Juliette Venel Lamav
email: juliette.venel@univ-valenciennes.fr

Numerical scheme for a whole class of sweeping process

Keywords:

The aim of this paper is to study a whole class of first order differential inclusions, which fit into the framework of perturbed sweeping process by uniformly prox-regular sets. After obtaining well-posedness results, we propose a numerical scheme based on a predictioncorrection algorithm and we prove its convergence. Finally we apply these results to a problem coming from the modelling of crowd motion.

Introduction

The study of first order differential inclusions started in the 1960s with the theory of the maximal monotone operators (see e.g. [START_REF] Brezis | Opérateurs Maximaux Monotones et Semi-groupes de contractions dans les espaces de Hilbert[END_REF]). Later J.-J. Moreau considered a problem involving a timedependent multivalued operator in [START_REF] Moreau | Evolution problem associated with a moving convex set in a Hilbert space[END_REF]. He dealt with the first sweeping process by convex sets C(t) included in a Hilbert space:

dq dt (t) ∈ -∂I C(t) (q(t)), (1)
where ∂I C is the subdifferential of the characteristic function of a convex set C. Such a situation may be visualized as a point q(t) moving inside C(t) and being pushed by the boundary of this convex set when contact is established. This problem can also be written :

dq dt (t) ∈ -N(C(t), q(t)),
where N(C, q) is the proximal normal cone to C at q (see Definition 2.1). He developped a so called catching-up algorithm to build discretized solutions and so proved the well-posedness of (1) under some assumptions on the set-valued map C(•). More precisely, in considering some subdivision (J k ) k of the time-interval, the set-valued map C is approached by a piecewise constant multifunction taking value C k on J k . The associated discretized solution q defined by ∀t ∈ J k+1 , q(t) = qk+1 = P C k+1 (q k ), with q0 fixed to the initial value, converges to the unique solution of (1).

Since then, important improvements have been provided by weakening the convexity assumption and by considering a perturbed version of this problem: dq dt (t) + N(C(t), q(t)) ∋ f (t, q(t)).

In [START_REF] Valadier | Quelques problèmes d'entraînement unilatéral en dimension finie[END_REF], M. Valadier studied sweeping process by complements of convex sets in finite dimension without perturbation. Perturbations (even multivalued perturbations) have been taken into account by C. Castaing, T.X. Dúc Hā, M.D.P Monteiro Marques and M. Valadier in [START_REF] Castaing | Evolution equations governed by the sweeping process[END_REF][START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF].

Then the main concept, which appeared to weaken the convexity assumption of sets C(t), is the notion of uniform prox-regularity. A set C is said to be η-prox-regular (or uniformly proxregular with constant η) if the projection onto C is single-valued and continuous at any point whose the distance to C is smaller than η. Under this assumption, the sweeping process without perturbation was firstly treated by G. Colombo, V.V. Goncharov in [START_REF] Colombo | The sweeping processes without convexity[END_REF], by H. Benabdellah in [START_REF] Benabdellah | Existence of solutions to the nonconvex sweeping process[END_REF] and later by L. Thibault in [START_REF] Thibault | Sweeping process with regular and nonregular sets[END_REF] and by G. Colombo, M.D.P. Monteiro Marques in [START_REF] Colombo | Sweeping by a continuous prox-regular set[END_REF].

The perturbed problem was later studied by M. Bounkhel, J.-F. Edmond and L. Thibault in [START_REF] Bounkhel | Nonconvex sweeping process and prox-regularity in Hilbert space[END_REF][START_REF] Thibault | Sweeping process with regular and nonregular sets[END_REF][START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF][START_REF] Edmond | BV solutions of nonconvex sweeping process differential inclusion with perturbation[END_REF].

In this paper, we consider perturbed sweeping process by a set-valued map Q : [0, T ] ⇉ R d satisfying that for every t ∈ [0, T ], Q(t) is the intersection of complements of smooth convex sets. Let us first specify the set-valued map Q. For i ∈ {1, . . . , p}, let g i : [0, T ] × R d → R be a convex function with respect to the second variable. For every t ∈ [0, T ], we introduce the sets Q i (t) defined by: Q i (t) := q ∈ R d , g i (t, q) ≥ 0 ,

and the feasible set Q(t) (supposed to be nonempty) is

Q(t) := p i=1 Q i (t). (4)
The associated perturbed sweeping process can be expressed as follows:

   dq dt (t) + N(Q(t), q(t)) ∋ f (t, q(t)) for a.e. t ∈ [0, T ] q(0) = q 0 ∈ Q(0).

