of unit volume which is closest to the origin in the sense of irrigation? To give this a precise meaning, we embed everything in the space of probability measures; hence we want to minimize the d α distance between the unit Dirac mass at 0 ∈ R d and sets E of unit volume, seen as the uniform measure on E. This problem reads min {d α (δ 0 , 1 E L ) : |E| = 1}, (S α ) where L denotes the Lebesgue measure on R d . We relax this problem by minimizing on a larger set, which is the set of probability measures with Lebesgue density bounded by 1, thus getting:

min {X α (ν) : ν ≤ 1, ν ∈ Prob(R d )}, (R α )
where X α (ν) = d α (δ 0 , ν).

In the following, we will sometimes encounter positive measures which do not have unit mass, thus we extend the functional by setting X α (ν) := d α (|ν|δ 0 , ν) for any nite measure ν.

A key tool in the analysis of this problem lies in the following lemma.

Proposition 3.1.5 (First variation inequality for X α ). Suppose that ν ∈ Prob(R d ) with X α (ν) < ∞. Suppose also that η is an optimal irrigation plan between |ν|δ 0 and ν, with landscape function z η . The following holds:

X α (ν) ≤ X α (ν) + α ˆzη d(ν -ν)
for any ν ∈ Prob(R d ).

Notice also that the integral ´zη d(ν -ν) is well-dened since ´zη dν = I α (η) = X α (ν) < ∞ and z η is non-negative, though it may be innite.

Proof. If ´zη dν = ∞ then there is nothing to prove. Otherwise for ν-a.e. x, z η (x) is nite hence there are η-good curves reaching x and one can nd a measurable 3 map g : R d → Γ which associates with every x an η-good curve reaching x. Let us build an irrigation plan η ∈ IP(|ν|δ 0 , ν) which is concentrated on η-good curves, by setting η = g # ν, so that

ˆΓ Z η dη = ˆΓ z η (γ(∞)) dη(γ) = ˆRd z η (x) dν.
Then, by the rst variation inequality for I α , we get:

X α (ν) . = d α (|ν|δ 0 , ν) ≤ I α (η) ≤ I α (η)+α ˆΓ Z η d(η-η) = X α (ν)+α ˆRd z η d(ν -ν).
3 One can characterize η-good curves as those γ such that Zη (γ) < ∞ where Zη (γ) := ´∞ 0 |γ| t,η dt is a slight variation of Z η dened in [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF] which is also lower semicontinuous. Hence the multifunction associating to every x the set of η-good curves reaching x can be written as ∈Q {γ ∈ Γ : Zη (γ) ≤ , γ(∞) = x}, i.e. as a countable union of multifunctions with closed graph. This means that this multifunction is measurable and admits a measurable selection (see e.g. [START_REF] Castaing | Convex analysis and measurable multifunctions[END_REF]).

Existence and rst properties

We will often denote by C = C(α, d) or c = c(d) dierent positive constants which depend only on α, d or d respectively.

Existence of minimizers

Theorem 3.2.1. The relaxed shape optimization problem (R α ) admits at least a minimizer.

Proof. The existence of a minimizer follows from the lower semicontinuity and tightness. Indeed, any minimizing sequence ν n must have bounded rst moment since ˆ|x| dν(x) ≤ ˆzη (x) dν(x) = d α (δ 0 , ν).

A bound on the rst moment implies tightness of the sequence and, up to extracting a subsequence, one has ν n ν. The condition ν n ≤ 1 implies ν ≤ 1 and the lower semicontinuity of d α provides the optimality of ν.

For 1 > α > 1 -1 d
, we will denote the optimal value for the relaxed shape optimization problem (R α ) by: e α := min{d α (δ 0 , ν) : ν ≤ 1 and ν ∈ Prob(R d )}.

Lemma 3.2.2 (Scaling lemma). For any nite measure ν we have X α (ν) ≥ e α |ν| α+ 1 d .

Proof. For λ = |ν| -1/d , let ν = λ d ϕ # (ν) be a scaling of ν under the map ϕ(x) = λx in R d . Then, ´Rd dν = λ d ´Rd dν = λ d |ν| = 1 and ν ≤ 1. Thus,

e α ≤ d α (ν, δ 0 ) = λ αd+1 d α (ν, |ν|δ 0 ) = |ν| -(α+ 1 d ) X α (ν).
For any ν, we say that z is a landscape function of ν if it is the landscape function z η associated with some optimal irrigation plan η ∈ IP(δ 0 , ν).

Theorem 3.2.3. Let ν be a minimizer of (R α ) and z a landscape function of ν. Then ν is the indicator of a set A which is a sublevel set of z:

A = {x : z(x) ≤ z }, with z = e α α (α + 1 d ).
(3.2.1)

In particular, A is a solution to problem (S α ) and it is a compact and path-connected set.

Proof. We show that ν also minimizes the rst variation of X α , that is µ → ´z dµ. Take ν a competitor for (R α ). By Proposition 3.1.5, one has:

X α (ν) ≤ X α (ν) + α ˆz d(ν -ν), but X α (ν) ≤ X α (ν), thus ˆz dν ≤ ˆz dν
for any ν. So as to minimize this quantity, ν must concentrate its mass on the points where z takes its lowest values. More precisely, there is a value z ∈ [0, ∞] such that ν(x)

     = 1 if z(x) < z , ∈ [0, 1] if z(x) = z , = 0 if z(x) > z .
Indeed, we just take z = sup{t ∈ R : |{z(x) ≤ t}| < 1}. Since ´z dν = e α > 0, necessarily z > 0. This kind of arguments is typical in optimization problems under an upper density constraint, as it was for instance done for crowd motion applications in [START_REF] Maury | Congestion-driven dendritic growth[END_REF].

Step 1: z ≤ eα α α + 1 d For 0 ≤ k < z , we consider the competitor ν = 1 {z≤k} and set |ν| = 1 -m, noting that m > 0 by denition of z . Using Lemma 3.2.2 and Proposition 3.1.5, one gets

e α (1 -m) α+ 1 d ≤ X α (ν) ≤ X α (ν) + α ˆz d(ν -ν) = e α -α ˆ{z>k} z dν. Since ν({z > k}) = 1 -|{z ≤ k}| = m, it follows that e α (1 -m) α+ 1 d ≤ e α -αkm. (3.2.2) As α + 1 d > 1, the map t → t α+ 1 d is (strictly) convex, thus e α 1 -α + 1 d m ≤ e α (1 -m) α+ 1 d ≤ e α -αkm,
hence forgetting the middle term, subtracting e α and dividing by m:

αk ≤ e α α + 1 d .

Taking the limit k → z yields:

z ≤ e α α α + 1 d .

(3.2.3)

Step 2: ν = 1 A where A = {z ≤ z } Take the competitor ν = 1 {z≤z } and set |ν| = 1 + m, m ≥ 0. Using again the scaling lemma and the rst variation of X α one gets: Step 3: Compactness and connectedness

e α (1 + m) α+ 1 d ≤ e α + α ˆz=z z d(ν -ν) = e α + αz m.
A is closed since z is lower semicontinuous and bounded since z(x) ≥ |x| for all x ∈ R d . It is path-connected since any point x with z(x) ≤ z is the endpoint of an η-good curve γ starting from 0 and γ ⊆ A because z is increasing along this curve.

Step 4: z ≥ eα α α + 1 d Take x 0 ∈ A with maximal Euclidean norm. Then the half ball H r (x 0 ) := B r (x 0 ) ∩ {x : x -x 0 , x 0 > 0} is included in A c . We consider the competitor ν = 1 A Hr(x 0 ) , with mass |ν| = 1 + m, where m = |H r (x 0 )| = cr d for some constant c = c(d). To irrigate ν, we pay at most the cost of irrigation of ν, plus the price for moving an extra mass m from 0 to x 0 along the irrigation plan, plus the cost for moving this mass to B r (x 0 ) \ A, which we can bound by Cm α r, as follows:

X α (ν) = d α ((1 + m)δ 0 , ν) ≤ d α ((1 + m)δ 0 , ν + mδ x 0 ) + d α (ν + mδ x 0 , ν + 1 Hr(x 0 ) ) = X α (ν + mδ x 0 ) + d α (mδ x 0 , 1 Hr(x 0 ) ) ≤ e α + αmz(x 0 ) + Cr 1+dα ,
where C = C(α, d) is some positive constant. Since x 0 is not a Lebesgue point of A and x 0 ∈ A, by Lemma 3.2.4 below, it follows that z(x 0 ) = z . Putting this in the previous inequality, combining it with the convexity inequality

X α (ν) ≥ e α (1 + m) α+ 1 d ≥ e α 1 + α + 1 d m ,
and dividing by m > 0, one gets:

e α α + 1 d ≤ αz + Cr 1+dα-d .
Passing to the limit r → 0, we obtain z ≥ e α α α + 1 d .

Lebesgue points

For any set K, let us set Θ K (x, r) :

= |K∩B(x,r)| |Br(x)|
the fraction of mass of B r (x) lying in K. We also set

β = d α -1 - 1 d = 1 + dα -d,
a number which is strictly between 0 and 1 as 1 > α > 1 -1 d , which will appear as in the rst chapter as a Hölder exponent of the landscape function.

Lemma 3.2.4. If z(x) < z , then x is a Lebesgue point of A.

Proof. Consider the competitor ν = 1 A∪Br(x) with mass |ν| = 1 + m where m = |B r (x) \ A|. To irrigate ν from 0 one may irrigate ν + mδ x from 0, then 1 Br(x)\A from x. The rst cost may be estimated by the rst variation of X α , and the second one may be bounded knowing that irrigating a mass m at distance r costs less than Cm α r for some constant C = C(α, d). This writes rigorously as:

X α (ν) = d α ((1 + m)δ 0 , ν) ≤ d α ((1 + m)δ 0 , ν + mδ x ) + d α (ν + mδ x , ν + 1 Br(x)\A ) = X α (ν + mδ x ) + d α (mδ x , 1 Br(x)\A ) ≤ e α + αmz(x) + Cm α r,
On the other hand, by the scaling lemma and by convexity, one has X α (ν) ≥ e α 1 + α + 1 d m ≥ e α + αmz , the last inequality resulting from Theorem 3.2.3. Combining this with the previous series of inequalities yields:

m 1-α (z -z(x)) ≤ Cr, which rewrites Θ 1-α A c (x, r)(z -z(x)) ≤ Cr β , (3.2.4)
where we recall β = 1 + dα -d is a number strictly between 0 and 1. As a consequence, if z(x) < z then Θ A c (x, r) r→0 --→ 0 and x is a Lebesgue point of A.

Hölder continuity of the landscape function

We set A r (x) := A ∩ B r (x), zr (x) the central median of z on the set A r (x) and z r (x) its mean. We are going to show that z is β-Hölder continuous using Campanato estimates (Campanato spaces were introduced in [START_REF] Campanato | Proprietà di hölderianità di alcune classi di funzioni[END_REF], see [Giu03, Section 2.3] for a modern exposition), as it is done in [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF]. More precisely, we are going to prove the following inequality, for arbitrary r > 0:

Ar(x) |z -z r (x)| ≤ Cr β , (3.3.1) |z r (x) -z r/2 (x)| ≤ Cr β , (3.3.2) z r (x) -z(x) ≤ Cr β , (3.3.3) |z(x) -z r (x)| ≤ Cr β , (3.3.4) |z |y-x| (x) -z |y-x| (y)| ≤ C|y -x| β . (3.3.5)
Notice that the two last inequalities imply that z is indeed β-Hölder continuous:

|z(y) -z(x)| ≤ |z(y) -z |y-x| (y)| + |z |y-x| (x) -z |y-x| (y)| + |z(x) -z |y-x| (x)| ≤ C|y -x| β .
The main diculty we will encounter is that we will quite easily obtain estimates of the form

• • • ≤ C r β Θ A (x, r) 1-α ,
and will need to get rid of the term Θ A (x, r) 1-α , i.e. treat the case when it becomes small.

Main lemmas

We will make use of the following lemmas.

Lemma 3.3.1 (Maximum deviation).

There is a constant C = C(d, α) > 0 such that the following holds:

∀y ∈ A r (x), z -z(y) ≤ C r β Θ A c (x, r) 1-α .
(3.3.6) Remark 3.3.2. One can see that if Θ A (x, r) becomes small, then Θ A c (x, r) is large (close to 1), and actually all values of z in A r (x) become close to the same value z up to Cr β .

Proof. We consider the competitor ν = 1 A∪Br(x) , with mass |ν| = 1 + m where m = |B r (x) \ A|. For any y ∈ A r (x), let us irrigate ν from 0 by irrigating ν from 0, moving an extra mass m from 0 to y along the irrigation plan, then irrigating 1 Br(x)\A from this mass at y. Using Lemma 3.2.2, we have

e α (1 + m) α+ 1 d ≤ X α (ν) ≤ X α (ν) + α ˆz d(mδ y ) + Crm α .
By convexity,

e α 1 + α + 1 d m ≤ (1 + m) α+ 1 d e α ≤
z -z(y) ≤ Cr(r d Θ A c (x, r)) α-1 = C r β Θ A c (x, r) 1-α . Lemma 3.3.3 (Mean deviation). There is some constant C = C(d, α) > 0 such that Ar(x) |z(y) -z r (x)| dy ≤ Cr β
for all r > 0 and all x ∈ A.

Proof. We will rst show that

Ar(x) |z(y) -z r (x)| dy ≤ C r β Θ A (x, r) 1-α . There is a disjoint union A r (x) = A -A + such that |A -| = |A + | = |Ar(x)|
2 and z ≤ zr (x) on A -, z ≥ zr (x) on A + . Let us consider the competitor ν = 1 A -1 A + + 1 A -. By the rst variation lemma: Consequently:

X α (ν) ≤ X α (ν) + α ˆz d(ν -ν). α ˆz d(ν -ν) ≤ d α (δ 0 , ν) -d α (δ 0 , ν) ≤ d α (ν, ν). We know that d α (ν, ν) ≤ C|ν -ν| α diam(supp(ν -ν)) ≤ C|A r (x)| α r for some C = C(α, d) > 0.
Ar(x) |z(y) -zr (x)| dy ≤ C|A r (x)| α-1 r ≤ C |A r (x)| α-1 |B r (x)| α-1 r 1+d(α-1) = C r β Θ A (x, r) 1-α . Moreover, one has |z r (x) -zr (x)| = Ar(x) z(y) -zr (x) dy ≤ Ar(x) |z(y) -zr (x)| dy ≤ C r β Θ A (x, r) 1-α which leads to Ar(x) |z(y) -z r (x)| dy = Ar(x) |z(y) -zr (x)| dy + |z r (x) -zr (x)| ≤ C r β Θ A (x, r) 1-α .
Now we get rid of Θ A (x, r) 1-α . If Θ A (x, r) ≥ 1/2, we get the desired inequality. On the other hand, if Θ A c (x, r) ≥ 1/2, by Lemma 3.3.1, we have

0 ≤ z -z(y) ≤ Cr β , ∀y ∈ A r (x),
which also implies that

0 ≤ z -z r (x) ≤ Cr β .
By these two inequalities, we have

|z(y) -z r (x)| ≤ Cr β , ∀y ∈ A r (x).
Now, taking the mean over A r (x) y leads to the wanted inequality as well:

Ar(x)
|z(y) -z r (x)| dy ≤ Cr β .

Remark 3.3.4. Notice that the estimate

Ar(x) |z(y) -z r (x)| dy ≤ C r β Θ A (x, r) 1-α
is valid in general: we only use the fact that ν is an indicator function (a density bounded from below would suce). The optimality of ν comes into play to to get rid of Θ A (x, r).

Hölder regularity

Proposition 3.3.5 (Small-scale dierence). For all x ∈ A and all r > 0 one has

|z r (x) -z r/2 (x)| ≤ Cr β . Proof. First we show that |z r (x) -z r/2 (x)| ≤ C r β Θ A (x, r) 1-α . Indeed, by Lemma 3.3.3, |z r (x) -z r/2 (x)| ≤ ´Ar/2 (x) |z(y) -z r (x)| dy |A r/2 (x)| ≤ |A r (x)| |A r/2 (x)| Ar(x) |z(y) -z r (x)| dy ≤ 2 d Θ A (x, r) Θ A (x, r/2) Cr β ≤ C r β Θ A (x, r/2) .
As before, if Θ A (x, r/2) ≥ 1/2 we get the desired estimate. Otherwise, we have

Θ A c (x, r/2) ≥ 1/2 and Θ A c (x, r) ≥ 2 -d Θ A c (x, r/2) ≥ 2 -1-d . Now, by Lemma 3.3.1, 0 ≤ z -z(y) ≤ C r β Θ A c (x, r) ≤ Cr β , ∀y ∈ A r (x). Consequently 0 ≤ z -z r/2 (x) ≤ Cr β and 0 ≤ z -z r (x) ≤ Cr β which implies that |z r (x) -z r/2 (x)| ≤ Cr β .
Lemma 3.3.6 (Lower deviation to the mean). There is a constant C = C(d, α) > 0 such that for all x ∈ A and all r > 0 one has:

∀y ∈ A r (x), z r (x) -z(y) ≤ Cr β .

(3.3.7)

Proof. First we show that

z r (x) -z(y) ≤ C r β Θ A (x, r) 1-α .
Remove the mass m = |A r (x)| going to A r (x) from the irrigation plan, make it travel along the plan to any xed y ∈ A r (x) and then send it to A r (x): this should cost more. This implies αmz(y) -α

ˆAr(x) z + Cm α r ≥ 0,
which may be rewritten as

z r (x) -z(y) ≤ Cm α-1 r ≤ C r β Θ A (x, r) 1-α . Now if Θ A (x, r) ≥ 1/2 one gets the desired result. Otherwise Θ A c (x, r) ≥ 1/2 and Lemma 3.3.1 yields: 0 ≤ z -z(y) ≤ Cr β , ∀y ∈ A r (x).
Thus 0 ≤ z -z r (x) ≤ Cr β and for any xed y ∈ A r (x),

|z r (x) -z(y)| ≤ |z r (x) -z | + |z -z(y)| ≤ Cr β ,
from which we also get the wanted inequality.

Lemma 3.3.7 (Deviation to the mean). For all x ∈ A and all r > 0, one has

|z(x) -z r (x)| ≤ Cr β .
Proof. By Proposition 3.3.5, one has

|z(x) -z r (x)| ≤ |z(x) -z r/2 (x)| + |z r/2 (x) -z r (x)| ≤ |z(x) -z r/2 (x)| + Cr β ,
which means by setting f (r) = |z(x) -z r (x)| for r > 0 that:

f (r) ≤ f (r/2) + Cr β . Consequently for all k ∈ N f (r) ≤ f (r • 2 -(k+1) ) + Cr β k i=0 2 -iβ thus f (r) ≤ lim sup ε→0 f (ε) + Cr β ∞ i=0 2 -iβ ≤ lim sup ε→0 f (ε) + Cr β .
Now let us prove that f (ε) → 0 when ε → 0, i.e. z ε (x) ε→0 --→ z(x). We already know that z is lower semi-continuous hence z(x) ≤ lim inf ε→0 z ε (x). Moreover using (3.3.7), we have

lim sup ε→0 z ε (x) ≤ lim sup ε→0 (z(x) + Cε β ) = z(x),
which implies that z ε (x) → z(x) when ε → 0. Therefore the inequality f (r) ≤ Cr β holds, that is to say:

|z(x) -z r (x)| ≤ Cr β .
Lemma 3.3.8 (Large scale dierence). For any x, y ∈ A, one has:

|z |y-x| (x) -z |y-x| (y)| ≤ C|y -x| β . Proof. Set r = |y -x|, and ∆ r = B r (x) ∩ B r (y). Notice that, ∆ r being a xed fraction of B r (x) (independent of r), |∆ r | = c|B r | for some c = c(d) ∈]0, 1[. If both Θ A c (x, r) ≥ c 2 and Θ A c ( y, r) ≥ c
2 , then by Lemma 3.3.1 one has:

0 ≤ z -z r (x) ≤ C r β Θ A c (x, r) 1-α ≤ Cr β , and 0 ≤ z -z r (y) ≤ C r β Θ A c (y, r) 1-α ≤ Cr β , which implies the desired inequality |z r (x) -z r (y)| ≤ Cr β . (3.3.8) On the other hand, if either Θ A c (x, r) or Θ A c (y, r) is less than c/2, say Θ A c (x, r) ≤ c 2 ,
we claim the desired inequality (3.3.8) still holds. Indeed, for all u ∈ A r (x)

∩ A r (y) one has |z r (x) -z r (y)| ≤ |z r (x) -z(u)| + |z r (y) -z(u)| thus integrating over A r (x) ∩ A r (y) in u one gets: |z r (x) -z r (y)| ≤ 1 |A r (x) ∩ A r (y)| ˆAr(x) |z(u) -z r (x)| du + ˆ|Ar(y)| |z(u) -z r (y)| du ≤ Cr β |A r (x)| + |A r (y)| |A r (x) ∩ A r (y)| ,
the last inequality resulting from Lemma 3.3.3. Note that

|A r (x) ∩ A r (y)| = |∆ r ∩ A| ≥ |∆ r | -|B r (x) \ A| = c|B r (x)| -|B r (x) \ A| which implies that |A r (x)| + |A r (y)| |A r (x) ∩ A r (y)| ≤ 2|B r (x)| c|B r (x)| -|B r (x) \ A| = 2 c -Θ A c (x, r) ≤ 4 c .
Thus, in this case, we still have

|z r (x) -z r (y)| ≤ Cr β |A r (x)| + |A r (y)| |A r (x) ∩ A r (y)| ≤ Cr β .
Theorem 3.3.9 (Hölder continuity). The function z is β-Hölder continuous on A. Let us state a consequence of this result which is a renement of the previous proposition on Lebesgue points. Now we may quantify the minimal size of a ball one can put inside A around x in terms of z -z(x). Proposition 3.3.10 (Interior points). For some constant C = C(α, d) the following holds:

∀x ∈ A, B r(x) (x) ⊆ A, (3.3.9) where r(x) = C(z -z(x)) 1/β ≥ 0. In particular {x ∈ A : z(x) < z } ⊆ • A and ∂A ⊆ {x ∈ A : z(x) = z }.
Proof. It suces to prove (3.3.9) for x 0 ∈ A satisfying z(x 0 ) < z . Consider a point x ∈ A c . Take a point y ∈ A which is closest to x : it is possible since A is compact. By construction y is not a Lebesgue point of A, thus by Lemma 3.2.4, z(y) = z . By the Hölder continuity of z stated in Theorem 3.3.9,

z -z(x 0 ) = |z(y) -z(x 0 )| ≤ C|y -x 0 | β ≤ C|x -x 0 | β ,
where the last inequality follows from the fact that |y-

x 0 | ≤ |y-x|+|x-x 0 | ≤ 2|x-x 0 | because y minimizes the distance from x. Hence, for all x ∈ A c , |x -x 0 | ≥ C(z - z(x 0 )) 1/β = r(x 0 )
, which implies the desired result.

