Hamza Ennaji
email: hamza.ennaji@unilim.fr

Noureddine Igbida
email: noureddine.igbida@unilim.fr.

NOUREDDINE IGBIDA † Van Thanh Nguyen
email: nguyenvanthanh@qnu.edu.vn.

Thanh Van

Nguyen

Hamilton-Jacobi Quasi-Convex

Equations

QUASI-CONVEX HAMILTON-JACOBI EQUATIONS VIA LIMITS OF FINSLER p-LAPLACE PROBLEMS AS p → ∞

published or not. The documents may come

Introduction

Let Ω be a smooth bounded subset of R N . Consider a continuous Hamiltonian F : Ω×R N → R such that, for all x ∈ Ω,

• Z(x) := {ξ ∈ R N : F (x, ξ) ≤ 0} is a convex and compact subset of R N .

• 0 ∈ int(Z(x)). Our main aim concerns the Hamilton-Jacobi (HJ for short) equation of first order F (x, ∇u) = 0 in Ω.

(1.1)

The class of HJ PDE is central in several branches of mathematics, both from theoretical, numerical and application points of view. The applications in classical mechanics, optics, Hamiltonian dynamics, semi-classical quantum theory, Riemannian and Finsler geometry as well as the optimal control theory are very important.

In addition to its connection with Hamilton's equations, in the case where the Hamiltonian has sufficient regularity, further connection with common PDEs was established in the literature. For instance, it appears in the classical limit of the Schrödinger equation (see e.g. [START_REF] Bardos | On the classical limit of the Schrödinger equation[END_REF]). Its connection with the discount HJ equation λu + F (x, ∇u) = 0 as λ → 0 was established in the seminal paper [START_REF] Lions | Homogenization of Hamilton-Jacobi equation[END_REF] and generalized in [START_REF] Davini | Convergence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted solutions[END_REF]. The vanishing viscosity method for first order HJ equations establishes the connection of HJ equations with the second order PDE -∆u + F (x, ∇u) = 0 as → 0 (see for instance [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF][START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF]). The celebrated paper of Varadhan [START_REF] Varadhan | On the behavior of the fundamental solution of the heat equation with variable coefficients[END_REF] shows that the heat kernel in a Riemannian manifold can be approximated by a Gaussian kernel, and thus makes the link between the heat equation and the HJ equation. This connection can be also done via Hopf-Cole transformations as showed in [START_REF] Dolcetta | The Hopf solution of Hamilton-Jacobi equations[END_REF]. This kind of transformations also allows recovering the HJ equation in the large scale hyperbolic limit of a class of kinetic equation (see e.g. [START_REF] Bouin | A kinetic eikonal equation[END_REF]).

Recently, the connection between HJ equation, optimal mass transport and Beckmann's problem was established in [START_REF] Ennaji | Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations[END_REF][START_REF] Ennaji | Beckmann-type problem for degenerate Hamilton-Jacobi equations[END_REF] with a flavor of variational approach. In particular, these connections work out a nonlinear divergence-form PDE, called Monge-Kantorovich equation, that we can associate definitively with the HJ equation. The connection is not straightforward since the optimal mass transportation, the Beckmann's problem as well as the associate divergence formulation are not standard. Roughly speaking, the offset is connected to some unknown distribution of mass concentrated on the boundary which would both, counterbalance the involved optimal mass transportation phenomena and describe the normal-trace of the allowed flux in the divergence formulation (see [START_REF] Ennaji | Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations[END_REF][START_REF] Ennaji | Beckmann-type problem for degenerate Hamilton-Jacobi equations[END_REF] for the details). The approach blends sophisticated tools from variational analysis, convex duality and trace-like operator for the so called divergence-measure field. To strengthen the connection with divergence equation and to shape the "pretending diffusive taste" of HJ equation, we propose in this paper how to achieve the solutions of HJ equation using an elliptic PDE of Finsler p-Laplace type. The Finsler structure associated with the Hamiltonian F takes part in the PDE in a common way bringing out some kind of anisotropic p-Laplace PDE, that we call here Finsler p-Laplace equation. We treat the equation (1.1) with a double obstacle on the boundary. Moreover, thanks to the substantial link of HJ equation with the optimal mass transport as well as the Beckmann problem, these problems will be concerned in their turn with the approach using the Finsler p-Laplace equation.

To describe roughly the approach, we consider the peculiar case of eikonal equation with Dirichlet boundary condition:

   |∇u| = k in Ω u = g on ∂Ω, (1.2)
where k is a positive continuous function in Ω and ∂Ω denotes the boundary of Ω. It is well known by now that the intrinsic distance defined by

d k (x, y) := inf ζ∈Γ(x,y) 1 0 k(ζ(t)) | ζ(t)|dt,
where Γ(x, y) is the set of Lipchitz curves joining x and y, describes the maximal viscosity subsolution through the following formula

u(x) = min y∈∂Ω {d k (y, x) + g(y)} . (1.3)
Here g : ∂Ω → R is assumed to be a continuous function satisfying the compatibility condition g(x) -g(y) ≤ d k (y, x), for all x, y ∈ ∂Ω.

Since (1.3) is likewise the unique solution of the following maximization problem

max z∈W 1,∞ (Ω) Ω z(x)dx : |∇z(x)| ≤ k(x) and z = g on ∂Ω , (1.4)
we know (see [START_REF] Ennaji | Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations[END_REF][START_REF] Ennaji | Beckmann-type problem for degenerate Hamilton-Jacobi equations[END_REF]) that a dual problem of (1.4) reads

min φ∈M b (Ω) N , ν∈M b (∂Ω) Ω k d|φ| + ∂Ω gdν : -div(φ) = χ Ω -ν in D (R N ) , (1.5)
which constitute actually a new variant of Beckmann's problem with boundary cost g. Here M b is used to denote the set of finite Radon measures. In particular, this is connected to the Monge optimal mass transport problem

inf Ω d k (x, T (x))dx : ν ∈ M b (∂Ω), T χ Ω = ν
as well as to the Monge-Kantorovich relaxed problem

min Ω×Ω d k (x, y)dγ(x, y) : ν ∈ M b (∂Ω), γ ∈ M + (Ω × Ω), (π x ) γ = χ Ω , (π y ) γ = ν .
Even if here the so called target measure ν is an unknown parameter of the problem, one sees that the problem aims certainly an optimal mass transportation between ρ 1 := χ Ω and ρ 2 := ν, and moreover u, given by (1.3) (the unique solution of (1.2)) is an Kantorovich potential of transportation. Since the pioneering work of Evans-Gangbo (cf. [START_REF] Evans | Differential equations methods for the Monge-Kantorovich mass transfer problem[END_REF]) in the case where k ≡ 1, it is known that key information concerning u may be given by the uniform limit of u p , the solution of the modified p-Laplace equation

-∆ p u p k = ρ 1 -ρ 2 in Ω u = g on ∂Ω.
(

Following the results of [START_REF] Evans | Differential equations methods for the Monge-Kantorovich mass transfer problem[END_REF], one can guess this limit to be given by the so-called Monge-Kantorovich system:

   -div(Φ) = ρ 1 -ρ 2 , |∇u| ≤ k in Ω Φ = m ∇u, m ≥ 0, m(|∇u| -k) = 0 a.e. u = g on ∂Ω.
(1.7)

Notice here that, a part a few special cases out of the scope of our situation (cf. [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF] Chap. 4.3 for discussions and references about regularity properties of Φ under extra assumptions), in general the flux Φ is a vector valued measure, and it is closely connected to the solution of Beckmann problem (1.5). Coming back to the HJ equation (1.2), it is clear now that the Monge-Kantorovich system is a suitable divergence equation for the solution of (1.2). Moreover, the limit of the flux of (1.6) converges weakly to Φ picturing thereby some kind of "nonlinear diffusion" phenomena behind the Hamilton-Jacobi equation.

Contributions.

In this paper, we are interested in studying the connection between the HJ equation, coupled with inequality constraints on the boundary,

   F (x, ∇u) = 0 in Ω φ ≤ u ≤ ψ on ∂Ω (1.8)
and an elliptic problem of Finsler p-Laplace type that we will introduce below.

We show how to recover the maximal viscosity subsolution to the class of HJ equations of the type (1.8) using a family of Finsler p-Laplace problems (with boundary obstacles) as p → ∞. Moreover, since the solution of (1.8) is intimately linked to the so called Kantorovich-Rubinstein problem in optimal transport, an appropriate Beckmann's transportation problem is derived and its solution is provided. Essentially, this will be the content of Theorem 3.6 whose proof relies on the results and estimates of Propositions 2.2 and 3.5. Finally, we show in Proposition 4.12 that the limit as p → ∞ of solutions of the p-Laplace problems is a Kantorovich potential for a classical Kantorovich problem involving the normal trace on the boundary of the optimal flow of Beckmann's problem. Our work illustrates some kind of "nonlinear diffusion" phenomena behind the Hamilton-Jacobi equation.