(

) 5
This differential inclusion can be thought as follows: the point q(t), submitted to the perturbation f (t, q(t)), has to stay in the feasible set Q(t). To obtain well-posedness results for (5), we will make the following assumptions which ensure the uniform prox-regularity of Q(t) for all t ∈ [0, T ]. We suppose there exists c > 0 and for all t in [0, T ] open sets

U i (t) ⊃ Q i (t) verifying d H (Q i (t), R d \ U i (t)) > c, (A0)
where d H denotes the Hausdorff distance. Moreover we assume there exist constants α, β, M > 0 such that for all t in [0, T ], g i (t, •) belongs to C 2 (U i (t)) and satisfies

∀ q ∈ U i (t) , α ≤ |∇ q g i (t, q)| ≤ β, (A1) ∀ q ∈ R d , |∂ t g i (t, q)| ≤ β, (A2)
∀ q ∈ U i (t) , |∂ t ∇ q g i (t, q)| ≤ M, (A3)
and

∀ q ∈ U i (t) , |D 2 q g i (t, q)| ≤ M. (A4)
For all t ∈ [0, T ] and q ∈ Q(t), we denote by I(t, q) the active set at q I(t, q) := {i ∈ {1, . . . , p} , g i (t, q) = 0} [START_REF] Castaing | Evolution equations governed by the sweeping process[END_REF] and for every ρ > 0, the following sets:

I ρ (t, q) := {i ∈ {1, . . . , p} , g i (t, q) ≤ ρ} .

In addition we assume there exist γ > 0 and ρ > 0 such that for all t ∈ [0, T ],

∀ q ∈ Q(t) , ∀ λ i ≥ 0, i∈Iρ(t,q) λ i |∇ q g i (t, q)| ≤ γ i∈Iρ(t,q) λ i ∇ q g i (t, q) . ( A5
)
We will use the following weaker assumption too:

∀ q ∈ Q(t) , ∀ λ i ≥ 0, i∈I(t,q) λ i |∇ q g i (t, q)| ≤ γ i∈I(t,q) λ i ∇ q g i (t, q) . (A5')
In particular, this last assumption implies that for all t, the gradients of the active inequality constraints ∇ q g i (t, q) are positive-lineary independent at all q ∈ Q(t), which is usually called the Mangasarian-Fromowitz constraint qualification (MFCQ). Conversely the MFCQ condition at a point q yields a local version of Inequality (A5').

The notion of uniform prox-regularity allows to adapt the catching-up algorithm because the projection onto a uniformly prox-regular set is well-defined in its neighbourhood. However, from a numerical point of view, it may be difficult to perform this projection. Here, we study a numerical scheme avoiding this difficulty, which is adapted from the one proposed in [START_REF] Maury | A time-stepping scheme for inelastic collisions[END_REF][START_REF] Maury | Un modèle de mouvement de foule[END_REF][START_REF] Maury | A microscopic model of crowd motion[END_REF]. The idea is to replace Q(t) with a convex set Q(t, q) (depending on the position). This substitution is convenient because classical methods can be employed to compute the projection onto a convex set. Yet this replacement raises some difficulties for the numerical analysis which are solved in proving that for every q ∈ Q(t), the set Q(t, q) is a good local approximation of Q(t) around q.

The paper is structured as follows: In Section 2, we describe the mathematical framework to study the differential inclusion [START_REF] Brezis | Opérateurs Maximaux Monotones et Semi-groupes de contractions dans les espaces de Hilbert[END_REF]. By justifying that the set-valued map Q is Lipschitzian and takes uniformly prox-regular values (Propositions 2.11 and 2.9), we obtain well-posedness results for (5) in Theorem 2.12. Then in Section 3 we propose a prediction-correction scheme [START_REF] Thibault | Sweeping process with regular and nonregular sets[END_REF] and prove its convergence in Theorem 3.4. Finally we apply these results in two examples in Section 4. The first situation is a case in point and the second one comes from the modelling of crowd motion in emergency evacuation.

2 Mathematical framework and well-posedness results

Preliminaries

Firstly we recall some definitions and properties to specify the mathematical framework. Here we consider a finite dimension space R d equipped with its Euclidean structure although these notions have been extended in a Hilbertian context. For more details, we refer the reader to [START_REF] Clarke | Proximal smoothness and the lower-C 2 property[END_REF][START_REF] Bounkhel | On various notions of regularity of sets in nonsmooth analysis[END_REF]. Note that for all x ∈ int(S) := S \ ∂S, N(S, x) = {0}. This concept extends in a certain way the notion of normal outward direction for a smooth manifold, as specified in the next proposition.