On the dimension of the boundary

We are interested in the dimension of the boundary ∂A, our guess being that it should be non-integer, and lie between d -1 and d. Here we look at the Minkowski dimension (also called box-counting dimension). Given a set X, we denote by N ε (X) the maximum amount of disjoint balls of radius ε centered at points of X. When these coincide we just call it the Minkowski dimension and denote it by dim M (X).

We shall get an upper bound on the upper Minkowski dimension. We say that X is of dimension smaller than

δ if dim M X ≤ δ. Lemma 3.4.2. There is a constant C = C(α, d) such that for all k ≤ z , |{x ∈ A : k < z(x) ≤ z }| ≤ C(z -k).
-αmz + e α 2 (α + 1 d )(α + 1 d -1)m 2 ≤ -αkm Thus m ≤ C(z -k) with 1/C = e α (α + 1/d)(α + 1/d -1)/(2α).
Theorem 3.4.3. The set ∂A is of dimension less than d -β.

Proof. For ε > 0 xed, take disjoint balls (B i ) i∈I of radius ε, where N := |I| = N ε (∂A).

We set

B + i = B i \ A, B - i = B i ∩ A.
We split the set of balls into two parts: those which have a larger intersection with A rather than A c , and vice-versa. Namely, we set

I + = {i ∈ I : |B + i | ≥ |B i |/2}, N + = |I + |, I -= {i ∈ I : |B - i | ≥ |B i |/2}, N -= |I -|,
so that I = I + ∪ I -and N ≤ N + + N -. We are going to bound N + and N -by some power of ε.

Step 1: Bound on N - Since z is β-Hölder continuous on A, one has for each

B i = B ε (x i ): ∀x ∈ B i ∩ A, |z(x) -z | < Cε β ,
since the center x i lies in ∂A ⊆ {z = z } according to Proposition 3.3.10. Consequently

(∂A) ε ∩ A ⊆ {z -Cε β < z ≤ z },
thus because of Lemma 3.4.2:

|(∂A) ε ∩ A| ≤ |{z -Cε β < z ≤ z }| ≤ Cε β .
Using the previous inequality and the fact that

|B - i | ≥ |B i |/2 ≥ Cε d for i ∈ I -, one has: CN -ε d ≤ i∈I - |B - i | ≤ |(∂A) ε ∩ A| ≤ Cε β ,
which implies:

N -≤ Cε -(d-β) .
(3.4.1)

Step 2: Bound on N + We consider the competitor ν = 1 à where à = A ∪ i∈I + B +

(1 + m) α+1/d ≥ 1 + (α + 1/d)m + 1/2 • (α + 1/d)(α + 1/d -1)(1 + m) α+1/d-2 m 2 ≥ 1 + (α + 1/d)m + Cm 2
because for ε small, 1 + m is less than 2 for example. Consequently one may say:

e α 1 + (α + 1/d)m + Cm 2 ≤ e α + αmz + i∈I + Cε|B + i | α .
Recall that αz = e α (α + 1/d), thus after simplifying one gets for some C > 0:

m 2 ≤ C i∈I + |B + i | α ε ≤ CN + ε 1+αd . (3.4.2) Notice that for i ∈ I + , |B + i | ≥ |B i |/2 ≥ Cε d , so that m = i∈I + |B + i | ≥ CN + ε d .
Injecting this into (3.4.2), one gets:

(N + ε d ) 2 ≤ CN + ε 1+αd , thus N + ≤ Cε 1+αd-2d = Cε -(d-β) .
(3.4.3) Putting (3.4.1) and (3.4.3) together yields:

N ε (∂A) = N ≤ N + + N -≤ C ε d-β , and dim M (∂A) = lim sup ε→0 log(N ε (∂A)) -log(ε) ≤ d -β,
which means that ∂A is of dimension smaller than d -β.

dim H (∂A) = dim M (∂A) = d -β.
Proving this requires to establish the inequality dim H (∂A) ≥ d -β, for which we do not have a working strategy yet.

Numerical simulations

Our goal now is to compute solutions to our shape optimization problem numerically.

To perform numerical simulations, we use the Eulerian framework of branched transport, rst dened by Xia in [START_REF] Xia | Optimal paths related to transport problems[END_REF]. This framework is based on vector measures with a measure divergence, i.e. measures v

∈ M d (R d ) such that ∇ • v ∈ M (R d
), the set of such measures being denoted by M div (R d ). The cost is the so-called α-mass:

M α (v) =    ˆ dv dH 1 (x) α d H 1 (x) if v is 1-rectiable,

+∞

otherwise.

An elliptic approximation of this functional was introduced by Oudet and Santambrogio in [START_REF] Oudet | A Modica-Mortola approximation for branched transport and applications[END_REF], in the spirit of Modica and Mortola [START_REF] Modica | Un esempio di Γ --convergenza[END_REF]. The approximate functional is dened for ε > 0 by:

M α ε (v) = ε -σ 1 ˆ|v(x)| σ dx + ε σ 2 ˆ|v(x)| 2 2 dx
for suitably chosen σ, σ 1 , σ 2 . It is proven in [START_REF] Oudet | A Modica-Mortola approximation for branched transport and applications[END_REF] that M α ε Γ-converges to M α as ε goes to 0, for a suitable topology on M div (R d ). Moreover, the Γ-convergence result also holds imposing an equality constraint on the divergence ∇ • v = f ε , for a suitable sequence f ε f , as proven in [START_REF] Monteil | Uniform estimates for a Modica-Mortola type approximation of branched transportation[END_REF]. The results of [START_REF] Oudet | A Modica-Mortola approximation for branched transport and applications[END_REF] are proven in dimension d = 2, but in [START_REF] Monteil | Elliptic approximations of singular energies under divergence constraint[END_REF] there is a proof of how to extend to higher dimension, in the case α > 1 -1/d (in dimension d = 2 there is also a version of the Γ-convergence result for α ≤ 1/2). Also note that, recently, other phase-eld approximations for branched transport or other network problems have been studied, see for instance Here we adapt the approach of [START_REF] Oudet | A Modica-Mortola approximation for branched transport and applications[END_REF] to our shape optimization problem by adding this time an inequality constraint on the divergence.

Recall that the Lagrangian and Eulerian frameworks are equivalent [START_REF] Pegon | On the Lagrangian branched transport model and the equivalence with its Eulerian formulation[END_REF], so that the irrigation distance may be computed in the following way:

d α (µ, ν) = inf v {M α (v) : ∇ • v = µ -ν}.
Consequently the shape optimization problem (R α ) rewrites, in relaxed form, as:

min v {M α (v) : µ -1 ≤ ∇ • v ≤ µ} where µ = δ 0 . (ES) Setting a = µ -1, b = µ and some mollied versions a ε = µ ε -1, b ε = µ ε ,
for example a convolution of µ with the standard mollier of suitable size r ε (e.g. ε σ 2 r -d ε = o(1) as in [START_REF] Monteil | Uniform estimates for a Modica-Mortola type approximation of branched transportation[END_REF]), we dene the following approximate problem, for ε > 0:

min v {M α ε (v) : a ε ≤ ∇ • v ≤ b ε }. (AS)
Let us remark that the above-mentioned Γ-convergence results do not allow us to say that this problem approximates (ES), as the inequality constraint on the divergence is not directly in these works. We leave this question for further investigation, as our aim is for now to make a rst attempt to compute numerically an optimal shape for the original problem (S α ).

Optimization methods

We tackle problem (AS) by descent methods. Two diculties arise: rst of all, the functional M α ε is not convex hence there is no guarantee that the methods converge, and if they do, they may converge to a local minimizer which is not necessarily a global minimizer ; secondly, this is a constrained problem, hence we will need to compute projections or resort to proximal methods to handle the constraint. The simplest approach is to use a rst-order method, for instance to perform a projected gradient descent on the functional M α ε for ε xed (but small):

The projected gradient method.

v 0 ∈ C v n+1 = p C (v n -τ n ∇M α ε (v n )), where C = {v : a ε ≤ ∇ • v ≤ b ε }
is the convex set of admissible vector elds for (AS).

Computing the projection p C is not an easy task, even more so as we want fast computations since this projection should be done at each step of the algorithm. Actually, this projection step will be quite costly (at least in our approach), hence we need to pass to a higher order method to get to an approximate minimizer in a reasonable number of iterations.

Recall that the projected gradient method is a particular case of the proximal gradient method, which we describe briey. Consider a problem of the form

min v f (v) + g(v)
where f is smooth and g proximable, in the sense that one may easily compute its proximal operator

prox τ g (v) = argmin v g(v ) + 1 2τ |v -v| 2 .
The proximal gradient method consists in doing at each step an explicit descent for f and an implicit descent for g:

The proximal gradient method.

v 0 given v n+1 = prox τn g (v n -τ n ∇f (v n )).
The projected gradient method corresponds to the case

g(v) = 0 if v ∈ C, +∞ otherwise.
If there was no function g, we recover the classical gradient descent method. Notice that there is an implicit choice in this method, since we compute gradients which depend on the scalar product. There is no reason that the canonical scalar product is well adapted to the function we want to minimize. Following the work of Lee, Sun and Saunders [START_REF] Lee | Proximal Newton-type methods for minimizing composite functions[END_REF] on Newton-type proximal methods, one may twist the scalar product, leading to the more general method:

A twisted proximal gradient method.

v 0 given v n+1 = prox τn,Hn g (v n -τ n ∇ Hn f (v n )), (3.5.1)
where ∇ H f (x) is the gradient of f with respect to the scalar product x, y H = Hx, y for H an invertible self-adjoint operator, and

prox τ,H g (v) = argmin v g(v ) + 1 2τ v -v 2 H .
The best quadratic model of f around a point x 0 is

Qf x 0 (x) = f (x 0 ) + ∇ H f (x 0 ), x H + 1/2 x, x H , with H = H f (x 0 )
being the Hessian of f at x, thus it is natural to consider (3.5.1) with H n = H f (x n ). Notice indeed that if g is zero, the proximal operator is the identity and that ∇ H f (v) = H -1 ∇f , so that one recovers Newton's method:

v n+1 = v n -τ n H -1 n ∇f (v n ),
which is known to converge quadratically for smooth enough f . This is why this method is called proximal Newton method. However, for large-scale problems, computing and storing the Hessian is very costly, thus an alternative is to set H n to be an approximation of the Hessian of f at v n , thus leading to proximal quasi-Newton methods. These methods were introduced in [LSS14], which we refer to for further detail and theoretical results of convergence.

A very popular choice for H n is given by the L-BFGS method (see [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF]), which is a quasi-Newton method building in some sense the best approximation of the Hessian at v n using only the information of the points v k and the gradients ∇f (v k ) for a xed number of previous steps k = n, n -1, . . . , n -L + 1. The interest is that no matrix is stored, and there is a very ecient way to compute the matrix-vector product H -1 n • v using simple algebra. Therefore, we decided to implement a proximal L-BFGS method, which in our case reads:

Projected L-BFGS method.

v 0 given v n+1 = p Hn C (v n -τ n H-1 n ∇f (v n )), (3.5.2)
where Hn is the approximate Hessian computed with the L-BFGS method with L steps and p Hn C is the projection on C with respect to the norm • Hn . The algorithm to compute the matrix-vector product H-1 n • x is given in Section 3.5.3.

Computing the projection

The diculty lies in the computation of the projection, that is on the proximal operator.

A box constraint on the variable is very easy to deal with, but here we are faced with box constraints on ∇•v, that is on a linear operator applied to v. Moreover, we want to compute a projection with respect to a twister scalar product

min v -v 0 2 H 2 : a ≤ ∇ • v ≤ b, v//∂Ω . (P)
Note that, when one considers the divergence operator as an operator acting on vector elds dened on the whole R d (extended to 0 outside Ω), the Neumann boundary condition above exactly corresponds to the fact that the divergence has no mass on ∂Ω, which can be considered as included in the inequality constraints.

As a convex optimization, such a problem admits a dual problem, which we are going to use. We set

ψ(w) = 0 if a ≤ w ≤ b, +∞ if not,
.

whose Legendre transform is

g(u) = ψ (u) = ˆbu + -ˆau -,
so that ψ = ψ = g . Let us derive formally the dual problem by an inf -sup exchange:

inf v//∂Ω v -v 0 2 H 2 : a ≤ ∇ • v ≤ b = inf v 1 2 v -v 0 2 H + ψ(∇ • v) = inf v 1 2 v -v 0 2 H + sup u -∇u, v -g(u) = inf v sup u 1 2 v -v 0 2 H -∇ H u, v H -g(u) ≥ sup u -g(u) + inf v 1 2 v -v 0 2 H -∇ H u, v H = -inf u g(u) + sup v ∇ H u, v H - 1 2 v -v 0 2 H = -inf u g(u) + ∇ H u 2 H 2 + ∇ H u, v 0 H = -inf u g(u) + 1 2 ˆH-1 ∇u • ∇u -ˆu(∇ • v 0 ).
Hence the dual problem reads:

min u 1 2 ˆH-1 ∇u • ∇u -ˆu(∇ • v 0 ) f (u) + ˆbu + -ˆau - g(u)
.

(D)

The inf -sup exchange can be justied with equality via Fenchel's duality [Bre11,

Chapter 1] in a well-chosen Banach space. Hence there is no duality gap:

min (P) + min (D) = 0.
As a consequence solving the dual problem provides a solution to the primal one.

Indeed if u is optimal for (D) then v = v 0 + ∇ H u is optimal for (P). Now let us justify why it was interesting to pass by the resolution of a dual problem. Such a problem is of the form

min u f (u) + g(u), (3.5.3)
where f is smooth, with gradient ∇f (u) = -∇ • (H -1 ∇u) -∇ • v 0 , and g is proximable:

prox τ g (u)(x) =      u(x) -τ a if u(x) < τ a, 0 if τ a ≤ u(x) ≤ τ b, u(x) -τ b if u(x) > τ b.
We know how to compute the proximal operator and the gradient of f , since L-BFGS provides a simple method to compute the product H -1 x. Problems of the form (3.5.3) with f smooth (and computable gradient) and g proximable can be tackled with rstorder methods such as the proximal gradient method described in the previous section (also called ISTA) or a fast proximal gradient method called FISTA, introduced in [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF]. We opted for the latter, which is a slight modication of the proximal gradient method using an intermediary point:

u 0 ∈ H 1 (R d ), ũn = u n + λ n (u n -u n-1 ), u n+1 = prox τ g (ũ n -τ ∇f (ũ n )), (FISTA)
where λ n is given by some recursive formula (we refer to [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF] for the details). It enjoys a theoretical and practical rate of convergence which is higher than ISTA and which is that of the classical gradient method:

f (u n ) -f opt ≤ 2L f |u 0 -u opt | 2 (n + 1) 2 .

Algorithms and numerical experiments

Following the work of [START_REF] Oudet | A Modica-Mortola approximation for branched transport and applications[END_REF], we discretize our problem on a staggered grid : we divide the cube Q = [-1, 1] 2 into M 2 subcubes of side 2/M , the functions U are dened at the center of the small cubes, while the x component V x of a vector elds V is dened on the vertical edges of the grid and the y component V y on the horizontal edges of the grid. This is quite convenient to compute the discrete divergence of a vector eld and the discrete gradient of a function.

• Unknowns: (V x i,j ) 1≤i≤M 1≤j≤M +1 , (V y i,j ) 1≤i≤M +1 1≤j≤M , with V x 1,j = V x M +1,j = V y i,1 = V y 1,M +1 = 0,
which means that V is parallel to the boundary.

• Objective function:

F (V ) = ε -σ 1 h 2 i,j N ( Vi,j ) σ + ε σ 2 h 2 /2 i,j |∇ i,j V x | 2 + i,j |∇ i,j V y | 2 .
There are several denitions to give to make sense of F . First of all N is a smooth approximation of the norm, of the form

N (x) = (|x| 2 + ε 2 s ) 1/2 for ε s small.
The discrete vector eld Vi,j = ( V x i,j , V y i,j ) is an interpolation of (V x , V y ) dened at the centers of the cubes:

V x i,j = V x i,j + V x i+1,j 2 , V y i,j = V y i,j + V y i,j+1 2 , 1 ≤ i, j ≤ M.
Finally the discrete gradient is dened as usual by

∇ i,j V x = ((V x i,j+1 -V x i,j )/h, (V x i+1,j -V x i,j )/h), 1 ≤ i ≤ M -1, 1 ≤ j ≤ M, ∇ i,j V y = ((V y i,j+1 -V y i,j )/h, (V y i+1,j -V y i,j )/h), 1 ≤ j ≤ M -1, 1 ≤ i ≤ M.
We may now give the main algorithm and its sub-methods.

Algorithm 1 Proximal L-BFGS for F Data: tolerance tol, initial vector eld V 0 , step τ 0 , source δ

V ← V 0 , U ← U 0 compute error while error > tol do τ ← τ 0 repeat G ← MultiplyBFGS(∇F (V )) V, U ← Project(V -τ G, U, δ, τ ) τ ← τ /2 until F (V ) has decreased update L-BFGS data compute error end while
The update step for L-BFGS data consists in storing in Y, Z, r the points and gradients of the L previous steps, so that at step n:

Y L-k = ∇F (V n-k ) -∇F (V n-k-1 ), Z L-k = V n-k -V n-k-1
for all k = 0, . . . , L -1, and r k = 1/(Y k • Z k ) for all k = 0, . . . , L -1. Notice here that we do a simple backtracking line search by reducing the step size τ until the energy has decreased, for example until it has suciently decreased and satises the Armijo rule.

Also, notice that the potential U computed at step n is used at the next step as initial data ; this trick extensively speeds up the computation of the projection. Finally, we took as error measurement some relative dierence between two consecutive steps. Now, as stated in Section 3.5.2, the projection on C with respect to • H is computed via the FISTA method, as follows:

Algorithm 2 Project V 0 on C with respect to • H Data: tolerance tol p , step τ p function Project(V 0 , U 0 , δ, τ ) D 0 ← ∇ • V 0 U ← U 0 while error > tol p do t p ← t; t ← (1 + 1 + 4t 2 p )/2; s ← (t p -1)/t G ← MultiplyBFGS(∇U ) U i ← U + s(U -U old ) U old ← U U ← Prox(U i -τ p (∇ • G -D 0 ), δ, τ ) compute error end while V ← V 0 + MultiplyBFGS(∇U ) return V, U end function
The Prox function is just the proximal operator associated with the discrete counterpart of g : u → ´bu + -´au -where a = δ -1, b = δ. Thus P = Prox(U, δ, τ ) is dened by:

P i =      U i -τ (δ i -1) if U i < τ (δ i -1), 0 if τ (δ i -1) ≤ U i ≤ τ δ i , U i -τ δ i if U i > τ δ i .
For the sake of completeness, we give a simple method to compute the L-BFGS multiplication H -1 X (see [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF][START_REF]Updating quasi-Newton matrices with limited storage[END_REF] for details).