Related works. Concerning limits as p → ∞ for the p-Laplace equations, one of the first mathematical studies is [START_REF] Bhattacharya | Limits as p → ∞ of δpup = f and related extremal problems[END_REF] with particular interest in torsional problems and ∞-harmonic functions, followed by the celebrated work of Evans and Gangbo [START_REF] Evans | Differential equations methods for the Monge-Kantorovich mass transfer problem[END_REF]. Similar problems were considered in [START_REF] García-Azorero | The Neumann problem for the ∞-Laplacian and the Monge-Kantorovich mass transfer problem[END_REF][START_REF] Garcia Azorero | Limits for Monge-Kantorovich mass transport problems[END_REF] for transport problems with masses supported on the boundary. Variants of Monge-Kantorovich problems with boundary costs were addressed in [START_REF] Mazón | An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary[END_REF] where the boundary costs can be seen as some import/export taxes. In the same spirit, similar results were obtained in [START_REF] Dweik | Weighted Beckmann problem with boundary costs[END_REF] with some weighted Euclidean distance as a cost. The use of PDE techniques à la Evans-Gangbo in the Finsler framework was addressed recently in [START_REF] Igbida | Optimal mass transportation for costs given by Finsler distances via p-Laplacian approximations[END_REF]. It is well known that Finsler metrics generalise the Riemannian ones and are of main interest in the study of optimal transport and minimal flow problems since they allow considering anisotropy, obstacles...

Our work adds to these series of papers linking HJ equations to other PDE's, thanks to the variational approach (cf. [START_REF] Ennaji | Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations[END_REF]) and permits generalizing the works on mass transport recalled above. It shows once again the flexibility of the Evans-Gangbo method.

The rest of this paper is organized as follows. In section 2, we present assumptions and preliminary results concerning the notion of solution to the HJ equation coupled with obstacles on the boundary under consideration, Finsler p-Laplace equations as well as their existence and characterization of solutions. In section 3, we derive suitable estimates independent of p and show the convergence of Finsler p-Laplace equations as p → ∞. The existence and characterization of solutions to the limited variational problems are also studied in detail. Finally, the connection between the limited variational problems and a variant of Monge-Kantorovich transportation problem is derived in section 4.

Preliminaries

Maximal viscosity subsolution.

Consider the Hamilton-Jacobi equation of first order, coupled with some inequality constraints on the boundary

F (x, ∇u) = 0 in Ω φ ≤ u ≤ ψ on ∂Ω.
(2.9)

Here, φ, ψ ∈ C(∂Ω) satisfy the compatibility condition φ(x) -ψ(y) ≤ d σ (y, x) for all x, y ∈ ∂Ω, with d σ being the intrinsic metric associated to F (see below).

For each x ∈ Ω, we define the support function σ(x, .) of the 0-sublevel set of F by

σ(x, q) = sup p∈Z(x)
p, q for all q ∈ R N , which turns to be a Finsler metric (see subsection 2.2 below). Then, the intrinsic distance associated to F is defined through

d σ (x, y) := inf ζ∈Γ(x,y) 1 0 σ(ζ(t), ζ(t))dt,
where Γ(x, y) is the set of Lipchitz curves joining x and y. In the case where φ ≡ ψ = g : ∂Ω → R is a continuous function satisfying the compatibility condition

g(x) -g(y) ≤ d σ (y, x) for all x, y ∈ ∂Ω,
it is well known (see e.g. [START_REF] Fathi | PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians[END_REF][START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF]) that the maximal viscosity subsolution of

F (x, ∇u) = 0 in Ω u = g on ∂Ω (2.10)
is given by

u(x) = min y∈∂Ω {d σ (y, x) + g(y)} . (2.11)
Moreover, this solution coincides with the maximal volume solution. Indeed, using the fact that the set of all viscosity subsolutions of (2.10) coincides with the set of Lipschitz functions u satisfying

σ * (x, ∇u(x)) ≤ 1 a.e.,
where σ * is the dual of the support function σ defined through

σ * (x, q) = sup σ(x,p)≤1
p, q , we proved in [START_REF] Ennaji | Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations[END_REF] that (2.11) is the unique solution of the following maximization problem

max z∈W 1,∞ (Ω) Ω z(x)dx, σ * (x, ∇z(x)) ≤ 1 and z = g on ∂Ω .
Now, for the study of the general problem (2.9) with inequality constraints on the boundary, we make use of a similar notion of solution. Actually we have Proposition 2.1. Under the assumption (2.19), the problem (2.9) has a unique solution u in the sense of maximal volume, that is, u is the unique solution to the following maximization problem

max z∈W 1,∞ (Ω) Ω z(x)dx, σ * (x, ∇z(x)) ≤ 1 and φ ≤ z ≤ ψ on ∂Ω .
Moreover, u is the maximal viscosity subsolution satisfying φ ≤ u ≤ ψ on ∂Ω.

Finsler p-Laplacian equation.

Let Ω be a bounded open subset of R N , a Finsler metric is a continuous function

H : Ω × R N → [0, ∞) such that H(x, .
) is convex, and positively 1-homogeneous in the second variable, that is, H(x, tp) = tH(x, p) for every t ≥ 0.

We define the dual of a Finsler metric H (which is also a Finsler metric) by

H * (x, q) = sup H(x,p)≤1 p, q = sup p =0
p, q H(x, p) .

In this paper, we assume that H is a non-degenerate Finsler metric, that is, there exist a, b > 0 such that a|p| ≤ H(x, p) ≤ b|p| (2.12) for all (x, p) ∈ Ω × R N . In other words, one has

ã|q| ≤ H * (x, q) ≤ b|q| (2.13)
for some ã, b > 0. Moreover, we have the Cauchy-Schwarz like inequality p, q ≤ H(x, p)H * (x, q).

(2.14) Euler's homogeneous function theorem (see e.g. [START_REF] Nesterov | Introductory lectures on convex optimization: A basic course[END_REF]) says that

∂ ξ H * (x, p) • p = H * (x, p) for any p ∈ R N , (2.15)
and by convexity of H * , we have

∂ ξ H * (x, p) • q ≤ H * (x, q) for any p, q ∈ R N .
Thus, using (2.13) we get

|∂ ξ H * (x, p) • q| ≤ b|q| for any p, q ∈ R N . (2.16)
Finally, we have

H(x, ∂ ξ H * (x, p)) = 1 for any p ∈ R N .
(2.17) For details and additional properties we refer the reader to [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF].

Every Finsler metric induces a Finsler distance via the so called length (or action) functional. The action of a Lipschitz curve ξ ∈ Lip([0, 1]; Ω) is defined through

A H (ξ) = 1 0 H(ξ(s), ξ(s))ds.
(2.18)

The induced distance d H by the action functional (2.18) reads as

d H (x, y) = inf ξ∈Γ(x,y) A H (ξ).
Note that in general, H(x, p) is not even in p so that d H may be non-symmetric, i.e., it may happen that d H (x, y) = d H (y, x).

Assuming that H * (x, .) ∈ C 1 (R N \ {0}) and the compatibility condition

φ(x) -ψ(y) ≤ d H (y, x) for all x, y ∈ ∂Ω, (2.19)
we consider the following Finsler (also called anisotropic) p-Laplace problems

   -div(H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p )) = ρ in Ω φ ≤ u p ≤ ψ on ∂Ω, (2.20)
where p > N and ρ ∈ L 2 (Ω) are given, and ∂ ξ H * stands for the derivative of H * with respect to the second variable. To study this problem let us consider the set

W φ,ψ = {u ∈ W 1,p (Ω) : φ ≤ u ≤ ψ on ∂Ω}
and we denote by

Θ p = H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ). Proposition 2.2. Assume (2.19) is strict, that is, φ(x) -ψ(y) < d H (y, x) for all x, y ∈ ∂Ω. (2.21)
The problem (2.20) has a unique solution u p in the following sense: u p ∈ W φ,ψ and

Ω Θ p • ∇(u p -ξ) dx ≤ Ω ρ (u p -ξ) dx for any ξ ∈ W φ,ψ . (2.22)
Moreover, the distribution defined through

Θ p • n, η = Ω Θ p ∇ηdx - Ω ηρdx, η ∈ D(R N ), (2.23)
is a Radon measure concentrated on ∂Ω which satisfies

Ω Θ p • ∇ηdx = Ω ηρdx + ∂Ω ηd(Θ p • n) for all η ∈ W 1,p (Ω), (2.24)
and supp((

Θ p • n) + ) ⊂ {u p = φ} and supp((Θ p • n) -) ⊂ {u p = ψ}. (2.25)
Proof. We consider the following minimization problem of Finsler p-Laplace type

min u∈W φ,ψ F p (u) := Ω H * (x, ∇u) p p dx - Ω uρdx. (2.26)
Observe that W φ,ψ is a closed, convex subset of W

supp (Θ p • n) ⊂ {x ∈ ∂Ω : u p (x) = φ(x)} ∪ {x ∈ ∂Ω : u p (x) = ψ(x)} . Take a test function η ∈ C ∞ (Ω) whose support is disjoint from {x ∈ ∂Ω : u p (x) = φ(x)} ∪ {x ∈ ∂Ω : u p (x) = ψ(x)}.
There exists some > 0 so that u p + tη remains admissible for (2.26) for |t| < , i.e., φ ≤ u p + tη ≤ ψ. By optimality of u p , we get the variational inequality

Ω Θ p • ∇(v -u p )dx ≥ Ω (v -u p )ρdx for all v ∈ W φ,ψ .
In particular, for v = u p + tη, we get

t Ω Θ p • ∇ηdx ≥ t Ω ηρdx.
This holds for positive and negative t, such that |t| ≤ . Consequently

Ω Θ p • ∇ηdx = Ω ηρdx.
In other words, Θ p • n, η = 0 and supp(Θ p • n) ⊂ {u p = φ} ∪ {u p = ψ}. We are now in a position to show that Θ p • n is actually a Radon measure. Indeed, the inequiality (2.21) implies that the two compact sets {x ∈ ∂Ω :

u p (x) = φ(x)} and {x ∈ ∂Ω : u p (x) = ψ(x)} are disjoint. There exist η 1 , η 2 ∈ D(R N ) such that η 1 (x) =
1 on {u p = φ}, 0 on {u p = ψ}, and η 2 (x) = 1 on {u p = ψ}, 0 on {u p = φ}.