Proposition 2.2. Let S be a closed subset of R d whose boundary ∂S is an oriented C 2 hypersurface. For each x ∈ ∂S, we denote by ν(x) the outward normal to S at x. Then, for each x ∈ ∂S, the proximal normal cone to S at x is generated by ν(x), i.e.

N(S, x) = R + ν(x).
We refer the reader to Proposition 3.2 of [START_REF] Venel | Modélisation mathématique et numérique des mouvements de foule[END_REF] for a detailed proof.

Using [START_REF] Clarke | Proximal smoothness and the lower-C 2 property[END_REF], we recall the concept of uniform prox-regularity as follows:

η x S Figure 1: η-prox-regular set. Definition 2.3. Let S be a closed subset of R d . S is said η-prox-regular if for all x ∈ ∂S and v ∈ N(S, x), |v| = 1 we have: B(x + ηv, η) ∩ S = ∅. Equivalently, S is η-prox-regular if for all y ∈ S, x ∈ ∂S and v ∈ N(S, x), v, y -x ≤ |v| 2η |y -x| 2 .
In other words, S is η-prox-regular if any external ball with radius smaller than η can be rolled around it (see Fig 1). Moreover, this definition ensures that the projection onto such a set is well-defined and is continuous in its neighbourhood. Note that a closed convex set C ⊂ R d is ∞-prox-regular. We claim a technical lemma which will be useful later.

⇔ ∀ξ ∈ R d , w, ξ ≤ |w| d S (ξ + x) (c) ⇔ ∃k > 0, ∃δ > 0, ∀v ∈ R d , |v| < δ, w, v ≤ k d S (v + x) (d)
The elementary proof is left to the reader.

Uniform prox-regularity of sets Q(t)

This subsection is devoted to specify the proximal normal cone to Q(t) (defined by ( 4)) for t ∈ [0, T ] and to justify the uniform prox-regularity of this set under the assumptions (A1), (A4) and (A5'). We point out that the time-dependence plays no role in this study. First we deal with only one constraint. Fix i ∈ {1, . . . , p} and consider the set Q i (t) defined by [START_REF] Bounkhel | Nonconvex sweeping process and prox-regularity in Hilbert space[END_REF]. The smoothness of the function g i (t, •) allows us to apply Proposition 2.2 and to deduce the expression of the proximal normal cone to Q i (t).

Corollary 2.5. For all t ∈ [0, T ] and q ∈ ∂Q i (t),

N(Q i (t), q) = -R + ∇ q g i (t, q).
By Definition 2.3, the constant of prox-regularity equals to the largest radius of a "rolling external ball". In order to estimate its radius, tools of differential geometry can be used. More precisely, to show that the set Q i (t) is uniformly prox-regular, we can apply the following theorem proved in [START_REF] Delgado | Blaschke's theorem for convex hypersurfaces[END_REF].

Theorem 2.6. Let C be a closed convex subset of R d such that ∂C is an oriented C 2 hypersurface of R d . We denote by ν C (x) the outward normal to C at x and by ρ 1 (x), .., ρ d-1 (x) ≥ 0 the principal curvatures of C at x. We suppose that

ρ := sup x∈∂C sup 1≤i≤d-1 ρ i (x) < ∞. Then S = R d \ int(C) is a η-prox-regular set with η = 1 ρ . Proposition 2.7. For all t ∈ [0, T ] and i ∈ {1, . . . , p}, Q i (t) is η 0 -prox-regular, with η 0 = α M .
Proof. Let t ∈ [0, T ], by Assumption (A1), the complement of Q i (t) is obviously the interior of the closed convex set C = {g i (t, •) ≤ 0} which satisfies the assumptions of Theorem 2.6. The constant of prox-regularity of Q i (t) can be obtained by calculating its principal curvatures which are the eigenvalues of the Weingarten endomorphism. Let q ∈ ∂Q i (t) = ∂C, the outward normal to C at q is equal to -ν(q), where ν(q) = -∇ q g i (t, q) |∇ q g i (t, q)| .