Algorithm 3 L-BFGS multiplication

H -1 X function MultiplyBFGS(X ) G ← X for i = L, . . . , 1 do s i ← r i Z i • G G ← G -s i Y i end for G ← (Z L • Y )/(Y L • Y L ) G for i = 1, . . . , L do t ← r i Y i • G G ← G + (s i -t)Z i end for return G end function
We present some numerical results obtained with ε s = 10 -4 , on a M × M grid with M = 201 and ε = 3h where h = 2/M , the code being written in Julia. We have started with random initial values for V and a smooth approximation δ of the Dirac δ 0 . After some days of computation on a standard laptop, one gets the following shapes and underlying networks. approach. An attempt at dening the landscape function for multiple sources was made by Xia in [START_REF] Xia | On landscape functions associated with transport paths[END_REF] in the discrete case. The present chapter aims at dening the landscape function in the multiple source case but with non atomic measures, under an hypothesis on the source and target measures µ, ν, namely that they have disjoint support:

supp(µ) ∩ supp(ν) = ∅.
From this denition we are able to extend many nice properties already known in the single-source case: the landscape function is the rst variation of the function X α , and it exhibits Hölder regularity under some regularity hypotheses on the measures. A direct use of the landscape function can be made to obtain the uniform boundedness of the length of the bers. Other applications already known in the single source case hopefully also hold in the multiple source case, but this is left for further investigation:

the regularity of the bers (locally nite curvature) proved in [START_REF] Morel | The regularity of optimal irrigation patterns[END_REF], and the fractal branching behaviour established in [START_REF] Brancolini | Fractal regularity results on optimal irrigation patterns[END_REF].

Framework denition

We make slight modications to the standard framework described in the rst chapter, in order to simplify notations and proofs. The main dierence will be that curves are no longer dened on R + but on R instead.

The set of curves We set Γ = Lip 1 (R, R d ), and dene for all γ ∈ Γ:

T -(γ) = sup{t : γ is constant on ] -∞, t[}, T + (γ) = inf{t : γ is constant on ]t, +∞[}.
For a curve γ ∈ Γ, we write γ :

[T -, T + ] → R d to mean that T -(γ) = T -and
T + (γ) = T + , and we denote by γ -= γ(T -(γ)), γ + = γ(T + (γ)) the starting point and stopping point of γ. We will write x γ y if γ is a curve starting at x and stopping at y, and x γ y if either x γ y or x γ y.

Restrictions and inclusions If γ is simple and x ∈ γ, we denote by t x the unique t ∈

[T -(γ), T + (γ)] such that γ(t) = x. For x, y ∈ γ, γ[x, y] denotes the usual restriction 1 γ |[tx,ty] if t x ≤ t y or γ |[ty,tx] if t y ≤ t x , and γ[x, +], γ[x, -] denote respectively γ[x, γ + ] and γ[x, γ -].
We also dene the obvious variants γ]a, b[, γ[a, b[ and so on excluding some extremities. We say that γ 1 is included in γ 2 and we write γ 1 ⊆ γ 2 if the trajectory of γ 1 is part of that of γ 2 .

Paths A path p between x and y is a sequence of curves γ i of the form

p : x = x 1 γ 1 x 2 γ 2 . . . x n-1 γ n-1 x n = y,
and an oriented path p from x to y is a sequence of curves γ i of the form

p : x = x 1 γ 1 x 2 γ 2 . . . x n-1 γ n-1 x n = y.
For such an oriented path p we will denote by γ p : [0, T + ] → R d the curve, parameterized by arc length, obtained by concatenating the γ i 's, so that x γp y.

Curve multiplicity Let η be an irrigation plan. For γ ∈ Γ we set

θ η (γ) = η(γ : γ ⊆ γ),
and we say that γ is a η-branch if θ η (γ) > 0. An η-branch path, oriented or not, is a path made of η-branches. When η is xed and there is no confusion, we will omit the prex ηand talk about branches and branch paths.

Cycle-free property We recall that an irrigation plan η is said cycle-free if there is no loop sequence x 1 , x 2 , . . . , x n ,

x n+1 = x 1 such that η(Γ[x i , x i+1 ]) > 0 for 1 ≤ i ≤ n,
where Γ[x, y] is the set of curves which visit x and y, no matter the order. In particular if η(Γ[x, y]) > 0 then there is a unique2 curve γ η (x, y) joining x and y (either from x to y or the contrary), such that η-almost every curve γ ∈ Γ[x, y] follows γ η (x, y) in its trajectory.

Good curves and paths

We need a notion of curve which is more general than branches: we would like to consider curves that do not stay entirely in the network but can go to its boundary, but which nevertheless follow the irrigation plan. This is the role played by what we call η-good curves.

Denition 4.1.1 (η-good curve). We say that γ :

[T -, T + ] → R d is an η-good curve stemming from σ ∈ γ if ˆγ θ η (γ[σ, x]) α-1 dx < ∞.
Moreover, if γ starts at the stem σ (resp. if it stops at the stem), we say that γ is right-sided (resp. left-sided). We say that it is one-sided if it is left-or right-sided.

Remark 4.1.2. In the single-source case µ = δ s , η-good curves stemming from s correspond to the denition of η-good curves given in Denition 1.4.3. Moreover it is possible to show that if γ stems from another point, then it is the restriction of an η-good curves stemming from s.

The denition of η-good curves induces a notion of η-good path.

Denition 4.1.3 (η-good path). We say that a path p, oriented or not, is η-good if it is made of one-sided η-good curves.

Now

We would like to prove that optimal irrigation plans η with nite cost are concentrated on the set of η-good curves: in this way this set could be thought as some kind of support for η. For now, we are able to prove it in the case of optimal irrigation plans which have a nite root system (dened in Subsection 4.1.2), though it should be true in all generality. Let us establish rst a general lemma which does not require this extra hypothesis. Lemma 4.1.4. Let η be a simple irrigation plan which has nite I α cost and which satises the cycle-free property. Fix x 0 ∈ N η . Then for η-almost every curve γ ∈ Γ(x 0 ),

ˆγ θ η (γ[x 0 , x]) α-1 dx < ∞.
Proof. Let us denote by Γ(x 0 , x) the set of curves γ such that x 0 , x ∈ γ and

θ x 0 (x) = η(Γ(x 0 , x)).
Recall that η satises the cycle-free property. Consequently, for all x such that θ x 0 (x) > 0, almost all curves γ in Γ(x 0 , x) follow a common trajectory γ x 0 ,x , i.e. γ[x 0 , x] = γ x 0 ,x . This implies that for almost every curve γ ∈ Γ(x 0 , x):

θ x 0 (x) = θ η (γ[x 0 , x]).
Taking the power α -1 and integrating of Γ(x 0 , x) yields:

θ x 0 (x) α = ˆΓ(x 0 ,x) θ η (γ[x 0 , x]) α-1 dη(γ).
This equality also holds trivially if θ x 0 (x) = 0. Then integrating in x:

ˆRd ˆΓ(x 0 ,x) θ η (γ[x 0 , x]) α-1 dη(γ) d H 1 (x) = ˆRd θ x 0 (x) α d H 1 (x) ≤ I α (η) < ∞.
Thus by Fubini's theorem one gets

∞ > ˆΓ(x 0 ) ˆRd 1 x∈γ θ η (γ[x 0 , x]) α-1 d H 1 (x) dη(γ) = ˆΓ(x 0 ) ˆγ θ η (γ[x 0 , x]) α-1 dx dη(γ).
Consequently for η-almost every curve γ in Γ(x 0 ),

ˆγ θ η (γ[x 0 , x]) α-1 dx < ∞,
which is what we wanted to prove.

Remark 4.1.5. This shows in particular that if θ η (x 0 ) > 0, then either x 0 is an atom of both µ and ν, or one can nd a branch containing x 0 . Indeed, since Γ(x 0 ) has positive measure, either almost all curves are constant equal to x 0 and x 0 is an atom of µ and ν, or there is a positive quantity of non-trivial curves in Γ(x 0 ). Then for sure there is a non-trivial curve γ ∈ Γ(x 0 ) such that ˆγ θ η (γ[x 0 , x]) α-1 dx < ∞,

and γ[x 0 , x] for some x in the interior of the curve is a branch containing x 0 .

We end this subsection by giving a denition of irrigation basins using η-good curves.

Denition 4.1.6 (Irrigation basins). If η ∈ IP(µ, ν) and x ∈ N η we dene the irrigation basins of x as:

Bas + η (x) := {y : there is an η-good curve stemming from x s.t. γ + = y, Bas - η (x) := {y : there is an η-good curve stemming from x s.t. γ -= y, and

Bas η (x) := Bas + η (x) ∪ Bas - η (x).

Root systems

The role of the origin in the single-source case is replaced by the notion of root system. Denition 4.1.7 (Root system). We call root system for η any set of points R ⊆ N η such that η-almost every curve γ passes through some point ρ ∈ R. The elements ρ ∈ R are called roots.

Remark 4.1.8. In the single-source case µ = δ s , there is obviously a root system made of a single point: R = {s}.

The following lemma states that for disjointly supported source and target measures, optimal irrigation plans have a nite root system. It follows from the fact that the irrigation network has a nite graph structure away from the support of the measures, which is proven in [START_REF] Xia | Interior regularity of optimal transport paths[END_REF] where it is termed interior regularity.

Lemma 4.1.9 (Finite root system). If η ∈ IP(µ, ν) is an optimal irrigation plan with nite I α cost, it admits a nite root system R ⊆ N η . Moreover one can assume that all points ρ ∈ R are at positive distance from the supports of µ and ν.

Proof. We cut the curves of the irrigation plan η between their rst exit time from (supp µ) ε and their rst entry time into (supp ν) ε for ε small, getting a irrigation plan η ∈ IP(μ, ν). Since by optimality η is known to have a nite graph structure far from the supports of the measures (by [BCM09, Theorem 4.7]), the marginal measures μ and ν are atomic. We take R to be the set of atoms of μ for example. Now we are able to prove that optimal irrigation plans are concentrated on the set of good curves, provided that they have a nite root system. Proposition 4.1.10. Suppose η is a simple irrigation plan with nite I α cost which satises the cycle-free property and admits a nite root system R. Then η is concentrated on the set of η-good curves passing through points of R.

Proof. By Lemma 4.1.4, if ρ ∈ R is xed, outside an η-negligible set E ρ , all curves γ ∈ Γ(ρ) satisfy ˆγ θ η (γ[ρ, x]) α-1 dx < ∞.
Since almost all curves pass through an element ρ ∈ R, η is concentrated on ρ∈R (Γ(ρ)\ E ρ ) which proves the result.

However, considering η-good curves reaching a root will be too restrictive to give a denition of a landscape function which satises the desired properties (for example semicontinuity). This justies the following denition.

Denition 4.1.11 (Root network). We denote by R the set of points x which belong to a simple branch path connecting two roots, and call it the root network.

Remark 4.1.12. If η has the cycle-free property and R is nite, then R is a compact set since it is made of a nite union of compact curves. If γ is a curve intersecting R, we denote by ρ -(γ), ρ + (γ) the rst and last points of γ belonging to R. Next lemma will relate η-good curves stemming from arbitrary points σ ∈ R d to η-good curves stemming from points ρ of the root network R.

R ρ 1 ρ 2 µ ν
Lemma 4.1.13. Let η be simple irrigation plan with nite I α cost which is cycle-free and has a nite root system R. If γ is a right-sided η-good curve stemming from σ ∈ R d and reaching x, then we are in one of the three situations:

(i) γ intersects R: γ can be cut in two, a branch γ[σ, ρ + (γ)] and a right-sided η-good curve γ[ρ + (γ), x],
(ii) γ is after R: γ ∩ R = ∅ and there is a right-sided η-good curve γ starting at ρ = ρ + (γ) and stopping at x such that γ = γ[σ, x],

(iii) γ is before R: γ ∩ R = ∅ and there is a branch γ starting at σ and stopping at

ρ = ρ -(γ) such that γ = γ[σ, x].
The analogous statement for left-sided η-good curves holds as well. Proof. If γ intersects R and ρ + (γ) = x, then we are clearly in situation (i). Now we assume that that γ does not intersect R except possibly at x. We proceed in two steps.

Step 1. Let ξ be a branch such that ξ ⊆ R and consider the curves containing ξ.

We know almost all of them pass through a root. We want to show that almost all of them may only pass through root after following ξ, or almost all of them may only pass through a root before following ξ. Suppose on the contrary that there are ρ 1 , ρ 2 ∈ R and positive amounts of curves A 1 , A 2 such that all curves in A 1 pass through ρ 1 then ξ along a common trajectory γ 1 , and all curves in A 2 pass through ξ then ρ 2 along a trajectory γ 2 . Since η has no circuit, the branch path

ρ 1 γ 1 [ρ 1 ,ξ -] ξ -ξ ξ + γ 2 [ξ + ,ρ 2 ] ρ 2 is a simple branch path joining ρ 1 = ρ 2 ∈ R hence by denition ξ ⊆ R which is a contradiction.
This proves what we claimed.

Step 2. Consider y ∈ γ[σ, x[: it is clear that γ[σ, y] is a branch which does not intersect R, hence by the rst step, we know that either almost all curves containing this branch reach a root before, or they all reach a root after it. By taking a sequence y n ∈ γ[σ, x[ such that y n → x, one can state that their are two possibilities. The rst possibility is that except for a negligible set U , all curves γ satisfy the following: if γ contains a piece γ[σ, y] for some y ∈ γ[σ, x[ and a root ρ ∈ R, then ρ is reached before γ, following a unique trajectory γ ρ from σ to ρ. We set ρ the rst point where these trajectories part when we follow the curves backwards starting at σ, and set γ the common trajectory from ρ to σ. We set γ = γ ∪ γ the concatenation of γ and γ. By denition all the curves following γ[σ, y] contain γ[ρ, y], hence γ is a right-sided η-good curve stemming from ρ. Let us show that ρ ∈ R. If ρ ∈ R, we are done. Otherwise there are two trajectories γ ρ , γ ρ which parted at ρ. Since they could not meet again by the cycle-free property, ρ

γρ[ρ,ρ] ρ γ ρ [ρ,ρ ]
ρ is a simple branch path, and by construction ρ ∈ R. Then we replace γ with its restriction γ[ρ + (γ), x] and we are in situation (ii).

The second possibility is that except for a negligible set U , all curves γ satisfy the following: if γ contains a piece γ[σ, y] for some y ∈ γ[σ, x[ and a root ρ ∈ R, then ρ is reached after γ[σ, y]. Since the amount of roots is nite and the cycle-free property holds, there is a point y ∈ γ[σ, x[ close enough to x such that for all the trajectories between y and ρ of the curves containing γ[σ, y] cannot part from γ. Hence almost all curves containing γ[σ, y] contain γ and γ is actually a branch. If γ + ∈ R then we are in situation (i) (with γ[ρ + (γ), x] being trivial), while if γ + / ∈ R, we are in situation (iii). The case of γ left-sided is obtained in a similar manner. Remark 4.1.14. Notice that this shows that any η-good curve (not necessarily onesided) can be decomposed into a path made of curves which are restrictions of η-good curves stemming from points of R.

Next lemma will be essential, in that if a good curve stems from a point ρ ∈ R, there is a point after which (and a point before which) the multiplicity and the joint multiplicity coincide. Lemma 4.1.15 (Good stem). Let η be a simple irrigation plan satisfying the cycle-free property, and R be a nite root system for η. If γ is an η-good curve stemming from a point of the root network, then

∀y ∈ γ]ρ + (γ), +[, θ η (y) = θ η (γ[ρ + (γ), y]), and ∀y ∈ γ]-, ρ -(γ)[, θ η (y) = θ η (γ[y, ρ -(γ)]).
Proof. Take such a curve γ. The curve γ[ρ + (γ), +] is an η-good curve stemming from ρ + (γ) ∈ R which is right-sided. Take y ∈ γ]ρ + (γ), +[. For sure θ η (y) ≥ θ η (γ[ρ + (γ), y]) > 0. We look at the curves passing through y. We know by hypothesis that almost all curves pass through a root, and the amount of roots is nite. Take A a positive amount of curves passing through y and by a root ρ ∈ R, then all curves in A follow a common trajectory γ = γ y,ρ , and ρ + (γ)

γ[ρ + (γ),y] y γy,ρ ρ is a branch path joining ρ + (γ) and ρ. If one trajectory is included into the other, since ρ /

∈ γ]ρ + (γ), y[ it must be γ[ρ + (γ), y] ⊆ γ y,ρ . By the cycle-free property, these trajectories cannot part and meet again. Thus if we follow the curves γ[ρ + (γ), y] and γ from the common point y, if those trajectories part at some point y , we know that ρ + (γ) γ[ρ + (γ),y ] y γ [y ,ρ] ρ would be a simple branch path joining ρ + (γ) and ρ, thus they would belong to the root network, which is a contradiction to the denition of ρ + (γ). By way of consequence, almost every curve passing through y contains γ[ρ + (γ), y] and

θ η (y) = θ η (γ[ρ + (γ), y]).
The same reasoning shows that

∀y ∈ γ]-, ρ -(γ)[, θ η (y) = θ η (γ[y, ρ -(γ)]).

Denition of the landscape function

We assume that µ and ν have compact and disjoint supports:

supp µ ∩ supp ν = ∅.
By Lemma 4.1.9, the situation is the following: each point ρ ∈ R will give rise to a cost function Z ρ dened on curves with extremity ρ as the integral3 along the curve of θ α-1 η .

The goal will be to prove that with a suitable choice of an additive constant to each Z ρ , which amounts to assigning a reference value at ρ, these maps take the same value when needed: namely when a point is reached by dierent curves, possibly joining dierent roots. This will allow us to dene a global landscape function z : R d → R.

Recall that Z η (γ) = ´γ θ η (x) α-1 dx. We dene a signed variant when a reference point σ is xed. Denition 4.2.1. Let σ ∈ R d be xed. For all simple curve γ such that σ = γ ± we dene

Z σ (γ) =        + ˆγ θ η (x) α-1 dx if σ = γ -, -ˆγ θ η (x) α-1 dx if σ = γ + .

Balance property

The following proposition is the multiple source counterpart to the well-denedness property established in Proposition 1.4.5, and will be key to dening the landscape function.

Proposition 4.2.2 (Balance property). We assume that η is an optimal irrigation plan with nite I α cost. Consider a good loop, that is a loop made of one-sided 4 η-good curves:

p 1 γ 1 p 2 γ 2 . . . γn p n+1 = p 1
where p i γ i p i+1 is either p i γ i p i+1 or p i γ i p i+1 . Then one has Proof. We denote by σ i the stem of the i-th η-good curve, and x i its other extremity.

p i γ i p i+1 Z η (γ i ) = p i γ i p i+1 Z η (γ i ), or equivalently i Z p i (γ i ) = 0. γ 1 +ε γ 2 +ε γ 3 -ε p 1 p 3 p 2 x 1 x 2 x 3
Choose an orientation for the loop and a small ε > 0. We want to add a small mass ε to all the curves which are oriented accordingly, and remove ε from all the other ones. We cannot do it directly to the whole curves because they are not necessarily branches: there may be no mass left at the point x i . Therefore for all i, we consider a small ε i ≥ 0 and we cut the curve γ i at a time t i = T ∓ (γ i ) ± ε i before the extremity x i , setting x i = γ i (t i ). We are going to remove and add the mass to the restrictions γ i = γ i [σ i , x i ] of the original curves γ i . For example, say the chosen orientation is such that the loop should be read left to right. Then one adds ε to the curves γ i such that p i γ i p i+1 and one removes ε to the other curves γ i such that p i γ i p i+1 . Of course, ε must be small enough so as to be able to remove ε to those curves, but we should choose ε and ε i even more carefully. We do the following:

• if γ i is a branch we do not restrict the curve, i.e. we set ε i = 0,

• if γ i is not a branch, since θ η (γ i [σ i , x i ]) continuously decreases to 0 as ε i → 0, we may assume that θ η (γ i [σ i , x i ]) = ε, with ε smaller than all the masses θ η (γ j ) > 0 which correspond to the branches γ j 's.