Then we can write

Θ p • n = D 1 + D 2 ,
where D 1 , D 2 are distributions given by

D 1 , η = Θ p • n, ηη 1 and D 2 , η = Θ p • n, ηη 2 .
This being said, for any positive test function η, we have that supp(ηη 1 ) ∩ {u p = ψ} = ∅, and for 0 ≤ t < we have u p + t(ηη 1 ) ∈ W φ,ψ . Consequently

t Ω Θ p • ∇(ηη 1 )dx ≥ t Ω (ηη 1 )ρdx, i.e, D 1 , η ≥ 0.
(2.27) On the other hand, for any positive test function η, we have that supp(ηη 2 ) ∩ {u p = φ} = ∅ and for -< t ≤ 0, we have that u p + t(ηη 2 ) ∈ W φ,ψ . Consequently

t Ω Θ p • ∇(ηη 2 )dx ≥ t Ω (ηη 2 )ρdx.
In other words, D 2 , η ≤ 0.

(2.28) In conclusion, D 1 and -D 2 are positive distributions. Hence, they are positive Radon measures. It follows that the distribution Θ p • n is a Radon measure on ∂Ω. Moreover, (2.27) and (2.28) give (2.25).

Thanks to the proof of Proposition 2.2, we have the following description of the solution.

Corollary 2.3. If H * (x, .) ∈ C 1 (R N \ {0}), then u p is the unique solution of the problem            -div(H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p )) = ρ in Ω H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ) • n ≥ 0 on {u p = φ} H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ) • n ≤ 0 on {u p = ψ} H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ) • n = 0 in {φ < u p < ψ} φ ≤ u p ≤ ψ on ∂Ω, (2.29)
where n is the exterior normal to the boundary ∂Ω, in the sense that

u p ∈ W φ,ψ , Θ p ∈ L p (Ω) N , Θ p • n ∈ M b (∂Ω)
, and the triplet (u p , Θ p , Θ p • n) satisfies (2.24)-(2.25).

Remark 2.4. In order to simplify the presentation we have assumed that H * (x, .) ∈ C 1 (R n \ {0}). However, we do believe that all the results of this paper remain true without this assumption and one needs just to replace the derivative of H * with respect to the second variable by the subdifferential.

Limits of Finsler p-Laplacian as p → ∞

The strategy is to obtain some uniform bounds in p of ∇u p , then we show that the triplet (u p , Θ p , Θ p • n) converges (up to a subsequence) to optimal solutions of the corresponding Kantorovich-Rubinstein and Beckmann-type problems. The following result gathers main estimates, that we will need later.

Ω H * (x, ∇u p ) p p dx - Ω u p ρdx ≤ Ω H * (x, ∇v) p p dx - Ω vρdx ≤ |Ω| p - Ω vρdx. (3.33)
Thanks to Theorem 2.E in [START_REF] Talenti | Inequalities in rearrangement invariant function spaces[END_REF], there is a Morrey-type inequality independent of p u L ∞ (Ω) ≤ C Ω ∇u L p (Ω) for any u ∈ W 1,p 0 (Ω), p > N + 1, where the constant C Ω does not depend on p and u. Observing that we can apply the above inequality to (u p -max ∂Ω ψ) + and (u p -min ∂Ω φ) -which are in W 1,p 0 (Ω) to obtain

u + p L ∞ (Ω) ≤ C Ω ∇u p L p (Ω) + | max ∂Ω ψ|, and u - p L ∞ (Ω) ≤ C Ω ∇u p L p (Ω) + | min ∂Ω φ|. So u p L ∞ (Ω) ≤ C 1 ∇u p L p (Ω) + C 2 .
From (3.33) and the preceding inequality we deduce that

Ω H * (x, ∇u p ) p p dx ≤ |Ω| p - Ω vρdx + Ω u p ρdx ≤ C 3 (1 + ∇u p L p (Ω) ),
where C 3 is a positive constant not depending on p. Combining this with (2.12), we get

H * (x, ∇u p ) p L p (Ω) ≤ C 4 p(1 + H * (x, ∇u p ) L p (Ω) ) which implies that H * (x, ∇u p ) L p (Ω) ≤ (C 5 p) 1 p-1 (3.34)
for some constant C 5 independent from p. Again, by (2.12), we get Now, let us prove (ii). We consider as before v(x) = min y∈∂Ω ψ(y) + d H (y, x). We have

∇u p L p (Ω) ≤ C 6 . ( 3
∂Ω (u p -v)d(Θ p • n) = Ω Θ p • ∇(u p -v)dx - Ω (u p -v)ρdx.
In other words

Ω (u p -v)ρdx = Ω Θ p • ∇(u p -v)dx + {up=ψ} (ψ -v)d(Θ p • n) -- {up=φ} (φ -v)d(Θ p • n) + .
We see that φ < v ≤ ψ on ∂Ω so that ψ -v ≥ 0 and φ -v < 0, thus φ -v < -C 1 for some positive constant C 1 . So we obtain

Ω Θ p • ∇u p dx + C 1 ∂Ω d(Θ p • n) + ≤ Ω (u p -v)ρdx + Ω Θ p • ∇vdx. (3.37)
Since H * is a Finsler metric, we have by Euler's homogeneous function theorem (see e.g. [START_REF] Nesterov | Introductory lectures on convex optimization: A basic course[END_REF])

that ∂ ξ H * (x, ξ) • ξ = H * (x, ξ) for any ξ ∈ R N . Thus Ω Θ p • ∇u p dx = Ω H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ) • ∇u p dx = Ω H * (x, ∇u p ) p dx.
Using this fact in (3.37), we get

Ω H * (x, ∇u p ) p dx + C 1 ∂Ω d(Θ p • n) + ≤ C 2 + Ω Θ p • ∇vdx,
where C 2 > 0 is independent from p. On the other hand, thanks to (2.14) we have

Ω Θ p • ∇vdx ≤ Ω H(x, Θ p )H * (x, ∇v)dx = Ω H(x, H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ))H * (x, ∇v)dx = Ω H * (x, ∇u p ) p-1 H(x, ∂ ξ H * (x, ∇u p ))H * (x, ∇v)dx = Ω H * (x, ∇u p ) p-1 H * (x, ∇v)dx,
where we have used the homogeneity of H and (2.17). Using Hölder and Young's inequalities and the fact that H * (x, ∇v) ≤ 1 a.e., we get

Ω H * (x, ∇u p ) p-1 H * (x, ∇v)dx ≤ Ω H * (x, ∇u p ) (p-1)p dx 1 p |Ω| 1 p ≤ p -1 p Ω H * (x, ∇u p ) p dx + 1 p |Ω|.
We deduce that

1 p Ω H * (x, ∇u p ) p dx + C 1 ∂Ω d(Θ p • n) + ≤ C 2 + 1 p |Ω|. Therefore ∂Ω d(Θ p • n) + ≤ C 3 (3.38)
for some positive constant C 3 independent of p. Set w(x) = max y∈∂Ω φ(y) -d H (y, x). Observe that φ ≤ w < ψ and following the same lines we get that

∂Ω d(Θ p • n) -≤ C 4 . (3.39)
As for Θ p , we have

Ω H * (x, ∇u p ) p dx = Ω Θ p • ∇u p dx = ∂Ω u p d(Θ p • n) + Ω u p ρdx.
Keeping in mind (3.38) and (3.39), Hölder's inequality gives

Ω H * (x, ∇u p ) p-1 dx ≤ C 5 ,
this proves (iii).

Thanks to Proposition 3.5, we can state the main result.

Theorem 3.6. Let u p be a minimizer of F p . Then, up to a subsequence, u p ⇒ u on Ω, where u solves the following variant of Kantorovich-Rubinstein problem

(KR) H : max Ω udρ : H * (x, ∇u) ≤ 1 a.e., φ ≤ u ≤ ψ on ∂Ω .
Moreover, there exists a couple (Θ, θ) ∈ M b (Ω) N × M b (∂Ω), such that (i) Up to a subsequence

(Θ p , Θ p • n) (Θ, θ) in M b (Ω) N × M b (∂Ω) -weak * .
(ii) (Θ, θ) solves the Beckmann problem

(B) H : min Φ∈M b (Ω) N ν∈M b (∂Ω) Ω H(x, Φ |Φ| )d|Φ| + ∂Ω ψdν -- ∂Ω φdν + : -div(Φ) = ρ + ν in D (R N ) .
(iii) The couple (u, Θ) solves the PDE

           -div(Θ) = ρ in Ω Θ(x) • ∇u(x) = H (x, Θ) in Ω φ ≤ u ≤ ψ on ∂Ω, (3.40)
in the following sense: Proof. The case where the inequality (2.19) is strict.