We can specify the expression of the Weingarten endomorphism W q . For every tangent vector h ∈ T q (∂Q i (t)),

W q (h) := -Dν(q)[h] = 1 |∇ q g i (t, q)| P (ν(q)) ⊥ D 2 q g i (t, q)[h] , with P ν ⊥ (x) = x -x, ν ν.
By (A1) and (A4), for all q ∈ ∂Q i (t) ⊂ U i (t), the eigenvalues of W q are bounded by M α -1 , which ends the proof according to Theorem 2.6 . Now let us study the feasible set Q(t) that is the intersection of all sets Q i (t). We begin to determine its proximal normal cone.

Proposition 2.8. For all t ∈ [0, T ] and q ∈ Q(t),

N(Q(t), q) = N(Q i (t), q) = - i∈I(t,q) R + ∇ q g i (t, q).
We refer the reader to Proposition 2.16 in [START_REF] Maury | A discrete contact model for crowd motion[END_REF] for a detailed proof. We emphasize that the inclusion N(Q i (t), q) ⊂ N(Q(t), q) always holds. However, the given proof of the other inclusion requires the convexity of functions g i (t, •) and the boundedness of gradients ∇ q g i (t, •) (Assumption (A1)).

We now come to the main result of this subsection: the set-valued map Q takes uniformly proxregular values (with a time-independent constant) which rests on the inverse triangle inequality (A5').

Proposition 2.9.

For every t ∈ [0, T ], Q(t) is η-prox-regular with η = η 0 γ = α M γ .
Proof. Let t ∈ [0, T ] and consider q ∈ ∂Q(t) and v ∈ N(Q(t), q) \ {0}. By Proposition 2.8, there exist

α i ≥ 0 such that v = - i∈I(t,q) α i ∇ q g i (t, q).
By definition (6) of I(t, q), q ∈ ∂Q i (t) for every i ∈ I(t, q). By Proposition 2.7, for all i ∈ I(t, q), -α i ∇ q g i (t, q) ∈ N(Q i (t), q), so we have by Definition 2.3,

-α i ∇ q g i (t, q), q -q ≤ |α i ∇ q g i (t, q)| 2η 0 |q -q| 2 , ∀q ∈ Q i (t). Since Q(t) ⊂ Q i (t)
, by summing these inequalities for i ∈ I(t, q), we obtain

- i∈I(t,q) α i ∇ q g i (t, q), q -q ≤ i∈I(t,q) α i |∇ q g i (t, q)| 2η 0 |q -q| 2 , ∀q ∈ Q(t).
¿From (A5'), it follows that

- i∈I(t,q) α i ∇ q g i (t, q), q -q ≤ γ 2η 0 i∈I(t,q) α i ∇ q g i (t, q) |q -q| 2 , ∀q ∈ Q(t).
We deduce from Definition 2.3 that Q(t) is η-prox-regular with η = η 0 γ .

Lipschitz regularity of Q

Here we check that t → Q(t) varies in a Lipschitz way. Aiming that, we need this technical lemma:

Lemma 2.10. There exists δ > 0 such that for all t ∈ [0, T ] and q ∈ Q(t):

∃u ∈ R d , |u| = 1, ∀i ∈ I ρ (t, q) , ∇ q g i (t, q), u ≥ δ, (8)
Proof. We set the following cone

N ρ (Q(t), q) := - i∈Iρ(t,q) R + ∇ q g i (t, q)
and its polar cone

C ρ (Q(t), q) := N ρ (Q(t), q) • := {w ∈ R d , ∀v ∈ N ρ (Q(t), q) , v, w ≤ 0}.
According to the classical orthogonal decomposition of a Hilbert space as the sum of mutually polar cones (see [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF]), we have:

Id = P Nρ(Q(t),q) + P Cρ(Q(t),q) ,
where P denotes the Euclidean projection. So let us consider for i ∈ I ρ (t, q) the corresponding decomposition of ∇ q g i (t, q):

∇ q g i (t, q) = a i + b i ∈ N ρ (Q(t), q) + C ρ (Q(t), q).