To preserve the source and target measures µ, ν, we need to reconnect the point x i to x i for all i. There are several possible cases depending if x i is also a stem of the adjacent curve or not, and depending on their orientations. But in any case, this reconnection costs less than

C i ε α |x i -x i |.
For i such that p i γ i p i+1 , in which case we add ε, one should pay a cost no larger than

αε ˆγ i θ η (x) α-1 dx = αεZ η (γ i ),
and for i such that p i γ i p i+1 , we gain at least -αε

ˆγ i θ η (x) α-1 dx = αεZ η (γ i ).
Since the total cost increment should be nonnegative by optimality of η, one gets αε

p i γ i p i+1 Z η (γ i ) -αε p i γ i p i+1 Z η (γ i )) + Cε α i |x i -x i | ≥ 0, thus p i γ i p i+1 Z η (γ i ) ≤ p i γ i p i+1 Z η (γ i ) + Cε α-1 i |x i -x i |. (4.2.1) It is clear that for all i, Z η (γ i ) → Z η (γ i ) as ε → 0. Moreover since x → θ η (γ i [σ i , x]) decreases when x tends x i along the curve and θ η (γ i [σ i , x i ]) = ε, one has ε α-1 |x i -x i | ≤ ˆγi [x i ,x i ] θ η (γ i [σ i , x]) α-1 dx,
which tends to 0 when x i → x i because γ i is η-good. Passing to the limit ε → 0 in (4.2.1) yields:

p i γ i p i+1 Z η (γ i ) ≤ p i γ i p i+1 Z η (γ i ).
Since the converse inequality holds, had we chosen the converse orientation for the loop, we obtain

p i γ i p i+1 Z η (γ i ) = p i γ i p i+1 Z η (γ i ).
Since Z p i (γ i ) is +Z η (γ i ) in case p i γ i p i+1 and -Z η (γ i ) in case p i γ i p i+1 , we get the

equivalent equation i Z p i (γ i ) = 0.
Notice that if we have a good loop

ρ 1 γ 1 x 1 γ 1 ρ 2 γ 2 x 2 γ 2 ρ 3 . . . ρ n γn x n γ n ρ n+1 = ρ 1
where ρ 1 , . . . , ρ n ∈ R, one has by the balance property:

n j=1 (Z ρ j (γ j ) -Z ρ j+1 (γ j )) = 0. (4.2.2)
Notice that in this expression, each function Z ρ appears an even number of times, half the time with a + sign, and half the time with asign, hence the same balance equation holds if one replaces Z ρ by Zρ = c ρ + Z ρ for any choice of values c ρ ∈ R. Our goal now is to choose these constants c ρ suitably so as to guarantee that all terms in the sum of (4.2.2) are 0: Zρ (γ) -Zρ (γ ) = 0, for all good path ρ γ x γ ρ where ρ, ρ ∈ R and γ, γ are one-sided η-good curves.

Construction of the landscape function

For any pair of roots ρ, ρ , we write ρ ρ if ρ γ x γ ρ holds for some one-sided η-good curves γ, γ , and call it a link 5 . We say that ρ, ρ are connected if there is a

chain ρ = ρ 1 ρ 2 . . . ρ n = ρ .
The connectedness relation is an equivalence relation on R dening equivalence classes which we call connected components. The following lemma will provide us with a good enumeration of all possible links between roots.

Lemma 4.2.3 (Links enumeration). For any connected component R of R there is an enumeration of all links between elements of R : Proof. We simply proceed by induction. We start by choosing a link ρ 1 ρ 1 arbitrarily, then having chosen the rst k -1 links, we choose the next one ρ k ρ k , distinct from the previous ones, such that ρ k or ρ k is an extremity of one the previous links. We do this until there is none left. By construction the rst item is satised. Moreover, all links have been enumerated: if there was another one left, since R is a connected component, there would also be one which was not selected but which would share an extremity with one of the previous extremities. This contradicts the fact that we had completed the selection process.

ρ 1 ρ 1 , ρ 2 ρ 2 , . . . , ρ n ρ n , such that ∀j ≤ n, ρ j ∈ i<j {ρ i , ρ i }.
Next proposition will dene the numbers c ρ for ρ ∈ R using this enumeration, while Proposition 4.2.6 will extend this choice to all (c ρ ) ρ∈ R in such a way that Zρ = c ρ + Z ρ satises the desired property: for x xed, the quantity Zρ (γ) does not depend on the choice of ρ nor the curve γ with extremity ρ but only on the other extremity x of γ. Proposition 4.2.5. Let R be a connected component of R, t ∈ R. We denote by

ρ 1 ρ 1 , ρ 2 ρ 2 , . . . , ρ l ρ l
the links enumeration of R given by Lemma 4.2.3. For all i, we choose one-sided η-good curves ξ i , ξ i such that ρ i ξ i x i ξ i ρ i . Then there is a unique choice of values

(c ρ ) ρ∈R such that (i) c ρ 1 = t, (ii) Z ρ i (ξ i ) + c ρ i = Z ρ i (ξ i ) + c ρ i for all 1 ≤ i ≤ l.
5 Of course ρ ρ is a link.

Proof. We set c ρ 1 = t and proceed by induction. We consider the rst link ρ 1

ξ 1 x 1 ξ 1 ρ 1 . We want c ρ 1 + Z ρ 1 (ξ 1 ) = c ρ 1 + Z ρ 1 (ξ 1 ) ⇐⇒ c ρ 1 = c ρ 1 + (Z ρ 1 (ξ 1 ) -Z ρ 1 (ξ 1 )). If ρ 1 = ρ 1 , then ρ 1 ξ 1 x 1 ξ 1 ρ 1 is
Z ρ 1 (ξ 1 ) -Z ρ 1 (ξ 1 ) = 0.
Hence (ii) holds for the rst link, as c ρ 1 = c ρ 1 . If ρ 1 = ρ 1 , then one may choose c ρ 1 so as to enforce it:

c ρ 1 := c ρ 1 + (Z ρ 1 (ξ 1 ) -Z ρ 1 (ξ 1 )).
And we go on by induction: let us assume that (ii) holds for i ≤ k -1 < l.

c ρ k := c ρ k + (Z ρ k (ξ k ) -Z ρ k (ξ k )).
Now if both ρ k , ρ k ∈ i<k {ρ i , ρ i }, by hypothesis on the links enumeration, there is a chain made of the previous links with extremities ρ k , ρ k :

ρ k = ρ1 ξ1 x1 ξ 1 ρ 1 . . . ρl ξl xl ξ l ρ l = ρ k ,
where each link ρj ξj xj ξ j ρ j is some link of the enumeration ρ i ξ i x i ξ i ρ i for i < k.

We complete the chain with ρ k ξ k x k ξ k ρ k to obtain a loop:

ρ k = ρ1 ξ1 x1 ξ 1 ρ 1 . . . ρl ξl x l ξ l ρ l = ρ k ξ k x k ξ k ρ k
and we apply the balance relation (4.2.2):

0 = j≤l (Z ρj ( ξj ) -Z ρ j ( ξ j )) + Z ρ k (ξ k ) -Z ρ k (ξ k ), = j≤l (c ρj + Z ρj ( ξj )) -(c ρ j + Z ρ j ( ξ j )) + (c ρ k + Z ρ k (ξ k )) -(c ρ k + Z ρ k (ξ k )) ,
Now by the induction hypothesis, since in the big sum we have dierences of cost functions (Z ρ + c ρ ) -(Z ρ + c ρ ) associated to previous links ρ i ξ i x i ξ i ρ i for i < k, all these dierences are 0. Consequently:

0 = (c ρ k + Z ρ k (ξ k )) -(c ρ k + Z ρ k (ξ k )),
and (ii) holds for i ≤ k. We conclude that we can choose the c ρ 's inductively so that it holds for all i ≤ l, and this choice is unique by construction.

Proposition 4.2.6. There exists a choice of real numbers (c ρ ) ρ∈ R such that for all link ρ γ x γ ρ with ρ, ρ ∈ R, one has Zρ (γ) = Zρ (γ ),

where Zρ = Z ρ + c ρ .

Remark 4.2.7. This choice is unique up to choosing a value c ρ k for a selection (ρ k ) k of representatives of the connected components of R.

Proof. Consider a point ρ ∈ R. There is a branch path joining ρ to some root ρ ∈ R:

ρ = x 1 γ 1 x 2 γ 2 . . . x n+1 = ρ,
where we recall by denition that the γ i are branches. We would like to set

c ρ = c ρ + n i=1 Z x i (γ i ),
but rst we need to check that this is independent of the chosen branch path. Take

another one ρ = x 1 γ 1 x 2 γ 2 . . . x n +1 = ρ, which leads to a loop ρ = x 1 γ 1 x 2 γ 2 . . . x n+1 = ρ = x n +1 γ n . . . x 2 γ 1 x 1 = ρ.
By Corollary 4.2.2 one has

n i=1 Z x i (γ i ) = n i=1 Z x i (γ i ),
which is what we wanted. Now let us show that with this choice of (c ρ ) ρ∈ R, the desired equality holds. Take ρ, ρ ∈ R and two one-sided η-good curves γ, γ with extremities ρ, x and ρ , x respectively. We know that there are roots ρ, ρ ∈ R and branch paths such that: where for all i ≤ ñ, ρi ρi+1 is some link ρ j i ξ j i x j ξ j i ρ j i of the enumeration given by Proposition 4.2.5. Here again, we build a loop by concatenating the curves γ, γ and three last chains together, which we write in short:

ρ = x 1 γ 1 x 2 γ 2 . . . x n+1 = ρ and ρ = x 1 γ 1 x 2 γ 2 . . . x n +1 = ρ .
ρ γ i ρ γ x γ ρ γ i ρ ξ j i ξ j i ρ .
Proof. The result holds by denition for one-sided η-good curves joining a point ρ ∈ R too any other point x. By subtraction it is clear that it therefore holds for any restriction of such curves. Noticing a telescopic sum, this shows that it holds also for curves which can be decomposed as a concatenation of such restrictions, thus by Lemma 4.1.13, it holds for any one-sided η-good curve, no matter the stem. Since a two-sided η-good curve is the concatenation of two one-sided η-good curves, we get the result for any η-good curve.

As a corollary, we get the same property for η-good paths. Recall that an oriented path p induces a curve γ p obtained by concatenating the curves of p.

Z η (γ p ) . = ˆγp θ η (x) α-1 dx = z(γ + p ) -z(γ - p ).

Semicontinuity

We dene

Bas + η ( R) = ρ∈ R Bas + η (ρ) and Bas - η ( R) = ρ∈ R Bas - η (ρ).
We are going to prove that the landscape function z dened in the previous section is Lipschitz continuous on the root network R, lower semicontinuous on the positive basin Bas + η ( R) and upper semicontinuous in the negative basin Bas - η ( R). Proof. Take t n → t and γ n → γ. Fix ε > 0. Since θ η stays unchanged after reparameterization of η, we assume that for almost all curves γ, T -(γ) = 0 and γ is parameterized by arc-length. We know that for n large enough t n ≥ t -ε, so that lim sup

n θ η (γ : γ n |[0,tn] ⊆ γ) ≤ lim sup n η(γ : γ n |[0,t-ε] ⊆ γ).
We shall prove that lim sup

n η(γ : γ n |[0,t-ε] ⊆ γ) ≤ η(γ : γ |[0,t-ε] ⊆ γ).
Let us set

F k = {γ : ∃t 0 , ∀s ∈ [0, t -ε], |γ(t 0 + s) -γ(s)| ≤ 1/k}.
γ n |[0,t-ε] ⊆ γ) ≤ η(F k ).
Using the monotone convergence theorem and (4.3.1),

lim sup n η(γ : γ n |[0,t-ε] ⊆ γ) ≤ lim k→∞ η(F k ) = η(F ) = η(γ : γ |[0,t-ε] ⊆ γ), thus lim sup n η(γ : γ n |[0,tn] ⊆ γ) ≤ η(γ : γ |[0,t-ε] ⊆ γ).
Passing to the limit ε → 0 we get by the monotone convergence theorem :

lim sup n η(γ : γ n |[0,tn] ⊆ γ) ≤ η(γ : γ |[0,t] ⊆ γ) or equivalently lim sup n θ η (γ n |[0,tn] ) ≤ θ η (γ |[0,t] ),
which is what we wanted to prove.

ˆT+ (γ) 0 θ η (γ |[0,t] ) α-1 dt.
The map (t, γ) → θ η (γ |[0,t] ) α-1 being lower semicontinuous, as well as γ → T + (γ), this readily implies that f is lower semicontinuous.

Before proving the semicontinuity of the landscape function, we rst need a continuity result on the root network. Lemma 4.3.5. The landscape function z is Lipschitz continuous on the root network R.

Proof. We know that R is made of a nite union B of branches (those belonging to a simple path joining two roots). We set M > 0 the minimal multiplicity of those branches. Take two points x, y ∈ R. We know that there is a path

x = p 1 γ 1 p 2 γ 2 . . . p n+1 = y
where γ i is a restriction of a branch belonging to B for all i. We remove a mass M to all curves such that p i γ i p i+1 , which is possible because all branch have a multiplicity larger than M , and add a mass M to those for which p i γ i p i+1 , and sends a mass M for x to y along a straight line, with cost no larger than CM α |y -x|. Since the cost increment must be nonnegative, one has

-M i Z p i (γ i ) + CM α |y -x| ≥ 0.
Recalling that z(y) -z(x) = i Z p i (γ i ), we obtain: Proof. We assume that η is parameterized by arc length. Take a sequence x n → x ∈ Bas + η ( R). We want to prove that z(x) ≤ lim inf n z(x n ).

z(y) -z(x) ≤ CM α-1 |y -x|.
We assume that lim inf n z(x n ) < ∞, otherwise there is nothing to prove. Also, up to subsequence, we may assume that z(x n ) → = lim inf n z(x n ) and z(x n ) ≤ M < ∞. We know that for all n, x n is reached by some right-sided η-good curve γ n stemming from a point ρ n ∈ R. By Lemma 4.1.15 and by restricting the curve we may assume that ρ n = ρ + (γ n ). Since R is compact, up to subsequence we may assume that ρ n → ρ ∈ R.

Moreover up to reparameterization we may assume that T -(γ n ) = 0 and that γ n is parameterized by arc length. Consequently by the fundamental property of Proposition

4.3.1, one has z(x n ) -z(ρ n ) = Z η (γ n ).
Since z is continuous on R and Z η is lower semicontinuous one has

Z η (γ) ≤ lim inf Z η (γ n ) = lim inf n z(x n ) -z(ρ).
It is clear that γ -= ρ, γ + = x. Now we want to check that γ is an η-good curve. Since

θ η (y) = θ η (γ n [ρ n , y]) for y ∈ γ n ]ρ n , y[ by Lemma 4.1.15, one has Z η (γ n ) = ˆγn θ η (y) α-1 dy = ˆT+ (γn) 0 θ η (γ n |[0,t] ) α-1 dt. As z(ρ n ) → z(ρ) and z(x n ) ≤ M we have Z η (γ n ) ≤ C for some constant C > 0.
By the previous proposition, the right-hand side is lower semicontinuous with respect to

γ n thus ˆT+ (γ) 0 θ η (γ |[0,t] ) α-1 dt ≤ lim inf n Z η (γ n ) ≤ C.
Since η is concentrated on simple curves, γ is injective on [0, T + (γ)[, and if it is not injective on the whole [0, T + (γ)], we restrict γ to the rst time it reaches γ + , obtaining a curve γ such that ˆγ θ η (γ [ρ, y]) α-1 dy ≤

ˆT+ (γ) 0 θ η (γ |[0,t] ) α-1 dt ≤ C < ∞
and γ is η-good. Therefore by Proposition 4.3.1 we obtain:

z(x) -z(ρ) = Z η (γ) ≤ lim inf n z(x n ) -z(ρ)
and z is lower semicontinuous on Bas + η ( R). We can do the same reasoning to get the upper semicontinuity of z on Bas - η ( R), noticing that if γ is a left-sided η-good curve reaching x from a point ρ one has a minus sign:

z(x) -z(ρ) = -Z η (γ).

Formula for the optimal cost

Proposition 4.3.8. If η ∈ IP(µ, ν) is an optimal irrigation plan with nite I α cost and z is an associated landscape function, then z ∈ L 1 (µ) ∩ L 1 (ν) and

I α (η) = ˆRd z d(ν -µ).
Proof. We know by Proposition 4.1.10 that η-almost every curve γ is an η-good curve, hence by the fundamental property stated in Proposition 4.3.1, one has:

Z η (γ) = z(γ + ) -z(γ -).
Moreover for all ρ ∈ R:

ˆΓ(ρ) (z(γ + ) -z(ρ)) dη(γ) = ˆΓ(ρ) Z η (γ[ρ, +]) dη(γ) ≤ ˆΓ(ρ) Z η (γ) dη(γ) = I α (η) < ∞.
Since R is a nite root system, this implies that z • π ∞ is η-summable thus z ∈ L 1 (ν).

The same reasoning gives that z ∈ L 1 (µ). Finally, again by Proposition 4.1.10, we have:

I α (η) = ˆΓ Z η (γ) dη(γ) = ˆΓ z(γ + ) -z(γ -) dη(γ) = ˆRd z dν - ˆRd z dµ.

First variation formula

In the single-source case, z is the rst variation of the functional X α = d α (δ 0 , •). In the multiple source case, we want to show that it is in a way the rst variation of d α , when we do perturbations of µ or ν. However, we must restrict these perturbations to happen only in a given basin Bas η (σ). Indeed, to estimate the cost increment, we need to perturb the underlying irrigation plan η ∈ OIP α (µ, ν), but in the single-source case all curves passed by a same point s = 0, hence to perturb ν one could redirect the curves going to a set A to another set B using the common junction at s, without having to modify µ in consequence. In the multiple source case, this fails in general: changing the measure ν may force us to change the measure µ as well. If we impose that the changes are made inside a basin Bas η (σ), then we can use the common junction σ to do whatever we want with µ and ν independently.

Theorem 4.3.9 (First variation formula). Let η ∈ IP(µ, ν) be an optimal irrigation plan with nite I α cost and z an associated landscape function. If µ , ν ∈ M 1 + (R d ) are such that µ -µ and ν -ν are concentrated in an irrigation basin Bas η (σ) then:

d α (µ , ν ) ≤ d α (µ, ν) + α ˆRd z d((ν -ν)) -(µ -µ)), provided z ∈ L 1 (ν ) ∩ L 1 (µ ).
Proof. We are going to use the rst variation formula of I α stated in 1.2.1: η-good curve γ x , γ y respectively, which stem from σ. We can modify the irrigation plan η by removing, adding and redirecting mass which ows through curves visiting σ, using gluings as in the proofs of Proposition 1.2.16 and Theorem 1.2.18, to obtain an irrigation plan η ∈ IP(µ , ν ) which is concentrated on η-good paths. As a consequence, by the fundamental property, one has Z η

I α (η) ≤ I α (η) + α ˆΓ Z η d(η -η).
(γ) = z(γ + ) -z(γ -) for η -almost every γ. Since z ∈ L 1 (µ ) ∩ L 1 (ν ), it yields: ˆΓ Z η (γ) dη (γ) = ˆΓ(z(γ + ) -z(γ -)) dη (γ) = ˆRd z dν - ˆRd z dµ .
Finally, plugging this in (4.3.2), knowing that I α (η) = d α (µ, ν) and that d α (µ , ν ) ≤ I α (η ), one gets

d α (µ , ν ) ≤ I α (η ) ≤ I α (η) + α ˆΓ Z η d(η -η) = d α (µ, ν) + α ˆRd z dν - ˆRd z dµ -α ˆRd z dν - ˆRd z dµ .
This theorem will be the essential tool in proving the Hölder regularity of the landscape function with some extra hypotheses, although other ingredients will be needed. Indeed, as we will do general perturbations which do not occur in a single basin, we will need to perform small transfers between basins with controlled cost.

Hölder regularity

In this section, we want to prove the regularity of the landscape function we have dened under some extra hypotheses on the measures µ and ν. Our assumption will be that µ and ν are probability measures of the form:

µ = f L |U and ν = g L |V ,
where f and g are functions bounded from below by a constant c 0 > 0, and U, V are open connected sets which are Ahlfors regular in the sense that for some r 0 > 0 and all r ≤ r 0 :

∀x ∈ U, Θ U (x, r) ≥ c 0 , and ∀y ∈ V, Θ V (y, r) ≥ c 0 .

Moreover, for this whole section we assume that η ∈ OIP α (µ, ν) is an optimal irrigation plan with α ∈]1 -1/d, 1[ which has a nite root system R. We denote by z a landscape function as constructed in Section 4.2.

Before going further, we will need some new denitions. Recall that the root network R is the union of all simple branch paths joining pairs of roots ρ, ρ ∈ R. For each connected component P of R, we pick once and for all a root ρ P ∈ P ∩R and we denote by P the nite collection of these connected components. Each connected component P ∈ P is associated with a basin Bas(P ) and its signed counterparts Bas(P ) ± as follows:

Bas + (P ) = These basins, which we call rooted basins, provide a decomposition of the measures µ, ν:

(i) Bas + η ( R) = P ∈P Bas + (P ) and Bas - η ( R) = P ∈P Bas -(P ), (ii
) for all P ∈ P, µ(Bas -(P )) > 0 and ν(Bas + (P )) > 0,

(iii) µ and ν are respectively concentrated on Bas - η ( R) and Bas + η ( R). for r = r 0 , where Q r (x) is the cube centered at x with radius r. Since U is open and connected, there is a curve γ : [0, 1] → U which joins x to y. If the radius r 0 is small enough, the cubes Q r 0 (γ(t)) stay inside U as well, and the map

f (t) = Θ A (γ(t), r 0 )
is continuous on [0, 1], with f (0) > 1/2 and f (1) < 1/2. Consequently there is a time t 0 and x 0 = γ(t 0 ) such that

Θ A (x 0 , r 0 ) = Θ B (x 0 , r 0 ) = 1/2.
Now we take r = r 0 /2 and we want to nd a cube Q r (x) ⊆ Q r 0 (x 0 ) such that the same equality holds: Θ A (x, r) = Θ B (x, r) = 1/2. Cut the cube Q r 0 (x 0 ) into 2 d subcubes Q i of radius r and centers x i . If the equality holds for one x i we set x = x i . Otherwise, one could nd two cubes Q i , Q j with i = j such that Θ A (x i , r) > 1/2 and Θ A (x j , r) < 1/2. Now we can reproduce the starting argument to nd an x lying in the segment [x i , x j ] such that Θ A (x, r) = 1/2. We set x 1 = x, r 1 = r, and we keep going to produce a sequence of cubes Q n = Q rn (x n ) such that each is included in the previous one, with r n = r 0 2 -n and such that

Θ A (x n , r n ) = Θ B (x n , r n ) = 1/2.
The sequence x n is Cauchy and converges to a point x which is the intersection of all these cubes. We claim that there is a constant c > 0 such that Θ A (x, r) ≥ c and Θ B (x, r) ≥ c for all r ≤ R (recall that this density is computed with respect to balls and not cubes). Take n such that r n+1 < r/2

√ d ≤ r n . One knows that x ∈ Q n+1 hence B r (x) ⊇ Q n+1 , thus: Θ A (x, r) ≥ |A ∩ Q n+1 | |B r (x)| = 1/2 |Q n+1 | |B r (x)| ≥ 1/2 2 d ω d r n /2 r d ≥ c > 0.
The same holds for B and we are done.