(u, Θ) ∈ W φ,ψ × M b (Ω) N , Θ • n = θ ∈ M b (∂Ω), Θ |Θ| • ∇ |Θ| u = H ., Θ |Θ| ,
First, we see that thanks to (3.30), we have by Ascoli-Arzelà's theorem, up to a subsequence, u p ⇒ u on Ω for some continuous function

u satisfying φ ≤ u ≤ ψ on ∂Ω. It is clear that u ∈ W 1,∞ (Ω).
We are now in a position to show that u solves (KR) H . To do so, we take any v ∈ W φ,ψ such that H * (x, ∇v(x)) ≤ 1 a.e.. Using the optimality of u p we see that

- Ω u p ρdx ≤ F p (u p ) ≤ F p (v) ≤ |Ω| p - Ω vρdx.
Taking the limit up to a subsequence, we get

sup Ω vρdx : H * (x, ∇v) ≤ 1, a.e., φ ≤ v ≤ ψ on ∂Ω ≤ Ω uρdx.
It remains to show that u is 1-Lipschitz with respect to d H , that is, H * (x, ∇u(x)) ≤ 1 a.e.. Recall that φ ≤ u ≤ ψ on ∂Ω. Again, using (3.34), we consider N < m ≤ p and we use Hölder's inequality to get

H * (x, ∇u p ) L m (Ω) ≤ (C 5 p) 1 p-1 |Ω| p-m
pm . Since u p ⇒ u uniformly in Ω, we can assume that up to a subsequence u p u weakly in W 1,m (Ω), and particularly, ∇u p ∇u weakly in L m (Ω, R N ). Mazur's lemma (see [START_REF] Ekeland | Convex analysis and variational problems[END_REF] for example) ensures the existence of a convex combination of ∇u p k converging in norm toward ∇u. More precisely, there exists {U i } such that

U i = n i k=i α i k ∇u p k
where

n i k=i α k i = 1, and α i k ≥ 0, i ≤ k ≤ n i and U i -∇u L m (Ω) → 0 as i → +∞. Since H * is continuous, we have H * (x, ∇u) L m (Ω) ≤ lim inf i→∞ H * (x, n i k=i α i k ∇u p k ) L m (Ω) ≤ lim inf i→∞ n i k=i α i k H * (x, ∇u p k ) L m (Ω) ≤ lim inf i→∞ n i k=i α i k (C 5 p k ) 1 p k -1 |Ω| p k -m mp k = |Ω| 1 m .
Taking m → ∞, we get H * (x, ∇u(x)) ≤ 1, a.e. x ∈ Ω. On the other hand, we see that (

(Ψ, ν) ∈ M b (Ω) N × M b (∂Ω) for (B) H . Since H * (x, ∇v) ≤ 1 for a.e. x ∈ Ω, we have Ω H(x, Ψ |Ψ| )d|Ψ| ≥ Ω H(x, Ψ |Ψ| )H * (x, ∇v)d|Ψ| ≥ Ω Ψ |Ψ| ∇vd|Ψ| ≥ Ω vdρ + ∂Ω φdν + - ∂Ω ψdν -
and consequently

Ω H(x, Ψ |Ψ| )d|Ψ| + ∂Ω ψdν -- ∂Ω φdν + ≥ Ω vdρ.
In particular, this implies that min(B) H ≥ max(KR) H . On the other hand, using Hölder's inequality combined with (2.15)-(2.16), we get

Ω H(x, Θ |Θ| )d|Θ| ≤ lim inf p Ω H x, H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ) dx = lim inf p Ω H * (x, ∇u p ) p-1 H(x, ∂ ξ H * (x, ∇u p ))dx ≤ lim inf p Ω H * (x, ∇u p ) p dx p-1 p = lim inf p Ω H * (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ) • ∇u p dx p-1 p = lim inf p Ω ∇u p dΘ p p-1 p = lim inf p Ω u p ρdx + ∂Ω u p d(Θ p • n) p-1 p = Ω uρdx + ∂Ω φdθ + - ∂Ω ψdθ -.
This implies that

min(B) H ≤ Ω H(x, Θ |Θ| )d|Θ| - ∂Ω φdθ + + ∂Ω ψdθ -≤ Ω uρdx = max(KR) H . Thus min(B) H = Ω H(x, Θ |Θ| )d|Θ| - ∂Ω φdθ + + ∂Ω ψdθ -= Ω uρdx = max(KR) H ,
which implies the optimality of u and (Φ, θ). Now it remains to show the results for the general case where the inequality (2.19) needs not to be strict.

We proceed by approximations. Consider two sequences {φ n } n and {ψ n } n of continuous functions on ∂Ω such that φ n (x) -ψ n (y) < d H (y, x) for all x, y ∈ ∂Ω, and φ n ⇒ φ and ψ n ⇒ ψ on ∂Ω. Then, thanks to the previous case, there exists a sequence of {u n } n ∈ W φn,ψn such that H * (x, ∇u n ) ≤ 1 a.e Ω. In addition, consider the corresponding solutions to the Beckmann problem (Θ n , θ n ). We then have

Ω u n dρ = Ω H(x, Θ n |Θ n | )d|Θ n | - ∂Ω φ n dθ + n + ∂Ω ψ n dθ - n = min(B) H . (3.42)
Then we deduce by the previous arguments that u n ⇒ u uniformly in Ω with H * (x, ∇u) ≤ 1 a.e. and φ ≤ u ≤ ψ in ∂Ω.

Next, we follow the main ideas of the proof of Proposition 3.5. Define v n (x) = min y∈∂Ω {ψ n (y) + d H (y, x)}.

Then

Ω Θ n • ∇u n dx + C 1 ∂Ω dθ + n ≤ Ω (u n -v n )ρdx + Ω Θ n • ∇v n dx, (3.43)
where C 1 is a positive constant independent from n. Using (3.40), we have

Ω Θ n • ∇u n dx = Ω H(x, Θ n |Θ n | )d|Θ n |.
On the other hand, since H * (x, ∇v n (x)) ≤ 1 a.e, we get

Ω Θ n • ∇v n dx ≤ Ω H(x, Θ n )H * (x, ∇v n )dx ≤ Ω H(x, Θ n |Θ n | )d|Θ n |.
Combining these facts in (3.43), and using (2.12) we get

∂Ω dθ + n ≤ C, with C > 0. (3.44)
Similarly, working with

w n (x) = max y∈∂Ω φ n (y) -d H (y, x) instead of v n , we get ∂Ω dθ - n ≤ C, with C > 0. (3.45)
As for Θ n , we deduce from (2.12), (

Ω |Θ n |dx ≤ C. 3.42), (3.44) and (3.45) that
Then, up to a subsequence, (Θ n , θ n ) (Θ, θ) weakly* as n → ∞. Thus, passing to the limit in (3.42), the proof is complete.

Finally, for the proof of the last item (iii), by passing to the limit, we recover the conditions supp(θ + ) ⊂ {u = φ} and supp(θ -) ⊂ {u = ψ},

and

Ω Θ • ∇η dx = Ω ηρ dx + ∂Ω η dθ for all η ∈ W 1,∞ (Ω).
The equation

Θ |Θ| • ∇ |Θ| u = H ., Θ |Θ| , |Θ| -a.e. in Ω
is due to the optimality of u and Φ (see for example [START_REF] Igbida | Optimal partial mass transportation and obstacle Monge-Kantorovich equation[END_REF][START_REF] Nguyen | Monge-Kantorovich equation for degenerate Finsler metrics[END_REF]).

By uniqueness of the maximal viscosity subsolution of (2.9) we easily deduce the following corollary.

Corollary 3.7. Let H = σ, with σ being the support function of the 0-sublevel sets of the Hamiltonian F in (2.9). Then the whole sequence {u p } p converges uniformly to the solution u of (2.9). Now let us state the PDE satisfied by the potential u and the flow Θ, which in particular will give a characterization of the HJ equation (2.9). Proposition 3.8. The couple (u, Θ) given by Theorem 3.6 is a solution of the following PDE

           -div(Θ) = ρ in Ω Θ ∈ ∂II B H * (x,.) (∇u) in Ω φ ≤ u ≤ ψ on ∂Ω,
in the sense that:

(u, Θ) ∈ W φ,ψ × M b (Ω) N , Θ • n = θ ∈ M b (∂Ω), Θ ∈ ∂II B H * (x,.) (∇ |Θ| u), |Θ| -a.e. in Ω, supp(θ + ) ⊂ {u = φ} and supp(θ -) ⊂ {u = ψ},
and

Ω Θ • ∇η dx = Ω ηρ dx + ∂Ω η dθ for all η ∈ W 1,∞ (Ω).
In particular, taking H = σ, with σ being the support function of the 0-sublevel sets of the Hamiltonian F , the maximal viscosity subsolution u of (2.9) is uniquely characterized by the existence of Θ ∈ M b (Ω) N such that the couple (u, Θ) is a solution of the PDE

           -div(Θ) = 1 in Ω Θ ∈ ∂II Z(x) (∇u) in Ω φ ≤ u ≤ ψ on ∂Ω.
Proof. The divergence and boundary constraints follow from Theorem 3.6 and

Θ ∈ ∂II B H * (x,.) (∇ |Θ| u)
is recovered by (3.41).