Assumption (A1) gives us: |b

i | ≤ |∇ q g i (t, q)| ≤ β. Since a i ∈ N ρ (Q(t)
, q), it can be written:

a i = -λ j ∇ q g j (t, q) , λ j ≥ 0 involving |b i | = |∇ q g i (t, q) -a i | = j =i λ j ∇ q g j (t, q) + (1 + λ i )∇ q g i (t, q) .
Then using the inverse triangle inequality (A5) and Assumption (A1), we get :

|b i | ≥ α γ λ j + 1 ≥ α γ .
As a consequence, it comes:

α γ ≤ |b i | ≤ β. (9)
Since 0 ∈ N ρ (Q(t), q) and a i = P Nρ(Q(t),q) (∇ q g i (t, q)), we obtain

2 b i , -∇ q g i (t, q) = |b i -∇ q g i (t, q)| 2 -|b i | 2 -|∇ q g i (t, q)| 2 = |a i | 2 -|b i | 2 -|∇ q g i (t, q)| 2 ≤ -|b i | 2 ≤ - α 2 γ 2 . (10)
Now we set

u := i∈Iρ(t,q) b i | i∈Iρ(t,q) b i | ∈ C ρ (Q(t), q). (11)
This is well-defined because [START_REF] Colombo | The sweeping processes without convexity[END_REF] and Assumption (A1) imply that for any j ∈ I ρ (t, q)

i∈Iρ(t,q) b i ≥ 1 β i∈Iρ(t,q) b i , ∇ q g j (t, q) ≥ 1 β b j , ∇ q g j (t, q) ≥ α 2 2βγ 2 . (12)
Then (8) follows from ( 9) and ( 12) with

δ = α 2 2γ 2 pβ .
Proposition 2.11. The set-valued map Q is Lipschitz continuous with respect to the Hausdorff distance. More precisely there exists

K L > 0 such that ∀t, s ∈ [0, T ] , d H (Q(t), Q(s)) ≤ K L |t -s|.
Proof. Consider t, s ∈ [0, T ] and q ∈ Q(t), let us construct a point closed to q belonging to Q(s). Let u given by Lemma 2.10, we introduce z(h) := q + hu with h > 0. We claim that for

h < h l := min(c, ρ/(β + δ)) , ∀i ∈ {1, . . . , p}, g i (t, z(h)) ≥ hδ
(where c and δ are introduced in (A0) and (2.10)). Indeed for h < c, due to the convexity of g i (t, •), it comes g i (t, q + hu) ≥ g i (t, q) + h ∇ q g i (t, q), u .

As a consequence, for i ∈ I ρ (t, q), g i (t, q + hu) ≥ hδ, by [START_REF] Ciarlet | Introduction à l'analyse numérique matricielle et à l'optimisation[END_REF]. Moreover for every i / ∈ I ρ (t, q), according to Assumption (A1),

g i (t, q + hu) ≥ ρ -hβ ≥ hδ if h < ρ β + δ
. Thus for h < h l , we have g i (t, q + hu) ≥ hδ for all i ∈ {1, . . . , p}. That is why we

deduce from Assumption (A2) that z(h) ∈ Q(s) if hδ ≥ β|t -s|. Setting ℓ := βh l δ , if |t -s| < ℓ, it can be written that d Q(s) (q) ≤ inf hδ≥β|t-s| |q -z(h)| ≤ β δ |t -s|.
Consequently we obtain if |t -s| < ℓ,

d H (Q(t), Q(s)) = max sup q∈Q(t) d Q(s) (q), sup q∈Q(s) d Q(t) (q) ≤ β δ |t -s|.
This inequality is actually satisfied for any t, s ∈ [0, T ]. To check it, it suffices to divide the corresponding interval into subintervals of length ℓ and to apply the triangle inequality. So the required result is obtained with K L = β/δ.

Well-posedness results

We now come to the main result.

Theorem 2.12. Let T > 0 and f : [0, T ] × R d → R d be a measurable map satisfying:

∀q ∈ s∈[0,T ] Q(s) , f (•, q) is Riemann-integrable on [0, T ] ( 13
)
∃K > 0 , ∀q ∈ s∈[0,T ] Q(s) , ∀t ∈ [0, T ] , |f (t, q) -f (t, q)| ≤ K|q -q| (14) ∃L > 0 , ∀q ∈ s∈[0,T ] Q(s) , ∀t ∈ [0, T ] , |f (t, q)| ≤ L(1 + |q|). (15)
Then, under Assumption (A5) for all q 0 ∈ Q(0), the following problem

   dq dt (t) + N(Q(t), q(t)) ∋ f (t, q(t)) for a.e. t ∈ [0, T ] q(0) = q 0 , (16)
has one and only one absolutely continuous solution q satisfying q(t) ∈ Q(t) for every t ∈ [0, T ].

Proof. As the set-valued map Q varies in a Lipschiz way by Proposition 2.11 and for all t ∈ [0, T ], Q(t) is η-prox-regular by Proposition 2.9, we can apply Theorem 1 of [START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF].