We set X -(resp. X + ) to be the collection of sets A made of a union of negative basins (resp. positive basins) associated to components P ∈ P, that is sets of the form A = P ∈P Bas -(P ) (resp. A = C∈P Bas -(P )) where P ⊆ P. In the rest of this section, all results will be stated and proved for negative basins but their positive counterparts hold as well ; hence for this section we write X = X -, X = Bas - η ( R) and X P = Bas -(P ) for P ∈ P.

while on the other hand by the rst variation formula of Theorem 4.3.9 one has: By upper semicontinuity of z on Bas - η ( R) one has the opposite inequality: Step 2:

d α (μ, ν) ≤ d α (µ, ν) + α ˆRd -z d(μ -µ) = d α (µ, ν) -αc 0 m r z(x) + αc 0 ˆBr(x)∩A z (
lim sup r→0 z r,A (x) ≤ z(x), hence z(x) = z A (x) . = lim r→0 z r,A (x).
ffl Br(x)∩A |z -z(x)| ≤ Cr β . We cut B r (x) ∩ A into two parts of equal measure: B r (x) ∩ A = A 1 A 2 where |A 1 | = |A 2 | = m r /2.
Removing c 0 1 A 1 and adding c 0 1 A 2 to µ, one gets by the same kind of estimates we used in the rst step: The rest is just standard Campanato estimates, which we do here for the sake of completeness. Let us compare the quantities z r,A (x) and z r/2,A (x):

Crm α r + c 0 α ˆA1 z -c 0 α ˆA2 z ≥ 0.
|z r/2,A (x) -z r,A (x)| = B r/2 (x)∩A (z -z r,A (x)) ≤ B r/2 (x)∩A |z -z r,A (x)| ≤ |B r (x) ∩ A| |B r/2 (x) ∩ A| Br(x)∩A |z -z r,A (x)
(r) = |z(x) -z r,A (x)| for r ∈]0, R] that: f (r) ≤ f (r/2) + Cr β .
Consequently for all k ∈ N,

f (r) ≤ f (r2 -(k+1) ) + Cr β k i=0 2 -iβ
thus by taking the limit k → ∞:

f (r) ≤ lim sup ε→0 f (ε) + Cr β ∞ i=0 2 -iβ ≤ lim sup ε→0 f (ε) + Cr β .
Now we know by (4.4.5) that f (ε) → 0 when ε → 0, which implies that f (r) ≤ Cr β , that is to say:

|z(x) -z r,A (x)| ≤ Cr β .

(i) for all P , m i,P = m i-1,P -(ξ i P (X P ) -χ i P (X P )),

(ii) m i := P |m i,P | ≤ (1 + K)m i-1 , (iii) m i,P j = 0 for all j ≤ i.

Notice that this is justied provided |m i | is always less that ε, which is guaranteed if m(1 + K) 1+#P ≤ ε. Then we dene a basin variation V = (µ , (χ P ), (ξ P )) by

χ P = 1/I • i χ i P , ξ P = 1/I • i ξ i P and µ = µ + P (ξ P -χ P ).
One has for all P ∈ P:

0 = m I,P = m I-1,P -(ξ I P (X P ) -χ I P (X P )) = . . . = m P - i≤I (ξ i P (X P ) -χ i P (X P )) hence (ξ P -χ P )(X P ) = m P . Moreover one has ˆz d(µ -µ) ≤ K(1 + K) I(1+β/d) m 1+β/d and d α (µ, µ ) ≤ K(1 + K) I(1+β/d) m 1+β/d .
Setting K = K(1 + K) #P(1+β/d) and ε = ε/(1 + K) #P , one gets the desired result.

A general rst variation inequality

Proposition 4.4.11. Let ν ∈ Prob(R d ) be a probability measure such that |ν -ν| is concentrated on Bas + η ( R) and has mass m ≤ ε. Then there is a probability measure µ and an irrigation plan η ∈ IP(µ , ν ) such that (i) η is concentrated on η-good paths,

(ii) ´z d(µ -µ) ≤ Cm 1+β/d , (iii) d α (µ, µ ) ≤ Cm 1+β/d .
Proof. Notice that one may write:

ν = ν + ξ - χ
where ξ, χ are nonnegative measures concentrated on Bas + η ( R) which are mutually singular and have same mass, with χ ≤ ν. Setting for all P ∈ P: m P = ξ(Bas + (P )) -χ(Bas + (P )), we know that P m P = 0. By the previous proposition, one can nd a basin variation (µ , (ξ P ), (χ P )) such that

A shape evolution problem

We want here to dene a dynamical counterpart to the shape optimization problem studied in Chapter 3, so as to obtain a shape evolving continuously towards the optimal shape. We try to give a sketch of what one would like to accomplish regarding this problem. The calculations will be quite formal and most of the claims will not be proven but are left open for now, as their study is currently ongoing. We introduce the shape evolution problem via an implicit scheme of what should be its time discretization, in the spirit of gradient ows discretization (see [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF]Chapter 8] or [START_REF] Santambrogio | Flots de gradient dans les espaces métriques et leurs applications (d'après Ambrosio-Gigli-Savaré)[END_REF][START_REF] Santambrogio | {Euclidean, metric, and Wasserstein} gradient ows: an overview[END_REF] for an overview of gradient ows, [START_REF] Ambrosio | Gradient ows in metric spaces and in the space of probability measures[END_REF] for the whole theory), or De Giorgi's minimizing movements [De 93; Amb95]. Actually, it may be seen as a geometric minimizing movement (see for example [START_REF] Braides | Local minimization, variational evolution and Γ-convergence[END_REF]Chapters 6,8] or [START_REF] Almgren | Curvature-driven ows: a variational approach[END_REF] on this topic).

Given λ > 0, we dene the functional F :

M + (R d ) → R ∪ {+∞} by F α,λ (µ) := X α (µ) -λ µ ,
and identifying Lebesgue measurable sets E with 1 E L , we also set

F α,λ (E) := X α (1 E L ) -λ|E|.
The term -λ|E| is a volume term which we take for simplicity as replacement for the constraint |E| = 1 which was imposed in Chapter 3.

Let T > 0 be a xed duration and τ > 0 be a small time step. Given a set E 0 , we look for a sequence of sets (E τ k ) k dened by

E τ 0 = E 0 ,
(5.1.1) and

E τ k+1 ∈ argmin E F (E) + ˆE∆E k d(x, ∂E k ) τ dx, (5.1.2)
for k ≤ T /τ . The integral term behaves like squared distance (divided by τ ), as it is more or less the integral of a distance on a small slice ; moreover if one considers a translation of a cube by a distance h this integral term indeed gives h 2 . Thus it is quite similar to minimizing movements, and very close to the variational approach for the motion by mean curvature (see [START_REF] Almgren | Curvature-driven ows: a variational approach[END_REF]). We would like to do the following:

(i) (Well-denedness) Prove that one can dene a sequence (E τ k ) k by (5.1.1)-(5.1.2), i.e. there exists minimizers at each step.

(ii) (Continuous motion) Dene a suitable set-valued map Ēτ (t) for t ∈ [0, T ] such that Ē(kτ ) = E τ k , for example Ēτ (t) = E τ t/τ and prove that it converges Ēτ τ →0 --→ Ē in some sense to some Ē.

(iii) (Evolution equation) Prove that the motion of Ē is described by the evolution equation:

v t (x) = λ -αz t (x) for x ∈ ∂ Ē(t),
(5.1.3) where v t (x) is the normal speed of Ē(t) at x and z t is a landscape function associated to E(t), which represents the marginal cost in irrigation. Recall that z is the rst variation X α , so that the term on the right is actually the opposite of δF α,λ , the rst variation of F α,λ . Of course, all these notions and equation

(5.1.3) should be given a precise meaning rst! Before entering the study, let us rewrite (5.1.2). Notice that

ˆE∆E k d(x, ∂E k ) dx = ˆE\E k d(x, E k ) dx + ˆEk \E d(x, E c k ) dx = ˆds (x, E k )(1 E\E k -1 E k \E ) dx = ˆE d s (x, E k ) - ˆEk d s (x, E k ) dx,
where d s (x, E) is the signed distance to E. Since the second term is a constant, (5.1.2) is equivalent to

E τ k+1 ∈ argmin E F α,λ (E) + ˆE d s (x, E k ) τ dx.
(5.1.4)

Now let us try to discuss the issues in dealing with each step and propose sketches for some proofs.

Well-denedness. We are going to take minimizers of (5.1.4) in the larger class of

positive measures ν such that ν ≤ 1. It is clear that ν → F α,λ (ν) + ´ds(x,E τ k ) τ
dν is lower semicontinuous since X α is lower semicontinuous and the remaining part

-λ|ν| + ˆds (x, E τ k ) τ dν(x) = ˆφ dν, where φ(x) = d s (x, E τ k ) τ -λ, is continuous because φ ∈ C b (R d ).
Assuming that E τ k compact, one may prove sucient properties (lower bounded, tightness of minimizing sequences...) to guarantee that there is a minimizer (5.1.6) By optimality of ν τ k+1 , the integral on the right must be nonnegative for all competitor ν, which means that ˆRd (φ(x) -λ) dν τ k+1 (x) = min ν≤1 ˆRd (φ(x) -λ) dν(x).

(5.1.7)

where we have set φ(x) = αz τ k (x) + ds(x,E τ k ) τ . Consequently, ν τ k+1 should be equal to 1 on {φ < λ} and to 0 on {φ > λ}. But using the strict concavity of the map x → x α it is possible to show that inequality (5.1.5), hence inequality (5.1.6) as well, is actually strict if ν = ν τ k+1 . As a consequence (5.1.7) admits a unique minimizer, and |{φ = λ}| = 0, which implies that ν τ k+1 = 1 φ≤λ L .

Therefore the set E τ k+1 := {αz τ k (x)+ ds(x,E τ k ) τ ≤ λ} is a solution of (5.1.4), or equivalently of (5.1.2). Notice φ being lower semicontinuous and coercive, E τ k+1 is a compact set (recall that we have assumed E τ k to be compact). We have proven if E 0 is compact, we may dene a sequence (E τ k ) k as wanted, and it is made of compact sets.

Continuous motion. Now we would like to pass to the limit τ → 0 in the sequence (E τ k ) k , but also in the sequence (z τ k ) k , as this function will play a role in describing the motion, to obtain a continuum of sets Ē(t) and functions z(t, •) for t ∈ [0, T ]. To do that, we should place ourselves in some well-suited functional spaces where we can prove enough compactness to pass to the limit (at least up to subsequence).

Let us investigate the sequence (E τ k ) k rst. We set K the set of compact subsets of R d and dene for all A, B ∈ K : (5.1.8)

Thus by summation over k, one gets

T /τ -1 k=0 τ D(E τ k+1 , E τ k ) τ 2 ≤ F α,λ (E 0 ) -F α,λ (E τ T /τ ) ≤ C,
(5.1.9) for some constant C > 0. It seems that the quantity D(A, B) behaves like a squared distance between A and B (as said before, considering a translation of a cube by a distance h gives h 2 ), thus (5.1.9) should be some kind of H 1 bound and give some compactness.

Let us investigate the sequence (z τ k ) k . The situation is perhaps clearer here: the regularity one should expect is Hölder-continuity of exponent β = 1 + dα -d. Indeed landscape functions associated to absolutely continuous measures µ are β-Hölder continuous as proven in [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF], with some extra regularity assumptions on µ. Moreover in Chapter 3 we prove the Hölder regularity (with same exponent) of the landscape function associated to optimal shapes. Thus what we are expecting here is that z τ k ∈ C 0,β (E τ k ), or perhaps z τ k ∈ C 0,β (X τ k ) for a large enough subset X τ k ⊆ E τ k . What we would like is an upper bound on the Hölder constant to get compactness by Ascoli's theorem. But this will not be enough for the z τ k to converge, up to subsequence, to a curve z(t) ∈ C 0,β (E(t)) for t ∈ [0, T ], as we need some regularity in t as well, that is we need a control of |z τ k+1 -z τ k | of some sort. This is something which may not be very easy for the following reason. We know that z τ k is a landscape function for ν τ k , that is a rst variation of X α at ν τ k , and usually the regularity in time of the rst variation is given by the fact that ν k+1 is close to ν k , and that the rst variation is unique and is somewhat regular as a function of ν (often it has an explicit in terms of ν). Here, we do not have uniqueness of the rst variation in general, hence the one chosen at step k + 1 may be very dierent from the one chosen at the previous step k. However since X α exhibits some kind of concave behaviour, we can hope that it is often unique (a concave function on R d being dierential everywhere but in small set) and continuous.

If uniqueness fails anyway in our case, a careful choice should be made to guarantee some continuity of the landscape functions between two consecutive time steps.

Evolution equation. As E τ k = {x : αz k (x) + d s (x, E k-1 )/τ ≤ λ}, it is reasonable to claim that ∂E τ k ⊆ {x : αz k (x) + d s (x, E k-1 )/τ = λ} as in the static shape optimization problem. Take a point x ∈ ∂E τ k , then one has:

d s (x, E τ k-1 ) τ = λ -αz k (x).
The quotient d s (x, E τ k-1 )/τ can be interpreted as the speed of the boundary at point x, thus the limit equation should be: v t (x) = λ -αz t (x) for x ∈ ∂E(t),

(5.1.10) where v t (x) is the normal speed of ∂E(t) at x. But what sense should we give to (5.1.10) ? This does not seem evident, as the sets E(t) have no reason to be regular (even BV , since we suspect they may have fractal boundary), and as the regularity in time that one should be expect is unclear. A possible weak formulation of equation (5.1.10) could be the following: a curve of compact sets E(t) moves with normal speed v if for all t > 0 and all x ∈ R d \ E(t), the following holds: d dt d(x, E(t)) = v(π E(t) (x)).

(5.1.11) Diculties arise because π E(t) may not be well-dened, even if x is taken close to the boundary ∂E(t), and the strategy to prove this from the discrete scheme has yet to be elaborated. However let us give a reasoning which goes in that direction for a particular geometric situation. Take a point x ∈ R d \ (E τ k ∪ E τ k+1 ), denote by x k = π E τ k (x), x k+1 = π E τ k+1 (x) (assuming they are well dened). Suppose that set has increased, in the sense that the segment [x, x k+1 ] does not intersect E τ k and that [x, x k ] intersects E τ k+1 (see Figure 5.1), and denote xk+1 the intersection point closest to x. (5.1.13) When passing to the limit τ → 0, equations (5.1.12)-(5.1.13) may lead to the considered relation (5.1.11).

Towards proving fractality of optimal shapes

In this section we try to provide some ideas and formal computations in the direction of proving that the optimal shapes of Chapter 3 have a fractal boundary of non-integer dimension d -β where β = 1 + dα -d. Again, we stress that the computations or proofs given here are quite sketchy and sometimes not proved in full detail as this is still ongoing work.

Let ν = 1 A be an optimizer of the shape optimization problem (R α ) and η ∈ OIP α (δ 0 , ν). We set Z = Z η and z = z η the landscape function associated to η. Then for any η-good curve γ such that Z(γ) ≥ t we dene π z≥t (γ) = γ z≥t := γ |[z -1 (t),+∞[ , and we consider the tail of η at level t, dened by η t = (π z≥t ) # (η {Z > t}).

Let us compute the initial and nal measures µ t := (π 0 ) # η t and ν t = (π ∞ ) # η t : Notice that µ t is atomic because if Z(γ) > t then θ(γ(z -1 (t))) > 0 hence there can only be countably many such γ(z -1 (t)), since they lie on dierent branches. Moreover the total mass of ν t is ν t = ν(x : z(x) > t), but in our shape optimization problem ν = 1 z≤z , hence ν t = Vol(x : z ≥ z(x) > t).

We know by Lemma 3.4.2 that we have the upper bound:

Vol(x : z(x) > t) ≤ C(z -t).

(5.2.1)

A conjecture on a lower bound of the volume In order to prove the lower bound on the Hausdor dimension of the boundary ∂E that we seek, that is

dim H (∂E) ≥ d -β,
which is stated in Conjecture 3.4.4, a key ingredient would be the reverse inequality of (5.2.1), that is a lower bound on the volume of {z ≥ z > t} as conjectured below.

Conjecture 5.2.1. There exists some c > 0 such that Vol(x : z ≥ z(x) > t) ≥ c(z -t).

We do not have a strategy to prove this (if true), but let us just relate this volume |x : z ≥ z(x) > t| to other expressions or quantities which might be used to estimate it from below. Let us compute the cost of η t , observing that if ∞ > z(x) > t then This is a rst simple expression for the cost of the tail of η cut for z > t. We may get another expression using Fubini-Tonelli's theorem:

I α (η t ) = ˆZ>t ˆγ 1 z(x)>t θ ηt (x) α-1 d H 1 (x) dη(γ) = ˆz(x)>t θ ηt (x) α d H 1 (x) = ˆz(x)>t θ η (x) α d H 1 (x).
We may relate the tail energy I α (η t ) to the volume Vol(t < z ≤ z ).

I α (η t ) ≥ c(z -t) 2 .
Lower dimension estimate from the lower estimate on the volume We want to show a way to prove dim H {z = z } ≥ d -β from Conjecture 5.2.1, that is from Vol(z > t) ≥ c(z -t), which we assume to be true in this paragraph. Notice that this would still not prove Conjecture 3.4.4 since we only know that ∂A ⊆ {z = z }.

The strategy is the following. The measure µ t is an atomic measure, which we renormalize as an atomic probability measure σ t = µ t / µ t . We write it as:

σ t = i∈N a i t δ x i t .
As t → z it converges weakly (up to subsequence) to a probability measure σ. We are going to prove that σ has a.e. a local dimension which is bounded from below by Lemma 5.2.7. There exists some ε 0 small such that for all ε < ε 0 and all x ∈ {x : z(x) = z }, the set B ε (x) only intersects a number of basins Bas(y) from points y such that z(y) = z -cε 1/β which is bounded from above by a constant N = N (α, d).

Proof. It comes from the fact that all these basins are disjoint and occupy a volume ≈ ε d , and they must all lie in some ball CB ε (x), consequently there number is bounded by such a constant.

Corollary 5.2.8. The level set of the landscape function L := {x : z(x) = z } is of dimension at least d -β, in the sense that:

dim H {x : z(x) = z } ≥ d -β.
Proof of Theorem 5.2.6. Fix ε > 0 and x such that z(x) = z . We consider the ball B ε (x) and we x a level t < z . For all y ∈ B ε (x) ∩ L, there is an ancestor y t , that is y t ≤ y, such that z(y t ) = t. We know that along a curve reaching

Further perspectives

The aim of this nal section is to formulate some remaining open questions, among many others of course, related to the work presented before and to propose some research perspectives in a few words.

The landscape function in the general case In Chapter 4 we have given a denition of landscape function when the optimal irrigation plan has a nite root system, and we have shown semicontinuity on the positive and negative basins, as well as Hölder regularity results with extra hypotheses on the source and target measures µ, ν. In general, there is only a countable (instead of nite) root system, hence there is still much work to be done to handle the general case. Many reasonings in Chapter 4 have been made by nite induction using the fact that there was a nite root system, thus some reasonings might be adapted to the general case by countable induction, perhaps to give a proper denition of the landscape function, although some more involved results like Hölder regularity or even semicontinuity should require more ideas (if even true).

A dierent approach to dene a landscape function may come from the following remark. Notice that the map H(x) = x → x α /α is continuous and concave on R + , hence it may be expressed as a Legendre transform: We have merely proceeded by an inf-inf exchange, and we stress that the inmum for w is among any nonnegative Borel maps. Now notice that the last term rewrites by Fubini-Tonelli's theorem: This question was one of the initial questions posed at the beginning of this thesis, although the apparently simpler question regarding the fractality of the optimal shapes considered in Chapter 3 could not be given a satisfying answer yet, hence much work and new ideas are needed in order to answer this question. What can only be proven for now is the fact that B ∩ B is Lebesgue-negligible.