For general H, it is labyrinthine to phrase the flow Θ explicitly in terms of the gradient of the potential u and the transport density alike Evans-Gangbo like formula in (1.7). The following result points out two particular situations showing how this is possible. and, we assume moreover that

H * (x, ∇ ω u) ≤ 1 ω -a.e. x ∈ Ω.
(3.47)

Then Θ = ω ∂ ξ H * (x, ∇ ω u),
and

H * (x, ∇ ω u) = 1 ω -a.e. x ∈ Ω. Proof. See that ∇ |Θ| u = ∇ ω u and H x, dΘ dω = 1 ω -a.e. Ω.
So, in one hand, using the fact that

∇ |Θ| u • Θ |Θ| = H x, Θ |Θ| |Θ| -a.e. Ω.
we have

∇ ω u • dΘ dω = ∇ |Θ| u • dΘ dω = 1 ω -a.e. Ω.
On the other, we see that

∇ ω u • dΘ dω ≤ H * (x, ∇ ω u) H x, dΘ dω = H * (x, ∇ ω u) ω -a.e. Ω.
So, assuming (3.47), we get

1 = ∇ ω u • dΘ dω = H * (x, ∇ ω u) H x, dΘ dω = H * (x, ∇ ω u) ω -a.e. Ω.
Thus the results follow by definition of H * .

Remark 3.11. Combining Theorem 3.6 and Corollaries 3.9-3.10, the couple (ω := H(x, Θ), u) solves the associated Monge-Kantorovich system to (KR) H and (B) H :

                   -div(ω∂ ξ H * (x, ∇ ω u)) = ρ in Ω ∂ ξ H * (x, ∇ ω u) • n ≥ 0 on {u = φ} ∂ ξ H * (x, ∇ ω u) • n ≤ 0 on {u = ψ} ∂ ξ H * (x, ∇ ω u) • n = 0 in {φ < u < ψ} φ ≤ u ≤ ψ on ∂Ω H * (x, ∇ ω u) ≤ 1 in Ω H * (x, ∇ ω u) = 1 ω -a.e.
(3.48)

In particular, given a positive continuous function k : Ω → R, and define the following Finsler metric H(x, p) = k(x)|p| for (x, p) ∈ Ω × R N . We easily see that its dual reads

H * (x, q) = |q| k(x) ,
and the systems (2.29)-(3.48) reduce the ones studied in [START_REF] Dweik | Weighted Beckmann problem with boundary costs[END_REF]. Moreover, if the Finsler metric is defined via the so called Minkowski functional (or gauge function)

g K (p) = inf{t > 0 : t -1 p ∈ K}, where K is a convex, closed and bounded set R N , then considering H * (x, p) = g K (p) and φ = ψ, we recover the Monge-Kantorovich system studied in [START_REF] Crasta | A nonhomogeneous boundary value problem in mass transfer theory[END_REF].

Connection with Monge-Kantorovich problem

Let us recall that we can derive a dual problem to (KR) H using perturbation techniques (as in [START_REF] Dweik | Weighted Beckmann problem with boundary costs[END_REF][START_REF] Ennaji | Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations[END_REF]), to get the following Kantorovich problem (K) H : min

γ∈Π(ρ + ,ρ -) Ω×Ω d H (x, y)dγ(x, y) + ∂Ω ψ(y)d(π y ) γ - ∂Ω φ(x)d(π x ) γ .
Here Π(ρ + , ρ -) = {γ ∈ M + (Ω × Ω) : (π x ) γ Ω = ρ + , (π y ) γ Ω = ρ -}, with π x and π y stand for the usual projections of Ω × Ω onto Ω, that is π x (x, y) = x and π y (x, y) = y for any (x, y) ∈ Ω × Ω and

(π x ) γ Ω = ρ + ⇔ γ(A × Ω) = ρ + (A) for any Borelean A ⊂ Ω, (π y ) γ Ω = ρ -⇔ γ(Ω × B) = ρ -(B)
for any Borelean B ⊂ Ω. The existence of optimal solution to (K) H can be obtained using the direct method of calculus of variations. Moreover, all the extremal values coincide:

min(B) H = min(K) H = max(KR) H . ( 4

.49)

Here φ and ψ play the role of import/export costs for the Kantorovich problem (K) as in [START_REF] Dweik | Weighted Beckmann problem with boundary costs[END_REF][START_REF] Mazón | An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary[END_REF] for the Euclidean and Riemannian costs. In addition, we show that the measure θ constructed in Theorem 3.6 will add to the measure ρ so that the potential u will be a Kantorovich potential for the classical transport problem on Ω between µ := ρ + L N Ω + θ + and ν :

= ρ -L N Ω + θ -, that is Ω ud(µ -ν) = min γ∈Γ(µ,ν) Ω×Ω d H (x, y)dγ(x, y),
where Γ(µ, ν) := {γ ∈ M + (Ω × Ω) : (π x ) γ = µ, (π y ) γ = ν} denotes the set of transport plans from µ to ν on Ω.

Proposition 4.12. Let u be the limit of the family of Finsler p-Laplace problems constructed in Theorem 3.6. Then u is a Kantorovich potential for the classical optimal transport problem between ρ + L N Ω + θ + and ρ -L N Ω + θ -. Moreover

Ω uρdx = min(K) H .
Proof. In the definition of Θ p • n in (2.23), we take as a test function η = u to get

∂Ω ud(Θ p • n) = Ω Θ p • ∇udx - Ω uρdx.
Thanks to Theorem 3.6, passing to the limit p → ∞ (up to a subsequence) we get

lim p→∞ Ω Θ p • ∇udx = ∂Ω udθ + Ω uρdx. ( 4
(u -w )d(Θ p • n) = Ω Θ p • (∇u -∇w )dx - Ω (u -w )ρdx.
Taking p → ∞ (again, up to a subsequence) and keeping in mind (4.50), we get as → 0. We first observe that

Ω uρdx + ∂Ω udθ = Ω (u -w )ρdx + ∂Ω (u -w )dθ + Ω Θ • ∇w dx = A + B , ( 4
Ω uρdx = lim →0 Ω w ρdx ≤ lim →0 Ω ∇w Θ |Θ| d|Θ| + ∂Ω ψdθ -- ∂Ω φdθ + ≤ lim →0 Ω H * (x, ∇w )H(x, Θ |Θ| )d|Θ| + ∂Ω ψdθ -- ∂Ω φdθ + ≤ Ω H(x, Θ |Θ| )d|Θ| + ∂Ω ψdθ -- ∂Ω φdθ +
where we have used Lemma 5.14 for the last inequality. Again we proceed as in the proof of Theorem To conclude, let us observe that taking v ∈ W 1,∞ (Ω) such that H * (x, ∇v(x)) ≤ 1, we have

Appendix

Let us recall some facts concerning the notion of tangential gradient which played an important role in the previous proofs. To give a glimpse on the necessity to introduce this notion, let us remember that Beckmann's transportation problem is an optimisation problem on measure space under a divergence constraint. More particularly, the flow satisfies -div(Φ) = µ ∈ M b (Ω). To do further analysis on such a problem and particularly to derive its dual problem we naturally attempt to integrate by parts in the divergence constraint and write, for some Lipschitz function u ∇u • σ dγ = udµ, where γ = |Φ| and σ = Φ |Φ| . Observe that ∇u may not be well-defined on a |Φ|-positive measure set and thus the previous formula may not have sense. Thanks to [START_REF] Bouchitte | Energies with respect to a measure and applications to low dimensional structures[END_REF] it is possible to give a sense to the previous formula using the notion of tangential gradient as follows. First we can define the tangent space to the measure γ

X γ (x) = γ -ess ∪ σ(x) : σ ∈ L 1 γ (Ω, R N ), div(σγ) ∈ M b (Ω) .
Then, the tangential gradient ∇ γ u(x) to a function u ∈ C 1 (Ω) at x with respect to the measure γ is the orthogonal projection of ∇u(x) onto X γ (x). Denoting by P γ (x) the orthogonal projection on X γ (x), it has been shown in [START_REF] Bouchitté | Completion of the space of measures in the Kantorovich norm[END_REF] that the linear operator u ∈ C 1 (Ω) → ∇ γ u(x) := P γ (x)∇u(x) ∈ L ∞ γ (Ω, R N ) can be uniquely extended to a linear continuous operator ∇ γ : u ∈ Lip(Ω) → ∇ γ u ∈ L ∞ γ (Ω, R N ). Moreover, we have the following useful integration by parts formula To end this section let us recall the following useful approximation result [START_REF] Igbida | Augmented Lagrangian method for optimal partial transportation[END_REF]Lemma A.1] (see also [START_REF] Nguyen | Monge-Kantorovich equation for degenerate Finsler metrics[END_REF]Lemma 3.1] for degenerate case of H). Lemma 5.14. Let H be a non-degenerate Finsler metric and u ∈ W 1,∞ (Ω) such that H * (x, ∇u(x)) ≤ 1 for a.e. x ∈ Ω. Then, there exists a sequence of u ∈ C 1 (Ω) such that u ⇒ u uniformly on Ω as → 0 and H * (x, ∇u (x)) ≤ 1 for all x ∈ Ω.