H(x) = inf
Numerical renements for the shape optimization problem Our numerical approach described in Chapter 3 to compute a candidate solution to our shape optimization problem does not allow us to rene the discretization grid and do the computations in a reasonable time. We would like to have a more precise picture of what an optimizer should look like, and to be able to compute a box dimension of the boundary so as to support Conjecture 3.4.4 stating that dim H (∂E) = d -β. In order to do that, one should use more sophisticated tools to rene the grid, perhaps by multi grid methods (to rene the grid near the boundary) or moving grid techniques. This is in project with E. Oudet. Also, one should at some point prove rigorously the Γ-convergence of our approximate elliptic functional towards the branched transport functional with inequality constraint on the divergence.

Variational approximation of the H-mass We would like to consider branched transport problems with more general costs than ´θη (x) α d H 1 (x), namely to replace the map x → x α with a function H : R + → R + leading to a so-called H-mass:

M H (v) =    ˆRd H(|v(x)|) d H 1 (x) if v is rectiable,

+∞

otherwise.

(5.3.8)

It seems the idea of considering such costs comes from the notion of at chains with coecients in normed groups, as considered by White in [START_REF] White | Rectiability of at chains[END_REF], the H-mass being introduced for the rst time, as far as we know, in [START_REF] Pauw | Size minimization and approximating problems[END_REF]. The standard hypotheses for H are the following:

(i) H(0) = 0,

(

  M), les solutions du problème de Monge-Kantorovich (MK) n'étant pas nécessairement concentrées sur le graphe d'une fonction. Une stratégie pour résoudre le problème originel de Monge a été proposée en 1979 par Sudakov [Sud79], puis mise en oeuvre avec succès 20 ans plus tard par Ambrosio dans [Amb03]; d'autres preuves ont été établies dans [EG99; TW01; CFM02]. Par ailleurs, de nombreuses variantes au problème de Monge-Kantorovich ont été étudiées, en particulier des généralisations de la forme : min π ˆRd ×R d c(x, y) dπ(x, y) : π ∈ Π(µ, ν) , (TO) où c : R d × R d → R + est une fonction continue. Une attention particulière a été portée aux cas c p (x, y) = |y -x| p , p un exposant convexe (p ≥ 1) ou concave (p ≤ 1). Un cas très naturel est le cas p = 2 (pour ses liens avec la physique), pour lequel Brenier a démontré en 1991 l'existence et l'unicité de la solution, et que celle-ci est concentrée sur un graphe. Plus précisément: Théorème de Brenier ([Bre91]). Soit K un compact de R d et µ, ν ∈ Prob(K) telles que µ L d . Alors il existe un unique plan de transport optimal pour le problème min π ˆRd ×R d |y -x| 2 dπ(x, y) : π ∈ Π(µ, ν) , et il est donné par le gradient d'une fonction convexe : π = (Id, ∇φ) # µ où φ est convexe.

[ BCS10 ;

 BCS10 CJS08]. À l'inverse, si plusieurs personnes veulent rejoindre une ville B à partir d'une ville A par la route, il leur sera moins coûteux de covoiturer pour mutualiser les frais d'essence et d'autoroute, auquel cas la concentration est favorisée : on entre le domaine du transport branché. Dans ce modèle, les particules peuvent se grouper et le coût d'un paquet de particules de masse totale m et voyageant conjointement est H(m) × où est la distance parcourue. Si H(m) = m, H est additive, i.e. H(a + b) = H(a) + H(b), et le coût des particules ensemble est égal au coût des particules considérées séparément : on retrouve en eet le problème de Monge. En revanche, si H est strictement sous-additive, i.e. H(a + b) < H(a) + H(b) si a, b > 0, alors il revient moins cher de se déplacer ensemble. Ce type de coût apparaît en réalité de manière très naturelle, dès lors qu'existe un phénomène d'économie d'échelle : transporter ou construire quelque chose de taille m coûte davantage lorsque m augmente mais proportionnellement moins, rapporté à la taille m. C'est le cas par exemple du transport routier, où un certain coût doit être payé pour construire les routes, mais il ne coûte pas beaucoup plus cher d'augmenter le trac et ou d'élargir le réseau. De manière générale, ce phénomène apparaît lorsqu'une partie du coût réside dans le coût de construction d'une certaine structure. Cette économie d'échelle se traduit par le fait que H(m)/m est décroissante en m, ce qui implique que H est sous-additive. Ces conditions sont vériées en particulier pour toute fonction H croissante, concave et telle que H(0) = 0, par exemple x → x α , α ∈ [0, 1]. CONTENTS (a) Transport de Monge. (b) Transport branché.

Figure 1 :

 1 Figure 1: Transport d'un point vers un segment.

  où |e| désigne la longeur de l'arête e. Le problème de transport branché discret consiste à trouver un minimiseur de cette énergie parmi tous les graphes envoyant µ sur ν, directe du calcul des variations, en faisant quelques réductions sur une suite minimisante, il n'est pas dicile de démontrer l'existence d'une solution ; une preuve est donnée dans le premier chapitre. Que peut-on alors dire sur les minimiseurs ? D'abord, qu'ils ne sont pas nécessairement uniques, comme le montre la Figure2, où le problème présente des symétries mais admet plusieurs solutions qui ne sont pas symétriques.

Figure 2 :

 2 Figure 2: Situation sans unicité (1/2 < α < 1).

Figure 3 :

 3 Figure 3: Graphes optimaux (α = 1/2).

  est pas sans ressemblance avec la α-masse. L'introduction de la première vient du fait qu'aucune contrainte de rectiabilité n'y apparaît a priori et qu'il est ainsi assez aisé de démontrer sa semicontinuité inférieure. Une preuve simple des propriétés de semicontinuité, compacité, coercivité menant à l'existence d'optimiseurs, et ne recourant pas à l'utilisation de paramétrisations des plans d'irrigation comme c'est souvent le cas dans la littérature existant, est donnée dans le Chapitre 1. Le problème d'irrigation lagrangien s'écrit alors min I α (η) . = ˆΓ ˆγ θ η (x) α-1 dx dη(γ) : µ

Figure 4 :

 4 Figure 4: Graphes optimaux dans le cas d'une source et deux puits de même masse.

  où σ, γ 1 , γ 2 sont bien choisis et ε → 0. Il s'agit donc de minimiser cette fonctionnelle sous contrainte inégalité sur la divergence de v pour notre problème d'optimisation de forme : 0 ≤ ∇ • δ ε -v ε ≤ 1 où δ ε est une certaine approximation de δ 0 . Dans le problème à source et destination xées, on imposait une contrainte égalité sur la divergence, dont la projection pouvait se faire très rapidement numériquement (par simple résolution d'un Laplacien). Dans notre approche, où la projection est plus compliquée, une méthode de descente de type quasi-Newton proximale a été utilisée.Des expériences numériques ont été menées, dont on donne un échantillon en Figure5.

Figure 5 :

 5 Figure 5: Mesure irriguée et réseau, α = 0.55

  discrete model . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Lagrangian model . . . . . . . . . . . . . . . . . . . . . 1.2.1 Notations and general framework . . . . . . . . . . . . . . . . 1.2.2 The Lagrangian irrigation problem . . . . . . . . . . . . . . . 1.2.3 Existence of minimizers . . . . . . . . . . . . . . . . . . . . . 1.2.4 Operations on irrigation plans . . . . . . . . . . . . . . . . . . 1.2.5 Irrigability and irrigation distance . . . . . . . . . . . . . . . 1.2.6 Basic structure properties of optimizers . . . . . . . . . . . . 1.3 The Eulerian model . . . . . . . . . . . . . . . . . . . . . . . 1.4 The single source case . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Tree property . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 The landscape function . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Optimal graphs with one source and two targets of equal mass.

  It is a way to restrict each curve γ to some interval [a, b], but each curve may be restricted in possibly innitely many ways, under a law given by χ γ . A possibility is for example to cut all curves γ between times a(γ) and b(γ) where a, b : Γ(X) → R + are Borel and a ≤ b. The resulting irrigation plan is denoted by η[a, b] and corresponds to η[χ] where χ γ = δ (a(γ),b(γ)) . If τ is a Borel map which associates to each γ ∈ Γ(X) a probability measure τ γ ∈ Prob(R + ), then the cuts denoted by η[0, τ ] and η[τ, +] associated to χ γ = δ 0 ⊗ τ γ and χ γ = τ γ ⊗ δ T (γ) respectively are called complementary cuts (respectively initial and nal cut) associated to τ .

  thus we must have |w(x)| = 0 H 1 -a.e., which means w = 0, thus v = v η and (i) holds. This implies |v η | ≤ i η ≤ |v| = |v η | and thus we have the equality |v η | = i η claimed in (ii).

Theorem 2. 4 . 1 (

 41 Equivalence theorem). If α < 1 and µ, ν ∈ Prob(K), the Eulerian problem (EI α ) and the Lagrangian problem (LI α ) are equivalent in the following sense: (i) the minima are the same min η∈IP(µ,ν) I α (η) = min v∈IF(µ,ν)

  We call irrigation plan any probability measure η ∈ Prob(Γ) satisfying the following nite-length condition L(η) := ˆΓ L(γ) dη(γ) < +∞, (3.1.1)

  ˆΓˆγ θ η (x) α-1 dx dη(γ),with the conventions 0 α-1 = ∞ if α < 1, 0 α-1 = 1 otherwise,and ∞ × 0 = 0. If µ, ν are two probability measures on R d , the irrigation (or branched transport) problem consists in minimizing the cost I α on the set of irrigation plans which send µ to ν, which reads min η∈IP(µ,ν)

  Irrigation distance Let us set d α (µ, ν) = min{I α (η) : η ∈ IP(µ, ν)} for any pair µ, ν of probability measures on R d . For any compact K ⊆ R d , it induces a distance on Prob(K) which metrizes the weak-convergence of measures in the duality with C (K). On non-compact subsets of R d , the distance d α is lower semicontinuous w.r.t. the weak-convergence of measures in the duality with bounded and continuous functions (narrow convergence) 2 . The following result comes from [BCM09; Xia03]. Proposition 3.1.4 (Scaling law). For any compactly supported measures µ, ν with equal mass, there is an upper bound on the irrigation distance depending on the mass and the diameter. We set µ = µ-µ∧ν, ν = ν -µ∧ν the disjoint parts of the measures and m = |µ | = |ν | their common mass. Then:

  Now by strict convexity of t → t α+ 1 d , if m > 0 then one has e α (1 + m)α+ 1 d > e α 1 + α + 1 d m , thus e α α + 1 d m < αz m,which contradicts (3.2.3). Consequently m = 0, hence ν = ν = 1 {z≤z } .

  e α + αmz(y) + Crm α , thus, knowing that e α α + 1 d = αz by (3.2.1): αmz ≤ αmz(y) + Crm α . By denition, m = ω d r d Θ A c (x, r) where ω d is the volume on the unit d-dimensional ball, hence

  Moreover notice that ˆz d(ν -ν) = ˆA+ z(y) dy -ˆAz(y) dy = ˆA+ (z(y) -zr (x)) dy + ˆA-(z r (x) -z(y)) dy = ˆAr(x) |z(y) -zr (x)| dy.

  More precisely: ∀x, y ∈ A, |z(y) -z(x)| ≤ C|y -x| β , for some constant C = C(α, d). Proof. By Lemma 3.3.7 and Lemma 3.3.8, |z(y) -z(x)| ≤ |z(y) -z |y-x| (y)| + |z |y-x| (y) -z |y-x| (x)| + |z(y) -z |y-x| (y)| ≤ 3C|y -x| β .

Denition 3. 4 . 1 (

 41 Minkowski dimension). We dene the upper Minkowski dimension of X by dim M (X) = lim sup ε→0 log(N ε (X)) -log ε , and the lower Minkowski dimension by dim M (X) = lim inf ε→0 log(N ε (X)) -log ε .

  Proof. Consider the competitor ν = 1 {z≤k} with total mass |ν| = 1 -m, where m = |{x ∈ A : k < z(x) ≤ z * }|. As in (3.2.2), one has e α (1 -m) α+1/d ≤ e α -αkm hence knowing that αz = (α + 1/d)e α and developing the term on the left-hand side at order 2, we obtain:

i

  . It has a mass |ν| = 1 + m where m = i∈I + |B + i |. To irrigate ν, we send an extra mass |B + i | to each center x i along the irrigation plan, which costs α|B + i |z , then we send this mass towards B + i , which costs at most C|B + i | α ε. But one should get a cost no less than e α (1 + m) α+1/d by the scaling lemma. Moreover, with a development of order 2 one has:

[ BOO16 ;

 BOO16 FCM16; BLS15].

  Figure 3.1: Algorithm output for dierent α's after ∼ 1500025000 iterations.

Figure 4 .

 4 Figure 4.1: A root network induced by a system of two roots.

Figure 4 .

 4 Figure 4.2: A right-sided η-good curve entering and leaving R.

Figure 4 .

 4 Figure 4.3: A good loop made of three η-good curves. Their stems are signaled by a large white circle.

Remark 4 .2. 4 .

 44 Obviously since all links ρ ρ are in this enumeration, all the roots ρ ∈ R appear in it.

Thus, the roots

  ρ, ρ are in a common connected component R , and one may nd a chain ρ = ρ1 ρ2 . . . ρñ = ρ ,

Corollary 4 .

 4 3.2. If p is an oriented η-good path, then

Lemma 4 .

 4 3.3. If η is an irrigation plan with nite I α cost, the map (t, γ) → θ η (γ |[0,t] ) is upper semicontinuous on R + × Γ.

Lemma 4 .3. 4 . 0 θ

 440 The map γ → ´T+ (γ) η (γ |[0,t] ) α-1 dt dened on curves Γ γ : [0, T + (γ)] → R d is lower semicontinuous. Proof. Set f (γ) :=

( 4 .

 4 3.2) Since µ -µ , ν -ν are concentrated in a common irrigation basin Bas η (σ), |µ -µ |almost every points x and |ν -ν |-almost every points y are reached by right-sided

  ρ∈P Bas + η (ρ), Bas -(P ) = ρ∈P Bas - η (ρ), and Bas(P ) = ρ∈P Bas η (ρ).

4. 4 . 1

 41 Transfer between basins Denition 4.4.1. Let A, B be two Borel sets of positive measure |A|, |B| > 0. We call x a transfer point from A to be B if for some constants c, R > 0 and all r ≤ R: Θ A (x, r) ≥ c, and x ∈ B.

Lemma 4 .

 4 4.2. If |U \ (A ∪ B)| = 0 with |A|, |B| > 0 and |A ∩ B| = 0, then there is a point x such that Θ A (x, r) ≥ c and Θ B (x, r) ≥ c, for r ≤ R and some constants c = c A,B > 0, R = R A,B > 0. In particular x is a transfer point from A to B and from B to A. Proof. Consider two points x, y which are Lebesgue points of A and B respectively, and a small radius r 0 such that Θ A (x, r) := |Qr(x)∩A| (2r) d > 1/2 and Θ B (x, r) := |Qr(x)∩B| (2r) d > 1/2

  y) dy.

  and (4.4.2) together yields:αc 0 m r z(x) -αc 0 ˆBr(x)∩A z(y) dy ≤ Crm α r β Θ A (x, r) 1-α ≤ Cr β ,(4.4.4) the last inequality holding for r ≤ R A,B (with C depending on c A,B ) because x is a transfer point from A to B. By (4.4.3) and (4.4.4), one has z(x) ≤ z r,A (x) + Cr β , then passing to the limit r → 0: z(x) ≤ lim inf r→0 z r,A (x).

( 4 .≤

 4 4.6)We denote by zr,A (x) the median of z on B r (x) ∩ A. Taking A 1 , A 2 of equal measure such that z ≤ zr,A (x) on A 1 and z ≥ zr,A (x) on A 2 , we obtain ˆA2 z -ˆA1 z = ˆA2 (z -zr,A (x)) -ˆA1 (z -zr,A (x)) = ˆBr(x)∩A |z -zr,A (x)|,and putting this into (4.4.6) implies: Br(x)∩A |z -zr,A (x)| ≤ Crm α-1 r Cr β , where we have used (4.4.4) in the last inequality. It is clear that up to changing again C, the median may be replaced by the mean: Br(x)∩A |z -z r,A (x)| ≤ Cr β .

≤

  Cr β , the last inequality following from (4.4.7), and the last but one from |B r/2 (x) ∩ A| ≥ cr d because x is a transfer point from A and B. Then one has |z(x) -z r,A (x)| ≤ |z(x) -z r/2,A (x)| + |z r/2,A (x) -z r,A (x)| ≤ |z(x) -z r/2,A (x)| + Cr β , which means by setting f

4. 4

 4 .4 A straightforward consequence Theorem 4.4.15. Let µ, ν be probability measures of the form µ = f L |U and ν = g L |V , where U, V are bounded, open and connected sets such that Ū ∩ V = ∅ and which are Ahlfors regular, and f, g ≥ c 0 > 0. If η is an optimal irrigation plan for the α-cost, where α > 1 -1/d, then η has equibounded ber lengths, in the sense that: η -ess sup γ∈Γ L(γ) < ∞. shape evolution problem . . . . . . . . . . . . . . . . . . . 124 5.2 Towards proving fractality of optimal shapes . . . . . . . . 128 5.3 Further perspectives . . . . . . . . . . . . . . . . . . . . . . . 134

F

  α,λ (ν) + ˆds (x, E τ k ) τ dν(x) =: G (ν).Now let us prove that ν k+1 is indeed the indicator of a set. Take a landscape function z τ kassociated to ν τ k = 1 E τ k. We know by Proposition 3.1.5 for any competitor ν ∈ M + (R d ),X α (ν) ≤ X α (ν τ k+1 ) + α ˆzτ k d(ν -ν τ k+1 ),(5.1.5) thus by adding -´λ dν + ´ds(x,

D

  (A, B) = ˆA∆B d(x, ∂A) dx = ˆRd d s (x, A)(1 B -1 A ) dx.By optimality of E τ k+1 , one has

F

  τ k+1 , E τ k ) ≤ τ (F α,λ (E τ k ) -F α,λ (E τ k+1 )).

Figure 5

 5 Figure 5.1: Projection on E τ k and E τ k+1 .

  hand, setting xk = π E τ k (x k+1 ), one has d(x, E τ k ) ≤ |x -xk | ≤ |x -x k+1 | + |x k+1 -xk | = d(x, E τ k+1 ) + d(x k+1 , E τ k ), but d(x k+1 , E τ k ) = d s (x k+1 , E τ k ) and αz k+1 (x k+1 ) + d s (x k+1 , E τ k )/τ = λ because x k+1 ∈ ∂E τ k+1 , which implies that d(x, E τ k+1 ) -d(x, E τ k ) τ ≤ λ -αz k+1 (x k+1 ).

  µ t , φ = ˆZ>t γ z≥t (0) dη(γ) = ˆZ>t γ(z -1 (t)) dη(γ). ν t , φ = ˆZ>t φ(γ z≥t (∞)) dη(γ) = ˆZ>t φ(γ(∞)) dη(γ) = ˆz>t φ(x) dν(x).

θ

  ηt (x) = θ(x). One hasI α (η t ) = ˆZ>t ˆγ 1 z(x)>t θ ηt (x) α-1 d H 1 (x) dη(γ) = ˆZ>t ˆγ 1 z(x)>t θ(x) α-1 d H 1 (x) dη(γ) = ˆZ>t (z(γ(∞)) -t) dη(γ) = ˆz>t (z(x) -t) dx.

  σ(B ε (x)) ≤ lim inf t→∞ σt (B ε (x)) ≤ Cε d-β .Passing to the log and sending ε to 0 yields:dim loc σ(x) = lim inf ε→0 log σ(B ε (x)) log ε ≥ lim inf ε→0 log(Cε d-β ) log ε = d -β.

ˆK

  w≥0wx -H (w) where H (w) = w p p where p < 0 is the conjugate exponent of α satisfying 1/α + 1/p = 1. The optimal irrigation problem may be rephrased formally as follows:inf η∈IP(µ,ν) η rectiable ˆK H(θ η (x)) d H 1 (x) θ η (x)w(x) d H 1 (x) -ˆK H (w(x)) d H 1 (x) = inf w>0 ˆK -H (w(x)) d H 1 (x) + inf η∈IP(µ,ν) η rectiableˆK θ η (x)w(x) d H 1 (x).