  .35) Now take some N < m ≤ p. Then by Hölder's inequality ∇u p L m (Ω) ≤ |Ω| p-m pm ∇u p L p (Ω) . (3.36) Thanks to (3.35), (3.36) and the Morrey-Sobolev embedding from W 1,m (Ω) to Hölder spaces, |u p (x) -u p (y)| ≤ C 7 |x -y| 1-α with α = N m .

  |Θ| -a.e. in Ω, (3.41) supp(θ + ) ⊂ {u = φ} and supp(θ -) ⊂ {u = ψ}, and Ω Θ • ∇η dx = Ω ηρ dx + ∂Ω η dθ for all η ∈ W 1,∞ (Ω).

  3.32) and (3.31) implies that Θ p and Θ p • n are bounded in M b (Ω) and M b (∂Ω) respectively. As a consequence, there exists Θ ∈ M b (Ω) N and θ ∈ M b (∂Ω) such that up to a subsequence Θ p Θ weakly* as p → ∞, and Θ p • n θ weakly* as p → ∞. Next, take any admissible potential v ∈ C 1 (Ω) for (KR) H and an admissible couple of flows

Corollary 3 . 9 .

 39 Let (u, Θ) be a solution of the PDE (3.40) in the sense of Theorem 3.6. If |Θ| L N , then, setting ω := H(x, Θ), (3.46) we have Θ = ω ∂ ξ H * (x, ∇u) L Na.e. x ∈ Ω, and ω (H * (x, ∇u) -1) = 0 L Na.e. x ∈ Ω. Proof. If |Θ| L N , then ∇ |Θ| u = ∇u, L Na.e.in Ω, and by taking ω as in (3.46), the relationship (3.41) implies that Θ • ∇u = ω L Na.e. in Ω. Since, moreover H * (x, ∇u) ≤ 1, then by definition of H * , we get Θ = ω ∂ ξ H * (x, ∇ ω u) and ω (H * (., ∇ ω u) -1) = 0, L Na.e. in Ω. Corollary 3.10. Let (u, Θ) be a solution of (3.40) in the sense of Theorem 3.6. We set again ω := H(x, Θ)

  .51) with A = Ω (u -w )ρdx + ∂Ω (u -w )dθ and B = Ω Θ • ∇w dx. Since w converges uniformly to u on Ω, we have that A → 0 as → 0. We claim that

  x, y)dγ(x, y), where γ is an optimal plan ofmin Ω×Ω d H (x, y)dγ(x, y) : (π x ) γ = ρ + L N Ω + θ + , (π y ) γ = ρ -L N Ω + θ -.Since (π x ) γ ∂Ω = θ + and (π y ) γ ∂Ω = θ -we deduce thatΩ uρdx = Ω×Ω d H (x, y)dγ(x, y) + ∂Ω ψdθ --∂Ω φdθ + = min(K) H .

Proposition 5 .

 5 13 ([4]).Given γ ∈ M + b (Ω) and υ ∈ L 1 γ (Ω, R N ) such that υ(x) ∈ X γ (x)for γ-a.e x. and div(γυ) := ρ ∈ M b (Ω). One then has Ω udρ = Ω υ∇ γ udγ, for any u ∈ Lip(Ω).

  It follows that the Θ p • n defined by (2.23) is a distribution supported on ∂Ω. Let us show moreover that

1,p

(Ω). The functional F p is coercive, strictly convex and lower semicontinuous on W φ,ψ . Therefore F p admits a unique minimizer on W φ,ψ which satisfies

(2.22)

. Now, to prove (2.24) we follow the main ideas of

[START_REF] Mazón | An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary[END_REF] Thereom 3.4

]. Clearly, (2.22) implies -div(Θ p ) = ρ in D (Ω).

  Proposition 3.5 (Main estimates). Assume (2.19) is strict, that is, φ(x) -ψ(y) < d H (y, x) for all x, y ∈ ∂Ω.

	Then, we have
	(i) estimate on u

p |u p (x) -u p (y)| ≤ C|x -y| r , for all x, y ∈ Ω;

(3.30)

(ii) estimates on Θ p • n: ∂Ω d(Θ p • n) + ≤ C 1 ,

and ∂Ω d(Θ p • n) -≤ C 2 ; (3.31) (iii) estimate on Θ p : Ω |Θ p |dx ≤ C, (3.32) where r, C, C 1 , C 2 are positive constants independent from p. Proof. First, we prove (i). Define v(x) = min y∈∂Ω ψ(y) + d H (y, x). Regarding the compatibility condition (2.19), we have φ ≤ v ≤ ψ on ∂Ω. It is not difficult to see that v is 1-Lipschitz with respect to d H and equivalently (see e.g. [13, Proposition 2.1]), we have that H * (x, ∇v(x)) ≤ 1 a.e. in Ω. Using the fact that u p is a minimizer of F p , we have

  3.6: since Θ p Θ, we have by Reshetnyak's lower semicontinuity theorem, we get (x, ∇u p) p-1 ∂ ξ H * (x, ∇u p ) dx (x, ∇u p ) p-1 H(x, ∂ ξ H * (x, ∇u p ))dx (x, ∇u p ) p dx (x, ∇u p ) p-1 ∂ ξ H * (x, ∇u p ) • ∇u p dx

	Ω	H(x,	Θ |Θ|	)d|Θ| ≤ lim inf p	Ω	H(x,	Θ p |Θ p |	)d|Θ p |
				= lim inf p H x, H = lim inf Ω p Ω H ≤ lim inf Ω p H p-1 p
				= lim inf p		Ω	H p-1 p
									p-1
				= lim inf p		Ω	∇u p dΘ p	p
				=	uρdx +	udθ
				Ω				∂Ω
				= lim →0 Ω	w ρdx +

* * * * ∂Ω w dθ where we have used Hölder's inequality combined with (2.15) and (2.17). Coming back to (4.51) we get Ω uρdx + ∂Ω udθ = Ω H(x, Θ |Θ| )d|Θ|.


domain: sde sdv

text:
Caroline Djian Caporalino

Ariane Fazari

Nathalie Marteau

Marie Offroy-Chave

Phytopathologia Mediterranea

Melvin Rodriguez-Heredia
email: m.rodriguez-heredia@qmul.ac.uk

Caroline Djian-Caporalino

Michel Ponchet

Laurent Lapeyre

Renaud Canaguier

Ari- Ane Fazari

Nathalie Marteu

Benoit Industri

Marie Offroy- Chave

Protective effects of mycorrhizal association in tomato and pepper against Meloidogyne incognita infection, and mycorrhizal networks for early mycorrhization of low mycotrophic plants

Keywords: Crop pests, symbiosis, agricultural management

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Phytoparasite nematodes are part of the soil microfauna with life cycles that are totally or partially within plants. Root-knot nematodes (RKNs) of the genus Meloidogyne are obligate endoparasites affecting a large number of plant species [START_REF] Sasser | A world prospective on nematology: the role of the society[END_REF]. They form characteristic galls in roots and block host plant conductive tissues, causing moisture stress [START_REF] Meon | Water relations of tomato (Lycopersicon esculentum Mill. Cv. Early Dwarf Red) infected with Meloidogyne javanica (Treub), Chitwood[END_REF], poor root development and growth, and significantly reduce crop productivity [START_REF] Hussey | Host-parasite relationships and associated physiological changes[END_REF][START_REF] Melakeberhan | The phenology of plant-nematode interaction and yield loss[END_REF].

RKNs are present in most farmlands, and can infect more than 5500 plant species, including vegetables, fruit crops, cereals and ornamentals [START_REF] Blok | Parasitism genes and host range disparities in bio-Root knot nematode infection in mycorrhized Solanaceae trophic nematodes: The conundrum of polyphagy versus specialisation[END_REF]. Economic losses due to RKN are estimated at tens of billions of euros per year [START_REF] Jones | Top 10 plant-parasitic nematodes in molecular plant pathology: Top 10 plant-parasitic nematodes[END_REF]. For example, more than 40% of economic crops in the southeast of France are affected by Meloidogyne spp. [START_REF] Djian-Caporalino | Nématodes à galles, des ravageurs de plus en plus préoccupants. Résultats de 3 ans d'enquête dans quinze régions françaises[END_REF][START_REF] Djian-Caporalino | Root-knot nematodes (Meloidogyne spp.), a growing problem in French vegetable crops: Root-knot and cyst nematodes in France[END_REF]. The usual methods for controlling RKNs included use of bromide/chloride/phosphorus-based products which are very concerning for the environment and human health, and therefore have been phased out in the European Union since 2009 (Council of the European Union, 2009). Physical approaches include prophylaxis, steam disinfection and solarisation, which are not always effective. Biological approaches for controlling RKNs are based on resistant plant varieties, nematode parasitic bacteria, toxins from nematicidal plants, biofumigation from plant oil cakes, and fungi that alter nematode life cycles [START_REF] Djian-Caporalino | Gestion des nématodes à galles : lutte conventionnelle et luttes alternatives. L'atout des plantes pièges[END_REF].