  θ η (x)w(x) d H 1 (x) = inf η∈IP(µ,ν) η rectiable ˆΓ L w (γ) dη(γ),where L w (γ) = ´γ w(x) dx is the weighted length of γ w.r.t. w. Set a pseudo-distance d w (x, y) = inf{L w (γ) : x γ -→ y} and notice that if we want the righthand side to be (almost) minimal then the curves γ should be (almost) d w -geodesics, henceinf η∈IP(µ,ν) η rectiable ˆΓ L w (γ) dη(γ) = inf π∈Π(µ,ν) ˆK×K d w (x, y) dπ(x, y),having dropped the rectiability requirement, which can be justied. Finally, the On the frontier between irrigation basins We place ourselves in the single source case. Take η ∈ OIP α (µ, ν) and recall the denition of basin from a point x ∈ N η :Bas η (x) = {y : there is an η-good curve γ s.t. x ∈ γ and y = γ(∞)}. One can show that it is a closed set. Consider two basins B, B from two points x, x such that x / ∈ B and x / ∈ B : what can we say on B ∩ B when the intersection is not empty? A conjecture by J.-M. Morel is that it should be fractal, perhaps in the sense that dim H (B ∩ B ) > d -1, a very tempting conjecture being dim H (B ∩ B ) = d -β.

(

  ii) H is lower semicontinuous and nondecreasing, (iii) H is subadditive, (iv) lim x→0 H(x) x = +∞.

  Des propriétés de régularité du réseau ont été démontrées par Xia [Xia04; Xia11] dans le modèle eulérien et[START_REF] Bernot | The structure of branched transportation networks[END_REF] dans le modèle lagrangien. Loin des mesures µ et ν, à l'intérieur du réseau, on retrouver la structure d'un graphe. Theorem 0.0.2(Xia [Xia04]). Si v est un ot optimal entre µ et ν et B est une boule à distance positive de supp µ ∪ supp ν alors v restreint à B est un graphe discret.

	Passons maintenant à la régularité au bord du réseau.

Theorem 0.0.3 ([BCM08; Xia11]). Si x est un point du réseau, le nombre de branches qui en partent est borné par une quantité Q = Q(d, α). Dans le cas d'une simple source µ = δ 0 , sous des hypothèses de régularité sur ν, on peut en dire davantage. Theorem 0.0.4 ([MS10]). Si γ est une courbe du réseau, alors elle est à courbure localement bornée. De plus, elle admet des demi-tangentes en chaque point, et une tangente en chaque point qui n'est pas un point de branchement. Dans [MS10], les auteurs montrent de plus que γ vérie une certaine EDO elliptique. Irrigabilité et distance d'irrigation Les mesures µ et ν étant xées, il n'est pas garanti qu'elles puissent être connectées avec un α-coût ni. Néanmoins, la relation µ et ν peuvent être connectées à α-coût ni est une relation d'équivalence, et tout couple de mesures peut être connecté si toute mesure peut se connecter à une masse de Dirac, auquel cas la mesure µ et dite α-irrigable. Un raisonnement dyadique a permis a Xia de démontrer un critère d'irrigabilité.

  Fonction paysage Dans le cas d'une seule source µ = δ 0 , on peut dénir une fonction paysage z η associée à un plan d'irrigation optimale η : δ 0 → ν, qui est devenue un outil fondamental en transport branché. Elle a été introduite par Santambrogio dans[START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF], et est inspirée de la fonction du même nom utilisée par les géophysiciens (voir[START_REF] Rodriguez-Iturbe | Fractal river basins: chance and selforganization[END_REF]). γ est une courbe du réseau (en un sens à dénir) de 0 à x, et z η (x) = +∞ s'il n'existe pas de telle courbe. Évidemment, il faut vérier que cette dénition est cohérente et ne dépend pas du choix de γ. Un intérêt de la fonction paysage est par exemple le fait qu'elle joue le rôle de variation première pour la fonctionnelle

	Elle est dénie comme
	z

η (x) = ˆγ θ η (x) α-1 dx si

  qu'on pourrait appeler les boules unité en transport branché. Ce sont les ensembles de volume 1 les plus proches de l'origine 0 au sens de l'irrigation. C'est l'objet du Chapitre 3. Pour donner un sens rigoureux à cela, tous les objets sont vus comme des mesures, 0 devenant δ 0 et un ensemble borélien E comme L E, ce qui donne lieu lorsque α > 1 -1/d au problème d'optimisation de forme

	min {d α (δ 0 , L E)	. = X α (L E) : Vol(E) = 1}.	(0.0.6)
	Le problème d'optimisation de forme est traité en considérant un problème relaxé,
	partant d'une branche de longueur	est de l'ordre de /ε. Cela peut être vu comme
	une propriété d'auto-similarité car d'une certaine façon la longueur total est préservée
	en regardant les sous-branches à toute échelle.

pour une certaine classe de compétiteurs ν . La fonction paysage est donc essentielle dans l'étude de problèmes faisant intervenir la distance d'irrigation (au départ d'une source), ce qui est l'objet du Chapitre 3. Par ailleurs, des propriétés de régularité Hölder peuvent être établies sous des hypothèses de régularité pour ν, une propriété utilisée pour démontrer la courbure localement bornée des branches, la bornitude uniforme des longueurs des branches, et des comportements fractals des structures optimales. Fractalité en transport branché En regardant les réseaux optimaux calculés numériquement dans [OS11] (dans les cas non-atomiques), or aux réseaux uviaux naturels et leurs bassins d'irrigation, on est tenté de les décrire comme fractals. Des travaux venant de la géophysique et l'hydrologie vont dans ce sens [RR01]. En fait, même si le réseau sous-jacent N η a une innité de branchements, il s'agit toujours d'un ensemble 1-rectiable, il n'est donc pas clair en quel sens la fractalité apparaît. Le terme fractals est usuellement associé ou bien à des propriétés d'autosimilarité d'objets non lisses, ou à la présence d'une dimension non entière. Un premier résultat rigoureux en transport branché en tombant d'une certaine manière dans la première catégorie a été prouvé par Brancolini et Solimini dans [BS14] : pour des mesures susamment diuses (par exemple la mesure de Lebesgue sur un ensemble régulier), le nombre de branches de longueur ∼ ε Et qu'en est-il pour l'autre versant de la fractalité, c'est-à-dire de la présence de dimension non entière (ou fractale) ? Plusieurs conjectures ont été émises quant aux ensembles pouvant présenter une dimension fractales ; ce peut être la frontière des bassins d'irrigation (une conjecture de J.-M. Morel), ou bien certains ensembles de niveaux de la fonction paysage introduite précédemment. On s'est intéressé avec F. Santambrogio et Q. Xia à un autre candidat potentiel relié au transport branché : le CONTENTS bord de ce remplaçant E par par des mesures à densité encadrée par 0 et 1, comme suit

  ) the set of irrigation plans on R d . Notice that any irrigation plan is concentrated on Γ 1 (R d ). Since this set is not closed in Prob(Γ(R d )), we will introduce some closed or compact subsets later. If µ and ν are two probability measures on R d , one says that η ∈ IP(R d ) irrigates ν from µ if one recovers the measures µ and ν by

			| dt,
	which are valued in [0, +∞]. Since curves are 1-Lipschitz, L(γ) ≤ T (γ). Moreover,
	one may prove that T and L are both lower semicontinuous functions and, as such, are
	Borel. We denote by Γ 1 (X) the set of curves of nite length.
	Irrigation plans We call irrigation plan on R d any probability measure η ∈ Prob(Γ(R d ))
	satisfying	L(η) :=	ˆΓ(R d )	L(γ) dη(γ) < +∞,	(1.2.1)
	and denote by IP(R d sending the mass of each curve respectively to its initial point and to its nal point,
	which means that

  Following Remark 1.2.11 we dene a gluing ηA ∈ η B [0, t x ] : η A [t x , t y ] : η B [t y , +], which is possible after checking the compatibility of the consecutive marginals. It is clear that the recovery cutting times are t x , t y . Then we set

  1.3. THE EULERIAN MODELRectiable irrigation ow Recall that if E is a 1-rectiable set, at H 1 -a.e x ∈ E there is an approximate tangent line (see[START_REF] Mattila | Geometry of sets and measures in Euclidean spaces[END_REF] Chapter 17]) denoted by Tan(x, E). An irrigation ow of the form v = E, τ, θ where E is 1-rectiable and τ (x) ∈ Tan(x, E) for H 1 -a.e. x ∈ E is termed rectiable 8 . From discrete to continuous Consider an irrigation graph G ∈ IG(R d ).One can check that ∇•v G = e∈E(G) θ e (δ e --δ e + ) ∈ M 1 (R d ) and that v G is a rectiable irrigation ow on R d . Also, both the cost E α and the constraint G ∈ IG(µ, ν) can be expressed solely in terms of v G . Indeed Kirchho 's law can actually be expressed in terms of the divergence

	One can
	dene the vector measure v G by
	v G :=
	e∈E(G)

e, ê, θ e .

  by denition η is rectiable if and only if j η is concentrated on a rectiable set, in which case it is concentrated on the rectiable network N η and one has j η = θ η H 1 using Fubini-Tonelli's theorem. . Let us call m the inmum of (LEN η ) and show that it admits a minimizer. Take a minimizing sequence(ζ n ) n such that every ζ n is normalized, in particular ζ n ∈ IP C (K) for some C > 0.Up to extraction we have convergence ζ n ζ, and since IP C (µ, ν) is closed by Proposition 1.2.2, ζ ∈ IP C (µ, ν). Moreover L(ζ) = m by lower semicontinuity of L . = I 1 on IP C (K), which we proved in Corollary 1.2.8. Now in order to show that ζ is a solution of (LEN η ) we only have to check the last constraint j ζ ≤ j η . Take any open set O. One has

Lemma 2.1.12 (Simple replacement). Let η ∈ IP(µ, ν) be an irrigation plan. Consider the minimization problem

min {L(ζ) : j ζ ≤ j η and ζ ∈ IP(µ, ν)} . (LEN η )

Then (i) this problem admits minimizers which are all simple, (ii) if η is rectiable, all minimizers ζ are also rectiable and j ζ ≤ j η rewrites θ ζ ≤ θ η H 1 -almost everywhere.

(2.1.3) Any minimizer of (LEN η ) is called a simple replacement of η.

Proof

  This is not true in general (because of the presence of cycles) but a Smirnov decomposition gives the result if v is optimal for (EI α ). Cycle If v ∈ IF(K), we say that w ∈ IF(K) is a cycle of v if |v| = |w| + |v -w| and ∇ • w = 0. It is easy to check that if v is rectiable then w and v -w are also rectiable. The following Smirnov decomposition is proved by Santambrogio via a Dacorogna-Moser approach in [San14].

  It all follows from Proposition 2.2.3 and Proposition 2.3.3. The equality i η = |v η |

	comes from

  Remark 2.4.2. Notice in particular that the equality i η = |v η | implies that curves of Extension to the H-mass We assume that H satises: These hypotheses are weaker than those needed to prove lower semicontinuity of the Lagrangian H-cost, as remarked in the last paragraph of Section 1.2.3. Under these hypotheses, it was proven in[START_REF] Colombo | On the lower semicontinuous envelope of functionals dened on polyhedral chains[END_REF] that the H-mass is lower semicontinuous and it is not dicult to show existence for the Eulerian problem with such H-mass. Without any changes other than replacing x α and x α-1 with H(x) and H(x)/x respectively, one may check that the equivalence between the Lagrangian and Eulerian irrigation problems hold. As a byproduct, we obtain existence of minimizers for the Lagrangian problem (built from a minimizer of the Eulerian problem), although we could not prove it directly since the lower semicontinuity of the H-cost with these hypotheses on H is not guaranteed, as noticed in[START_REF] Brancolini | Equivalent formulations for the branched transport and urban planning problems[END_REF]. In this paper, they prove in particular

	(i) H(0) = 0,
	(ii) H is lower semicontinuous and nondecreasing,
	(iii) H is subadditive,
	(iv) lim x→0	H(x) x = +∞.

η have the same tangent vectors when they coincide. To be more precise, one may use the proof of Proposition 2.2.2 to state that there is an H 1 -a.e. dened function τ : N η → S d-1 such that for η-a.e. γ ∈ Γ, for H 1 -a.e. x ∈ γ, γ(t) = | γ(t)|τ (x) whenever γ(t) = x. existence and equivalence between the Eulerian and Lagrangian problems in a more general case, dropping the last condition lim x→0 H(x) x

= +∞. In that case optimizers are not necessarily rectiable and the Eulerian cost can be expressed as

  H 1 +v s and v s is the singular part w.r.t H 1 .

	Looking at the optimal branching structures computed numerically in [OS11] (in
	some non-atomic cases), or at natural drainage networks and their irrigations basins,
	one is tempted to describe them as fractal (see [RR01]). Actually, even though the
	underlying network has innitely many branching points, it is still a 1-rectiable set,
	hence it is not clear in what sense fractality appears. Fractality is a notion which usu-Chapter 3 ally relates either to self-similarity properties of non-smooth objects, or to non-integer
	dimension of sets. A rst rigorous result which would fall in the rst category is proven
	by Brancolini and Solimini in [BS14]: for suciently diuse measures (for example the Lebesgue measure restricted to a Lipschitz open set), the number of branches of length ∼ ε stemming from a branch of length l is of order l/ε. This may read as a self-similarity A fractal optimal shape property since in a way the total length is preserved when looking at subbranches at
	all scales.
	The present chapter leans towards the other notion of fractality, that is towards
	fractal dimension. Some sets in branched transport have already been proposed

Contents 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The irrigation problem . . . . . . . . . . . . . . . . . . . . . . 3.1.2 The shape optimization problem . . . . . . . . . . . . . . . . 3.2 Existence and rst properties . . . . . . . . . . . . . . . . .

3.2.1 Existence of minimizers . . . . . . . . . . . . . . . . . . . . . 3.2.2 Lebesgue points . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hölder continuity of the landscape function . . . . . . . . . 3.3.1 Main lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Hölder regularity . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 On the dimension of the boundary . . . . . . . . . . . . . . 3.5 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Optimization methods . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Computing the projection . . . . . . . . . . . . . . . . . . . . 3.5.3 Algorithms and numerical experiments . . . . . . . . . . . . . as candidates to exhibit non-integer dimension, for instance the boundary of adjacent irrigation basins (an open conjecture by J.-M. Morel). Here we are interested in another candidate which is related to branched transport: the boundary of what we call unit balls for branched transport. With the results of the present chapter, we can only prove an upper bound on the dimension, which is non-integer, and conjecture that this upper bound is actually sharp.

  This result pushes us to propose the following conjecture: Conjecture 3.4.4. The boundary ∂A is of dimension d -β, in the sense that:

  We pass on to the next: ρ k ξ k x k ξ k ρ k . By hypothesis, either ρ k or ρ k belongs to the roots already covered, say ρ k . If ρ k / ∈ i<k {ρ i , ρ i } then one may enforce the desired property

	by setting:

  It is a decreasing sequence of subsets of Γ. Let γ ∈ Γ 1 belonging to k ↓ F k and being parameterized by arc-length, with length T . There is a sequencet k ∈ [0, T ] such that γ |[0,t-ε] ⊆ γ, which means that k ↓ F k = F := {γ : γ |[0,t-ε] ⊆ γ}.Since γ n → γ, we know that γ n ∈ F k for n large enough. Consequently

		(4.3.1)
	lim sup	η(γ :
	n
	∀s ∈ [0, t -ε], |γ(t k + s) -γ(s)| ≤ 1/k.

Up to subsequence t k → t ∈ [0, T ], so that ∀s ∈ [0, t -ε], γ( t + s) = γ(s),

and

  Interchanging the roles of x and y yields|z(y) -z(x)| ≤ CM α-1 |y -x|,and z is Lipschitz continuous on R. Remark 4.3.7. If µ and ν have disjoint compact supports, one may dene an extension of z in a neighbourhood of supp µ and supp ν which is upper semicontinuous on the neighborhood of supp µ and lower semicontinuous on the neighborhood of supp ν.

	Proposition 4.3.6 (Semicontinuity). The landscape function is lower semicontinuous
	on Bas + η ( R) and upper semicontinuous on Bas -η ( R).

  Proposition 5.2.2. The function t → I α (η t ) is Lipschitz and which shows that t → I α (η t ) is Lipschitz and implies that Now we may reformulate the desired lower bound on the volume as follows.Proposition 5.2.3. The following assertions are equivalent:(i) ∃c > 0 : Vol(z > t) ≥ c(z -t), (ii) ∃c > 0 : I α (η t ) . = ´z>t (z -t) ≥ c(z -t) 2 , (iii) ∃c > 0 : ´z>t (z -z) ≥ c(z -t) 2 .Proof. Let us prove (i) ⇔ (iii). We know by the rst variation formula for X α and by the scaling Lemma thate α -e α (α + 1/d)m + e α α(α + 1/d)(α + 1/d -1) 2 m 2 ≤ e α -αThis rewrites, recalling that e α (α + 1/d) = αz :

	d + dt I But	ˆt<z≤t+ε d + dt I α (η t ) = lim (z(x) -(t + ε)) ε ε→0 + I ˆz>t dx ≤ Vol(t < z ≤ t + ε) ε→0 --→ 0, ˆz>t cm 2 ≤ (z -z(x)) dx,	z(x) dx,

α (η t ) = -Vol(t < z ≤ z ) and d - dt I α (η t ) = -Vol(t ≤ z ≤ z ).

Proof. Take for example ε > 0 small. We have

I α (η t+ε ) -I α (η t ) = ˆz>t+ε (z(x) -(t + ε)) dx -ˆz>t (z(x) -t) dx = -ε Vol(t < z ≤ z ) + ˆz>t+ε (z(x) -(t + ε)) dx -ˆz>t (z(x) -(t + ε)) dx = -ε Vol(t < z ≤ z ) -ε ˆt<z≤t+ε (z(x) -(t + ε)) ε dx. α (η t+ε ) -I α (η t ) ε = -Vol(t < z ≤ z ).

A similar computation with ε < 0 shows that

d - dt I α (η t ) = -Vol(t ≤ z ≤ z ).

where m = Vol(z > t).

where c = c(α, d) is some constant. This inequality shows that (i) ⇒ (iii). On the other hand one has

ˆz>t (z -z(x)) dx ≤ ˆz>t (z -t) dx ≤ (z -t)m,

which shows that (iii) ⇒ (i).

Similarly

ˆz>t (z(x) -t) dx ≤ ˆz>t (z -t) dx ≤ (z -t)m,

which shows that (ii) ⇒ (i). It remains to prove (i) ⇒ (ii). But we know that d dt I α (η t ) = Vol(t < z ≤ z ) ≥ c(z -t), which yields:

  x ∈ L we have |y -x| β ≈ z -z(y) thus we may chose a t which is suciently close to z , namely z -t ≈ ε β to ensure that y t ∈ B ε (y), independently of y. Let us call Y t the set of such y t 's. Thanks to the previous lemma, this set has a cardinal bounded from above by a constantN = N (α, d). Thus we write Y t = (y i t ) i≤Nt , with N t ≤ N . Since B ε (x) ⊆ B ε (x)is open, passing to the limit as t → ∞ yields:

	Bas(y i t ),
	i≤Nt

we may deduce that

σt (B ε (x)) ≤ i≤Nt σt (Bas(y i t )) ≤ C i≤Nt diam(Bas(y i t )) d-β .

One has that diam(Bas(

y i t )) ≈ (z -z(y i t )) 1/β ≈ ε, which leads to σt (B ε (x)) ≤ CN t ε d-β ≤ Cε d-β .

Since

One iteratively add a new vertex to the path until there is no more, which happens because G is cycle-free. This last vertex is necessarily a source or target, i.e. a leaf.

, η y

) would do.

A rectiable irrigation ow is, in the language of currents, a rectiable normal 1-current on R d .

For us, if I is an interval γ |I is a map dened only on I, but it induces a unique Lipschitz extension to the whole R which is constant before and after I and we will make the slight abuse of not distinguishing the two.

Parameterized by arc length on an interval [0, ].

Actually, up to a sign depending on the orientation of the curve, just as in the denition of the integral on the real line ´b a f = ± ´[a,b] f depending on whether a ≤ b or b ≤ a.

Since two-sided good curves can be cut in two one-sided good curves, it does not matter.

Remerciements

Chapter 4

The landscape function with multiple sources By the balance property, one has:

(4.2.3) Now notice that in the sum Z ρ j i appears the same amount of time with a + sign than with asign, except for the extreme terms ρj i = ρ and ρj i = ρ : Z ρ appears once more with a + sign, and Z ρ once more with a -. Hence recalling that Zρ := c ρ + Z ρ, (4.2.3) rewrites Zρ (γ ) -Zρ (γ) =i ( Zρ j i (ξ j i ) -Zρ j i (ξ j i )), but we know by Proposition 4.2.5 that Zρ j i (ξ j i ) = Zρ j i (ξ j i ) for all i, hence:

Zρ (γ) = Zρ (γ ). if γ is a one-sided η-good curve joining ρ ∈ R and x.

Basic properties

In this section we prove the basic properties of the landscape function which were already known in the single-source case: semicontinuity, expression of the optimal cost and of the variation of d α in terms of z.

Fundamental property

Beforehand, we establish what we call the fundamental property of the landscape function: it is in a sense a primitive of θ α-1 η along η.