Arbuscular mycorrhizal fungi (AMF) are obligate biotrophs that exclusively colonise plant roots. These fungi form hyphae connections between roots from one or several host species to establish the Common Mycorrhizal Networks (CMNs) [START_REF] Simard | Mycorrhizal networks: Mechanisms, ecology and modelling[END_REF]. Plants associated with CMNs have improved assimilation of phosphate, macronutrients such as N, K and Mg, and some micronutrients [START_REF] Bhatia | Biomass production and changes in soil productivity during longterm cultivation of Prosopis juliflora (Swartz) DC inoculated with VA mycorrhiza and Rhizobium spp. in a semi-arid wasteland[END_REF][START_REF] Montaño | Micorrizas arbusculares en ecosistemas áridos y semiáridos (Arbuscular mycorrhizae in arid and semiarid ecosystems)[END_REF]. Colonising AMF receive organic carbon from the host plants [START_REF] Sanders | Mechanism of Absorption of Phosphate from Soil by Endogone Mycorrhizas[END_REF]. The review of Veresoglou and Rilling (2012) indicated that AMF have capacity to decrease losses caused by diverse plant pathogens, with the interaction between AMF and RKNs representing 28% of the listed reports. For example, early mycorrhization by mixtures of AMF species is effective against Meloidogyne spp. and Pratylenchus spp. [START_REF] Vos | Arbuscular mycorrhizal fungi induce systemic resistance in tomato against the sedentary nematode Meloidogyne incognita and the migratory nematode Pratylenchus penetrans[END_REF]. [START_REF] Sikora | Effect of vesicular-arbuscular mycorrhiza (Endogone mosseae) on the population dynamics of the root-knot nematodes (Meloidogyne incognita and M. hapla)[END_REF] reported that Funneliformis mosseae and Rhizophagus fasciculatus decreased M. incognita infection on tomato by, respectively, 13% and 50%. In pepper, Peregrin et al., (2012) assessed effects of AMF and Bacillus megaterium (simultaneously and individually) on Meloidogyne incognita, although the bioprotection conferred by the AMF was not clearly demonstrated.

Tomato and pepper have been frequently reported as mycotrophic plants [START_REF] Cress | Kinetics of Phosphorus Absorption by Mycorrhizal and Non mycorrhizal Tomato Roots[END_REF][START_REF] Al-Karaki | Growth of mycorrhizal tomato and mineral acquisition under salt stress[END_REF][START_REF] Schroeder | Phosphorus and intraspecific density alter plant responses to arbuscular mycorrhizas[END_REF][START_REF] Schroeder-Moreno | Intra-and inter-specific density affects plant growth responses to arbuscular mycorrhizas[END_REF][START_REF] Gashua | Occurrence of Arbuscular Mycorrhizal Fungi in Chilli peppers (Capsicum annuum L.) Grown in Sahelian Soil[END_REF][START_REF] Chialva | Understanding Changes in Tomato Cell Walls in Roots and Fruits: The Contribution of Arbuscular Mycorrhizal Colonization[END_REF]. Nevertheless, tomato plants do not rapidly develop intensive mycorrhization compared to highly mycotrophic species [START_REF] Schroeder | Phosphorus and intraspecific density alter plant responses to arbuscular mycorrhizas[END_REF][START_REF] Kubota | Cooccurrence of Arum-and Paris-type morphologies of arbuscular mycorrhizae in cucumber and tomato[END_REF][START_REF] Islas | Capacidad micotrófica y eficiencia de consorcios con hongos micorrícicos nativos de suelos de la provincia de Buenos Aires con manejo contrastante[END_REF]. This can be inconvenient for horticultural applications, as any probable benefit from AMF against RKN may only be obtained through early mycorrhization [START_REF] Jaizme-Vega | Interactions between the root-knot nematode Meloidogyne incognita and Glomus mosseae in banana[END_REF][START_REF] Molinari | Bio-control agents activate plant immune response and prime susceptible tomato against root-knot nematodes[END_REF]. Interconnection from mature plants by CMNs could improve establishment and growth of seedlings [START_REF] Van Der Heijden | Socialism in soil? The importance of mycorrhizal fungal networks for facilitation in natural ecosystems[END_REF]. [START_REF] Derelle | Association of highly and weakly mycorrhizal seedlings can promote the extra-and intraradical development of a common mycorrhizal network[END_REF] reported enhanced mycorrhization of Silene vulgaris (weakly mycotrophic) by mycorrhizal networks (MNs) previously developed by Medicago truncatula (highly mycotrophic) under in vitro conditions. Therefore, examining new strategies for tomato mycorrhization involving MNs could provide worthwhile new and practically valuable knowledge.

To test bioprotection of AMF against RKNs, the effect of mycorrhizal colonisation against development of M. incognita was examined in tomato and pepper, as two economically important crop plants. Additionally, the potential of MNs for accelerating mycorrhization was demonstrated on tomato plants using MNs previously established by a suitable highly mycotrophic host such as sorghum.

MATERIALS AND METHODS

Plant material

RKN-susceptible seedlings of tomato (Solanum lycopersicum 'Saint Pierre') and pepper (Capsicum annuum 'Doux long des Landes') were transplanted into 9 cm × 9 cm pots containing sterilised soil, and were inoculated with AEGIS powder from NIXE® (AMF inoculum containing Rhizophagus irregularis). Unless otherwise stated, all the plant cultures were in a growth chamber at 25°C (± 1°C) with a 16 h light / 8 h dark illumination cycle. A nutrient solution adapted for AMF development (200 mg L -1 Ca(NO 3 ) 2 , 300 mg L -1 KNO 3 ,

25 mg L -1 KH 2 PO 4 , 150 mg L -1 MgSO 4 7 H 2 O, 1.5 mg L -1 H 3 BO 3 , 0.05 mg L -1 (NH 4 ) 6 Mo 7 O 24 , 1 mg L -1 ZnSO 4 7 H 2 O, 2 mg L -1 MnSO 4 H 2 O, 0.25 mg L -1 CuSO 4 5 H 2 O, 225 mg L -1
Fe EDTA) was added every 3 d to the plants, at 1 mL per application for the first week of culture, then increased by 1 mL every week from the second to the fifth weeks. After 6 weeks of culture, the root mycorrhization rates were assessed for three plants of each species, and 5000 freshly hatched J2s of M. incognita suspended in water were inoculated onto each plant. The number of nematode egg masses was assessed on each plant at 6 weeks after J2 inoculation, by acid eosin staining. The same experiment was performed simultaneously with non-AMF treated plants as experimental controls.

Tomato mycorrhization by mycorrhizal networks consisted of growing an AMF inoculated sorghum sudangrass plant (Sorghum bicolor × Sorghum sudanense (Piper) Stapf) under equivalent conditions to the seedlings for the RKN experiments. After 6 weeks, the aerial part and roots were shredded and integrated to the pot soil. A tomato seedling was then transplanted into the sorghum/soil mix, and the mycorrhization rate was assessed after 2 and 4 weeks. Tomato plants inoculated directly with the commercial inoculum were used as experimental controls.

Assessment of root mycorrhization rates

Plant roots were bleached with 1% (w/v) KOH solution at 80° C for 1 h, and then incubated in a 5% solution of ink/lactic acid (80%) for 12 h. After rinsing, 1 cm root segments were mounted in glycerol/lactic acid (1/1 v/v) on glass microscope slides. Mycorrhization rates were evaluated by light microscopy, by considering the mycelium frequency inside and outside each root segment, the quantity of colonized root, and the abundance of developed arbuscules/vesicles [START_REF] Trouvelot | Mesure du taux de mycorhization VA d'un système radiculaire. Recherche et méthodes d' estimation ayant une signification fonctionnelle[END_REF].

Evaluation of Meloidogyne incognita development

The roots of experimental plants were rinsed thoroughly with water and the M. incognita egg masses were stained with an aqueous solution of 5% eosin B and 0.5% acetic acid. The egg masses (red spots on the roots) were quantified.

Statistical analyses

Results from each experiment were analysed by comparing measured parameters for mycorrhized and nonmycorrhized plants. Differences between experimental conditions were tested by Student's t-Test at significant levels (P < 0.05) using R Software.

RESULTS

Figure 1 shows effects of inducing AMF root colonisation of tomato from direct inoculation before seedling transplantation, compared to formation of a MN in the soil. Sorghum developed mycorrhization rates of 50-70% within 6 weeks before the tomato seedlings were transplanted. After 2 weeks of culture, tomato roots from the SB treatment had a mean of 22% mycorrhization, whereas the plants from the DI treatment had no mycorrhization. After 4 weeks, roots from the SB treated plants had a mean of 49 % of mycorrhization, while the DI treated plants had very low mycorrhization rates (<5%). The previous formation of a MN in the media accelerated the mycorrhization of tomato roots, which under the growth chamber conditions required 6 weeks for development of mycorrhization of approx. 40%.