Proposition 4.3.1 (Fundamental property). If γ is an η-good curve then Z η (γ) . = ˆγ θ η (x) α-1 dx = z(γ + ) -z(γ -).

Denition 4.4.3 (Basin variation of µ). For each P ∈ P, suppose that we have two nonnegative measures χ P , ξ P which are concentrated on X P and which represent a mass that we want to remove from µ and add to µ respectively. Assume that P χ P ≤ µ and set µ = µ + P (ξ P -χ P ) which is a nonnegative measure. Then the triple (µ , (χ P ) P , (ξ P ) P ) is called a basin variation of µ.

Denition 4.4.4 (Transfer relation). We dene a binary relation on X as follows: A B if there exists constants K = K A,B > 0, ε = ε A,B > 0 such that for all m ≤ ε one can nd a basin variation (µ , (χ P ), (ξ P )) of µ such that: (i) P χ P (X P ) + ξ P (X P ) ≤ Km, (ii) for all P such that X P ⊆ A ∪ B, χ P (X P ) = ξ P (X P ), (iii) P :X P ⊆A (ξ P -χ P )(X P ) = -m and P :X P ⊆B (ξ P -χ P )(X P ) = +m,

In that case we say that the basin variation (µ , (χ P ), (ξ P )) transfers a mass m from A to B with controlled cost. Lemma 4.4.5. Let x be a transfer point from A to B. The following holds:

Remark 4.4.6. The basin variation (µ , (χ P ), (ξ P )) satisfying the requirements for A B as in Denition 4.4.4 that we get from the proof is the following: χ P = c1 X P ∩Br(x) for all P ∈ P with r d ≈ m, ξ P 0 = mδ x , and ξ P = 0 if P = P 0 ; P 0 being such that x belongs to

Proof. We set z r,A (x) = ffl Br(x)∩A z(y) dy and m r = |B r (x) ∩ A|. The proof strongly relies on computations made in [START_REF] Santambrogio | Optimal channel networks, landscape function and branched transport[END_REF] to prove the Hölder continuity of the landscape function in the single-source case, and on calculations which are quite classical when working with Campanato spaces (see [Giu03, Section 2.3]). We divide the proof into two steps.

Step 1: z(x) = z A (x) := lim r→0 z r,A (x). Recalling that µ ≥ c 0 1 U , we remove c 0 1 Br(x)∩A from µ and we send a mass c 0 m r to x, then we spread it again to B r (x) ∩ A.

To be more precise, we set μ = µ -c 0 1 Br(X)∩A + c 0 m r δ x . Since diam(supp(µ -μ)) ≤ r and µ -μ ≤ 2c 0 m r , we know that

By way of consequence

Choose P 0 such that x ∈ X P 0 ⊆ A. Consider a family of disjoint sets XP ⊆ X P such that P XP = P X P . We dene χ P = t r c 0 1 XP ∩Br(x)∩A for all P ∈ P, ξ P 0 = mδ x and ξ P = 0 if P = P 0 , and nally µ = µ + P ξ P -

µ is a probability measure, and one may check that (µ , (χ P ), (ξ P )) is a basin variation satisfying (i), (ii) and (iii) of Denition 4.4.4. Let us check (iv) and (v):

A = X =⇒ ∃P ∈ P, (X P ⊆ A and A X P ).

Proof. We prove the three items successively.

Proof of (i) We treat for example the rst implication, the second one being handled symmetrically. Set ε A,C = 1/2 • min(ε A,B , ε B,C ) and take m ≤ ε A,C . We know there exists basin variations (µ 1 , (χ 1 P ), (ξ 1 P )) and (µ 2 , (χ 2 P ), (ξ 2 P )) transferring a mass 2m from A to B and from B to C respectively, with controlled cost. Let us build a new one as follows:

,

By construction P χ P ≤ µ and µ is a nonnegative measure, so that (µ , (χ P ), (ξ P )) is a basin variation of µ, and it is easy to check that it transfers a mass m from A to C. Obviously, for all P one has P χ P (X P ) + ξ P (X P ) ≤ (K A,B + K B,C )m. Moreover one can control the cost variation:

Moreover by subadditivity of the d α distance

) all the required properties to get A C are satised.

Proof of (iii) Consider a loop A 1 A 2 A 1 and B such that A := A 1 ∪ A 2 B. Let us establish for example A 1 B. We consider a basin variation (μ, ( χP ), ( ξP )) of µ which transfers a mass m from A to B with controlled cost. Set m i = ξ(A i ) -χP (A i ) ∈ R for i = 1, 2. We take a basin variation (μ, ( χP ), ( ξP )) which sends a mass m 2 from

) /2. Thus (µ , (χ P ), (ξ P )) is a mass transfer of mass m/2 (and not m) from A 1 to B with controlled cost for constants

We have proved that A 1 B, and the same goes for A 2 . The analogous statement for A B 1 ∪ B 2 holds as well.

Proof of (iv) Take a set A ∈ X such that A = X. If there is a P such that X P ⊆ A and |X P ∩ A| > 0 then clearly A X P . Otherwise set B = P :X P ⊆A X P . We know that |A|, |B| > 0, |A ∩ B| = 0 and |U \ (A ∪ B)| = 0, hence by Lemma 4.4.2 there is a frontier point x from A to B. Since B is closed, x actually belongs to B, and there is a basin X P contained in B such that x ∈ X P . By denition x is a frontier point from A to X P and by Lemma 4.4.5 one has A X P .

Remark 4.4.8. Notice that using (i) successively, if one has a loop

A i for all i, j, and using (iii) successively, it generalizes to arbitrary loops:

Proposition 4.4.9. The binary relation is total in the sense that for all A, B ∈ X , A B.

Proof. Let us dene some terminology which will render the proof very easy. We say that two elements A, B ∈ X are equivalent if A B and B A. We call blocks all elements A ∈ X which are unions of equivalent basins, that is sets of the form A = P ⊆P X P where P ⊆ P and all the (X P ) P ∈P are pairwise equivalent. We will consider block chains

which are simple in the sense that for i = j, A i and A j have no basin in common.

The length (C ) of C is n and its size s(C ) is the total number of basins appearing in the A j 's: s(C ) = card{P : ∃i, X P ⊆ A i }. We consider the simple chains of maximal size (it makes sense because there are only nitely many), and among them those with minimal length. Pick such a chain C . If card{P : X P ⊆ A n } < card P then there exists a P ∈ P such that A n → X P and X P ⊆ A n by (iv) of Proposition 4.4.7. Now since C has maximal size, X P must belong to a previous basin: X P ⊆ A i for i < n. By (ii) one has A n A i , thus the blocks A i , A i+1 , . . . , A n are equivalent. Consequently all the basins X P belonging to the A j 's for i ≤ j ≤ n are equivalent, using (iii) successively as in Remark 4.4.8. Hence Ãi := j≥i A j is itself a block and one has a block chain

which is a simple block chain with s( C ) = s(C ) and ( C ) < (C ): a contradiction. Consequently one had card{P : X P ⊆ A n } = card P and the chain was made of a single block A 1 = A n made of all basins. Therefore all basins are equivalent, and all elements A ∈ X as well by (ii). This is what we wanted.

We set K = max A,B∈X K A,B and ε = max A,B∈X ε A,B .

Proposition 4.4.10 (Basin transfers). There exists constants K, ε > 0 such that for any family of real numbers (m P ) P ∈P satisfying P ∈P m P = 0 and m := P |m P | ≤ ε, one may nd a basin variation (µ , (χ P ), (ξ P )) of µ such that (i)

Proof. Assume that the m P 's are not all 0, otherwise there is nothing to do. Take

0 and a basin variation which sends a mass |m P 1 | from X Q 1 to X P 1 . Then we set for all P ∈ P:

We know that m 1 := P |m 1 P | ≤ m + Km P 1 ≤ (1 + c)m and m 1,P 1 = 0. We consider the family of real numbers (m 1,P ) P ∈P . We continue by induction, building sequences P i , Q i , V i , m i,P until all the (m i,P ) P are 0, and set I the number of steps. We obtain a sequence satisfying for all i ≤ I:

(i)

It is possible to connect µ and ν using η-good paths, since for all P , ξ P -χ P and ( ξχ) Bas + (P ) are concentrated on a common basin Bas η (ρ P ) and have equal (signed) mass. Hence there exists an irrigation plan η ∈ IP(µ , ν ) obtained by adding, removing, and reconnecting η-good curves passing by a same connected component P , so that η is concentrated on η-good paths.

From this result, we are able to prove a rst variation formula for general variations of ν, with µ xed, at the cost of an extra error term, which is of lower order in our case α ≥ 1 -1/d.

for some constant C independent from ν and m.

Remark 4.4.13. Notice that since z is bounded from below on Bas η + ( R) and that z ∈ L 1 (ν), the integral ´z d(ν -ν) is well-dened and valued in R ∪ {+∞}.

Proof. If ´z d(ν -ν) = +∞ there is nothing to prove. Otherwise z ∈ L 1 (ν ) and we take µ and η ∈ IP(µ , ν ) given by the previous proposition. One has

hence using the triangle inequality and I α (η) = d α (µ, ν):

By the previous proposition, the two last terms are bounded by Cm 1+β/d , which implies that

The regularity theorem

Theorem 4.4.14. There exists a constant C > 0 such that for any r ≤ r 0 , x ∈ Bas - η ( R) and y ∈ Bas + η ( R), one has:

hence z is β-Hölder continuous on Bas - η ( R) and Bas + η ( R).

Proof. Consider a ball B = B r (x) ∩ Bas + η ( R) centered at some x ∈ Bas + η ( R). We partition it in two parts of equal measure:

Then we apply Theorem 4.4.12:

but by the triangle inequality, one has The same reasoning gives the corresponding inequality on Bas - η ( R). By the equivalence between Campanato and Hölder spaces, as U and V are Ahlfors regular, this means that z is β-Hölder continuous on U and V . d -β. Of course, this measure is concentrated on the set {z = z } thus getting a lower bound on the dimension of this set by a mass distribution principle ( [START_REF] Falconer | Techniques in fractal geometry[END_REF]).

Let us start with some notations. We write a ≈ b if there are constants c, C depending only on α, d such that ca ≤ b ≤ Ca. For all x we denote by Bas(x) the basin of x, by δ(x) its diameter and we set z (x) = sup{z(y) : x ≤ y}, ρ(x) = sup{|y -x| : x ≤ y}, where by x ≤ y we mean that y ∈ Bas(x). Finally we set (x, y) = sup{L(γ) : γ η-good from x to y}, and (x) = sup{L(γ) : γ η-good starting at x }.

Lemma 5.2.4. The following equivalences hold for all x ∈ E:

(5.2.5) Moreover if y ≥ x is a point such that either z(y) = z (x) or (x, y) = (x) then |y -x| ≈ (x).

(5.2.6)

Proof. Take a point y ≥ x. Let us bound ´y x θ α-1 from above and below, using the fact that z is Hölder-continuous and that θ(x) = |Bas(x)|:

Now take a point y ≥ x which has maximal z, that is z(y) = z (x). One may write

Take a point y which maximizes (x, y ), that is (x, y ) = (x). One may write

Consequently one has approximate inequalities in the previous chain of inequalities:

Moreover, for the chosen y and y , one has moreover:

|y -x| ≈ |y -x| ≈ (x). Now notice that for all i, µ t (x i ) = Vol(Bas(x i )) ≈ δ(x i ) d , therefore:

the last inequality following from the fact that

We are not going to look at σ t directly but at σt dened by

The calculation we did before gives:

When t → z , the measures σt converge (up to subsequence) to a measure σ. Notice that for all φ which is continuous:

thus σ t and σt have the same limit (up to subsequence).

Lemma 5.2.5. The measure σ is concentrated on {x : z(x) = z }.

Proof. Take x / ∈ L, that is to say z(x) < z . Consider z > t > z(x) + ε with ε small. Then for any y ∈ B Cε 1/β (x), one has z(y) < t by Hölder continuity of z and thus y does not belong to any basin of points x t which satisfy z(x t ) = t. Consequently σt (B Cε 1/β (x)) = 0, which is true for all t large enough, hence taking the limit t → z yields:

Consequently supp σ ⊆ {x : z(x) = z }.

Theorem 5.2.6. The measure σ has lower local dimension at least d -β, in the sense that for σ-a.e. x, one has:

optimal irrigation problems rewrites

.

(5.3.2)

The optimal irrigation problem can be seen as a Monge-Kantorovich problem with an unknown transport plan π ∈ Π(µ, ν) but the cost is also part of the unknown: it is a distance d w where w is not xed but is penalized by ´K -H (w(x)) d H 1 (x). This term forces w to be +∞ everywhere but on a 1-dimensional set because -H (w) = -w p /p with p < 0. This formulation might be worth looking at for itself, even more so as one can apply Monge-Kantorovich duality to the term MK w (µ, ν): but the meaning of this last problem is not so clear because d w is not a classical distance as it takes +∞ values and the choice of the functional space for φ is not so clear either.

A possibility would be to look for solutions

where

This setting gives a (proposed) a rigorous meaning to (5.3.4). But then what could we say about Kantorovich potentials, that is optimal φ's ? It turns out they could play the role of the landscape function. Indeed, in the single source case µ = δ 0 , the primal problem is trivial since there is only one transport plan, and the primal-dual problem (5.3.4) may be written as

ˆK φ(y) dν(y), and d w (0, •) is a solution. Now if η ∈ OIP α (δ 0 , ν), then w(x) = θ η (x) α-1 is a solution to (5.3.2) and one expects that η is concentrated on d w -geodesics so that d w (0, x) = z η (x) for ν-a.e. x.

The idea is the following: given η ∈ OIP α (µ, ν), set w η = θ α-1 η , consider the problem sup φ∈Lip 1,dw ˆφ d(ν -µ),

(5.3.7)

for example in the setting of (5.3.5), and dene landscape functions as solutions of this problem. The diculty is then to show that there are solutions ! In classical optimal transport theory, the dual problem sup ψ⊕φ≤c ˆφ dν + ˆψ dµ has solutions φ ∈ L 1 (ν), ψ ∈ L 1 (µ) when c is lower semicontinuous, bounded from below and real-valued, which is not the case here with c = d w . Notice that if φ was a solution to (5.3.7) and if we had duality in the sense of (5.3.4), then one should have

where π η ∈ Π(µ, ν) is the transport plan associated to η. Since |φ(y) -φ(x)| ≤ d w (x, y) and since integrals with respect to µ, ν, π η may be rewritten as integrals with respect to η, the previous equality should imply that L w (γ

In particular the rst variation formula for I α (Proposition 1.2.1) should yield a rst variation formula for d α , as done in the proof of Proposition 3.1.5 with µ = δ 0 xed, of the form

for all µ , ν . Notice however that such a general inequality for arbitrary µ , ν may not be true if η is not single rooted, as noticed in Theorem 4.3.9. With such a denition of landscape function one could hope to recover the fact that it is a kind of (super) rst variation for d α , which is the main property that is asked to tackle variational problems involving this distance. Moreover, this is also the only property that is used on the landscape function to prove Hölder continuity. Finally, notice that in the single source case, problem (5.3.2) rewrites as

where z w = d w (0, •). Since z w (x) is obtained by integrating w along a geodesic γ from 0 to x, w can be thought as some kind a derivative z w of z w along the network, and if one could rephrase the problem in terms of z = z w , it would read as

so that z would satisfy in some vague sense ∆ p z = ν but with negative p. This does not seem completely unreasonable since a connection between the branched transport problem and p-harmonic maps has been made in the discrete case by Xia in [START_REF] Xia | On landscape functions associated with transport paths[END_REF].

More general, the last hypothesis can be dropped, in which case one should take another denition of H-mass by lower semicontinuous envelope:

for at chains T , that is the completion of polyhedral chains with respect to the at norm F and M H is dened by expression 5.3.8 on polyhedral chains. The validity of (5.3.8) when (iv) is satised, as well as a general representation formula when it does not hold, was established recently in [START_REF] Colombo | On the lower semicontinuous envelope of functionals dened on polyhedral chains[END_REF]. A denition of H-cost in a Lagrangian setting was given by [START_REF] Brancolini | Equivalent formulations for the branched transport and urban planning problems[END_REF], together with the proof on general properties in this case.

We have already remarked in Chapter 2 that the equivalence with the Lagrangian and Eulerian models holds in the case of the H-mass with almost no changes, with extra hypotheses on H (those also assumed [START_REF] Brancolini | Equivalent formulations for the branched transport and urban planning problems[END_REF]). Notice that removing hypothesis (iv) does not guarantee that v is rectiable but it can have an absolutely continuous part.

In covers in particular the case of the urban planning studied in [START_REF] Buttazzo | Optimal urban networks via mass transportation[END_REF][START_REF] Brancolini | Equivalent formulations for the branched transport and urban planning problems[END_REF] where

What we would like to do now is perform numerical computations for such problems.

Can we adapt the variation model a la Modica-Mortola to the case of the H-mass? In dimension 2, given H, we would like to nd a function h : R

We expect that a rectiable current v of mass m concentrated on a segment of length should be approximated by a diuse vector eld v ε concentrated on a tube around this segment, oriented by the segment and invariant along it. The ux on a perpendicular section should be equal to m and v ε should optimize M h ε with cost H(m), hence it should be given by a radial prole u : R → R which optimizes:

Questions arise: can we invert this formula, i.e. nd an h such that it holds for all m, H being given? Is there a formula or an expression in terms of H? Can we prove Γ-convergence? Cette thèse est consacrée à l'étude du transport branché, de problèmes variationnels qui y sont liés et de structures fractales qui peuvent y apparaître. Le problème du transport branché consiste à connecter deux mesures de même masse par le biais d'un réseau en minimisant un certain coût, qui sera pour notre étude proportionnel à mL α an de déplacer une masse m sur une distance L. Plusieurs modèles continus ont été proposés pour formuler le problème, et on s'intéresse plus particulièrement aux deux grands types de modèles statiques : le modèle Lagrangien et le modèle Eulérien, avec une emphase sur le premier. Après avoir posé proprement les bases de ces modèles, on établit rigoureusement leur équivalence en utilisant une décomposition de Smirnov des mesures vectorielles à divergence mesure. On s'intéresse par la suite à un problème d'optimisation de forme lié au transport branché qui consiste à déterminer les ensembles de volume 1 les plus proches de l'origine au sens du transport branché. On démontre l'existence d'une solution, décrite comme un ensemble de sousniveau de la fonction paysage, désormais standard en transport branché. La régularité Hölder de la fonction paysage, obtenue ici sans hypothèse de régularité a priori sur la solution considérée, permet d'obtenir une borne supérieure sur la dimension de Minkowski de son bord, qui est non-entière et dont on conjecture qu'elle en est la dimension exacte. Des simulations numériques, basées sur une approximation variationnelle à la Modica-Mortola de la fonctionnelle du transport branché, ont été eectuées dans le but d'étayer cette conjecture. Une dernière partie de la thèse se concentre sur la fonction paysage, essentielle à l'étude de problèmes variationnels faisant intervenir le transport branché en ce sens qu'elle apparaît comme une variation première du coût d'irrigation. Le but est d'étendre sa dénition et ses propriétés fondamentales au cas d'une source étendue, ce à quoi l'on parvient dans le cas d'un réseau possédant un système ni de racines, par exemple pour des mesures à supports disjoints. On donne une dénition satisfaisante de la fonction paysage dans ce cas, qui vérie en particulier la propriété de variation première et on démontre sa régularité Hölder sous des hypothèses raisonnables sur les mesures à connecter. Title : Branched transport and fractal structures Keys words : branched transport, optimal transport, calcul of variations, fractals, geometric measure theory Abstract : This thesis is devoted to the study of branched transport, related variational problems and fractal structures that are likely to arise. The branched transport problem consists in connecting two measures of same mass through a network minimizing a certain cost, which in our study will be proportional to mL α in order to move a mass m over a distance L. Several continuous models have been proposed to formulate this problem, and we focus on the two main static models : the Lagrangian and the Eulerian ones, with an emphasis on the rst one. After setting properly the bases for these models, we establish rigorously their equivalence using a Smirnov decomposition of vector measures whose divergence is a measure. Secondly, we study a shape optimization problem related to branched transport which consists in nding the sets of unit volume which are closest to the origin in the sense of branched transport. We prove existence of a solution, described as a sublevel set of the landscape function, now standard in branched transport. The Hölder regularity of the landscape function, obtained here without a priori hypotheses on the considered solution, allows us to obtain an upper bound on the Minkowski dimension of its boundary, which is non-integer and which we conjecture to be its exact dimension. Numerical simulations, based on a variational approximation a la Modica-Mortola of the branched transport functional, have been made to support this conjecture. The last part of the thesis focuses on the landscape function, which is essential to the study of variational problems involving branched transport as it appears as a rst variation of the irrigation cost. The goal is to extend its denition and fundamental properties to the case of an extended source, which we achieve in the case of networks with nite root systems, for instance if the measures have disjoint supports. We give a satisfying denition of the landscape function in that case, which satises the rst variation property and we prove its Hölder regularity under reasonable assumptions on the measures we want to connect.