To evaluate the protective potential of the AMF under growth cabinet conditions, the effects of conventional mycorrhization against M. incognita development were assessed in tomato and pepper plants. Before M. incognita inoculation, tomato roots had 40% (± 18% S.D.) mycorrhization and pepper roots had 68% (± 21% S.D.) mycorrhization. At 4 weeks after M. incognita inoculation, the AMF treated tomato plants had longer stems and more abundant foliage than the uninoculated plants (Figure 2A). The tomato plants were severely affected during the last 2 weeks of RKN infection. At the end of the experiment, the non-mycorrhized plants had shorter stems and more withered leaves than the mycorrhized tomato (Figure 2B). AMF treatment gave no obvious on shoot dry matter of the plants (Figure 2E), but mycorrhization induced significantly increased plant height (Figure 2F).

The reproduction rates of the nematodes were evaluated by counting the numbers of egg masses produced by M. incognita females, which represents the population of nematodes able to complete life cycles within host roots. There was a significant decrease (almost 30%) of egg masses in roots of the AMF treated pepper plants (Figure 3A). Mycorrhized tomato plants had greater numbers of egg masses than pepper plants (Figure 3A). However, the relatively unfavourable state of the nonmycorrhized tomato for RKN infection could also indicate that tomato was a less suitable host than pepper for RKN development. Therefore, the numbers of egg masses were normalised to the shoot height of tomato plants (Figure 3B). The resulting values of mycorrhized tomato plants were on average 40% less than for non-mycorrhized plants.

DISCUSSION

Mycorrhization of tomato plants can be challenging due to their low mycotrophy compared to other species (Schroedder and Janos, 2004). The present study has demonstrated that developing a MN in soil before transplanting of seedlings is a promising procedure for improving tomato mycorrhization. We used sorghumsudangrass because it has rapid growth, low nutrient demand, high mycotrophy and biofumigant properties, and is a known nematode trap crop. The sorghum variety selected here is currently used as a cover crop for nematode control which can be interplanted with legumes such as soybean [START_REF] Djian-Caporalino | Evaluating sorghums as green manure against rootknot nematodes[END_REF][START_REF] Dover | Nematode Management Using Sorghum and Its Relatives[END_REF]. Therefore, the proliferation of AMF by a highly mycothrophic host, which is eventually harvested and blended with soil, could be a useful technique for improving plant fitness and for RKN control, if the host also has nematicidal properties, as for the sudangrass cultivar selected. Rapid mycorrhization using CMNs has been previously tested by [START_REF] Derelle | Association of highly and weakly mycorrhizal seedlings can promote the extra-and intraradical development of a common mycorrhizal network[END_REF]. These types of experiments under in vitro conditions, require barriers between plants to avoid competition and did not use relevant crop plants. In the present study, the MN developed by the sorghum plants could provide increased amounts and viability of AMF compared to more usual AMF inoculum. This approach probably bypasses the low mycotrophy of tomato. Increased activity and intact spores or hyphae could explain the acceleration of AMF colonisation on tomato. However, many aspects addressing the early mycorrhization of tomato by MN remain unclear. For example, a comparative study of the response to the commercial AMF and the MN inoculum to the chemical signalling (i.e. branching factors) from the potential host has not been carried out.

Protection of pepper plants due to AMF has been previously demonstrated for Fusarium, Phytophthora, and Rhizoctonia pathogens [START_REF] Sahi | In vitro biological control of Fusarium oxysporum-r causing wilt in Capsicum annuum[END_REF][START_REF] Ahmed | Effect of chitin on biological control activity of Bacillus spp. and Trichoderma harzianum against root rot disease in pepper (Capsicum annuum) plants[END_REF][START_REF] Ahmed | Evaluation of Trichoderma harzianum for controlling root rot caused by Phytophthora capsici in pepper plants[END_REF]. In other Solanaceae, protection of plants against phytoparasitic nematodes due to host mycorrhization has been evaluated for tobacco, with reductions of 25-35% of Heterodera solanacearum cysts [START_REF] Fox | Interaction of Heterodera solanacearum and Endogone gigantea on tobacco[END_REF], eggplant, with mycorrhized roots presenting 87% fewer galls of M. incognita than non-mycorrhized roots [START_REF] Horta | Interaction of Root knot nematode (Meloidogyne incognita) & AM fungus (Glomus fasciculatum) on brinjal[END_REF], and mycorrhized tomato, with a 13% reduction of M. incognita compared to controls without AMF [START_REF] Masadeh | Biological Control of Meloidogyne incognita (Tylenchida: Meloidogynidae) on Tomato using Arbuscular Mycorrhizal Fungi and Rhizobacteria[END_REF]. [START_REF] Castillo | Protection of olive planting stocks against parasitism of root-knot nematodes by arbuscular mycorrhizal fungi[END_REF] also evaluated the protection of olive plants using AMF, against M. javanica and M. incognita under controlled conditions.

Results from the present study indicate that a 6 week mycorrhization period prior to M. incognita infection considerably decreased RKN development in pepper. Host nutrition and root development were specifically restricted in this experiment, which probably explains the abnormal state of the plant shoots. Pepper plant were more tolerant than tomato under the experimental conditions of this study. This research is one of few studies reporting ability of AMF to decrease M. incognita development in pepper. In tomato we verified increased tolerance to M. incognita infection symptoms. The symptoms of infection by M. incognita on tomato were more attenuated when roots were treated with AMF. Even so, egg masses were more abundant in mycorrhized plants, so we considered plant height for adjusting numbers of nematode masses in roots. These indices are extensively used for estimating the whole plant infection, consider-ing RKN proliferation and general plant development [START_REF] Mateille | Sur tomates, la cartographie des indices de galles[END_REF]. The enhanced phenotype of AMF treated tomato shoots was possibly due to acclimation, which could help hosts to resist RKN infections for long periods. Hyphae replacing part of roots damaged by M. incognita could cause tomato tolerance to M. incognita infections. Additionally, shoot configuration from environmental stress may also be important, as photosynthetic activity and leaf development can be positively affected by the AMF colonisation [START_REF] Chastain | Leaf ontogeny strongly influences photosynthetic tolerance to drought and high temperature in Gossypium hirsutum[END_REF][START_REF] Chandrasekaran | Impact of Arbuscular Mycorrhizal Fungi on Photosynthesis, Water Status, and Gas Exchange of Plants Under Salt Stress-A Meta-Analysis[END_REF]. This means that more M. incognita reproductive cycles can occur in mycorrhized hosts, whereas non-mycorrhized plants will not withstand RKN development because roots and shoots grow poorly and the plants will rapidly die due to infections. Economically, AMF treatments may provide enhanced or more consistent production of tomato fruit despite M. incognita proliferation. The differences in results between tomato and pepper plants confirm that AMF bioprotection is dependent on host species [START_REF] Veresoglou | Suppression of fungal and nematode plant pathogens through arbuscular mycorrhizal fungi[END_REF].

Revitalisation of indigenous AMF by highly mycotrophic plants with nematicidal properties could be combined with resistant horticultural varieties for crop bioprotection and durable control of RKNs. The results of the present study on tomato also suggest that future research should assess leaf acclimation of mycorrhized plants under RKN biotic stress, using in vivo approaches such as monitoring of chlorophyll-a fluorescence.

Figure 1 .

 1 Figure 1. (Graph A) Mean mycorrhization rates (and standard errors: n ≥ 4) in tomato roots resulting from different AMF inoculation procedures at 2 and 4 weeks after inoculation. Before the tomato seedlings were transplanted to pots, a commercial inoculum was applied directly to their roots (direct inoculation: DI), or the soil was blended with leaves and roots of 6-week-old mycorrhized sorghum plants (micrograph D) (sorghum blend: SB). * and **, respectively, indicate differences at P < 0.05 and 0.01. Characteristic AMF colonisation of tomato roots grown for 4 weeks after SB (micrograph B) or DI (micrograph C) treatments. Arrows indicate typical AMF structures.

Figure 2 .

 2 Figure 2. Plants grown in growth chambers for evaluating AMF effects on root knot nematode (RKN) development. Tomato plants at 4 weeks (micrograph A) and 6 weeks (micrograph B) after RKN inoculation. Pepper plants (micrographs C and D) 6 weeks after inoculation with RKN. Means (and standard errors: n ≥ 9) of shoot dry weights (Graph E) and plant heights (Graph F) for 12-week-old tomato and pepper plants 6 weeks after inoculations with J2s of Meloidogyne incognita. Means accompanied by different letters are significantly different (P < 0.001 (***), Student's test).

Figure 3 .

 3 Figure 3. Mean numbers of Meloidogyne incognita egg masses in root segments of tomato or pepper plants (Graph A), and mean numbers of egg masses (normalised to plant height) in root segments of tomato plants (Graph B), with or without AMF inoculation treatments, 6 weeks after inoculations with J2 nematodes. Means accompanied by different letters are significantly different (n ≥ 9: P < 0.05 (*), 0.01 (**) or 0.001 (***), Student's test).

ACKNOWLEDGEMENTS

This study was funded by the INRAE project SMaCH 'REACTION ' (2014-2016), the SYSTEMYC project funded by l'Agence Française pour la Biodiversité and by NIXE SARL.