We now consider a constant set-valued map Q (the constraints g i are supposed to be timeindependent). In this particular case, we only require Assumption (A5') instead of Assumption (A5).

Theorem 2.13. Under Assumptions (A0), (A1), (A4), (A5') with time-independent set Q, let T > 0 and f : [0, T ] × Q → R d be a measurable map satisfying:

∀q ∈ Q , f (•, q) is Riemann-integrable on [0, T ] (17) ∃K > 0 , ∀q ∈ Q , ∀t ∈ [0, T ] , |f (t, q) -f (t, q)| ≤ K|q -q| (18) ∃L > 0 , ∀q ∈ Q , ∀t ∈ [0, T ] , |f (t, q)| ≤ L(1 + |q|). ( 19
)
Then, for all q 0 ∈ Q, the following problem

   dq dt (t) + N(Q, q(t)) ∋ f (t, q(t)) for a.e. t ∈ [0, T ] q(0) = q 0 , (20)
has one and only one absolutely continuous solution q taking values in Q. Moreover the map q is also solution of the following differential equation:

   dq dt (t) + P N(Q,q(t)) (f (t, q(t))) = f (t, q(t)) for a.e. t ∈ [0, T ] q(0) = q 0 , (21)
For this last point we refer the reader to Proposition 3.3 in [START_REF] Bernicot | Existence of sweeping process in Banach spaces under directional prox-regularity[END_REF].

Remark 2.14. The assumptions about the perturbation f can be weakened (see [START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF]). However they are required in the numerical analysis of the scheme proposed in the following section. More precisely, we need Assumption [START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF] to prove Lemma 3.6.

A numerical scheme 3.1 Presentation

We present in this section a numerical scheme to approximate the solution of ( 16) on the time interval [0, T ]. This scheme is adapted from the one proposed by B. Maury for granular media in [START_REF] Maury | A time-stepping scheme for inelastic collisions[END_REF]. Let n ∈ N ⋆ , h = T /n be the time step and t n k = kh be the computational times. We denote by q n k the approximation of q(t n k ) with q n 0 = q 0 . The next configuration is computed as follows:

q n k+1 := P Q(t n k+1 ,q n k ) (q n k + hf (t n k , q n k )) (22)
with

Q(t, q) := {q ∈ R d , g i (t, q) + ∇ q g i (t, q), q -q ≥ 0 ∀ i} for q ∈ U (t) := U i (t).
We recall that all the gradients ∇ q g i (t, q) are well-defined provided that q ∈ U (t). The set Q(t, q) can be seen as an inner convex approximation of Q(t) with respect to q. This scheme is a prediction-correction algorithm: predicted position vector q n k + hf (t n k , q n k ), that may not be admissible, is projected onto the approximate set of feasible configurations. Let us check that this scheme is well-defined for h < c K L with c and K L respectively given by Assumption (A0) and Proposition (2.11): Proposition 3.1. Assume that hK L < c, then for all k < n -1

q n k+1 ∈ Q(t n k+1 , q n k ) ⊂ Q(t n k+1 ) ⊂ U (t n k+2 ).
Thus every computed configuration is feasible .

q n k q n k + hf (t n k , q n k ) q n k + hf (t n k , q n k ) q n k+1 q n k+1 qn k+1 qn k+1 Q(t n k+1 ) Q(t n k+1 , q n k )
Figure 2: Theoretical and numerical projections.

Proof. Since q n 0 = q 0 ∈ Q(0) and d H (Q(0), Q(t n 1 )) ≤ K L h, we get q n 0 ∈ Q(t n 1 ) + hK L B(0, 1) ⊂ U (t n 1 ),
according to (A0) and Proposition 2.11. So Q(t n 1 , q n 0 ) is well-defined and is included in Q(t n 1 ) due to the convexity of the functions g i (t n 1 , •). Then as

q n 1 ∈ Q(t n 1 , q n 0 ), q n 1 ∈ Q(t n 1 ) ⊂ Q(t n 2 ) + hK L B(0, 1) ⊂ U (t n 2 )
. By iterating, we end the proof.

By Lemma 2.4 (a), it can be checked that

q n k+1 -q n k h + N( Q(t n k+1 , q n k ), q n k+1 ) ∋ f (t n k , q n k ), (23)
so that the scheme can also be seen as a semi-implicit discretization of [START_REF] Maury | Un modèle de mouvement de foule[END_REF], where the cone N( Q(t n k+1 , q n k ), q n k+1 ) stands for an approximation of N(Q(t), q(t)).

In Figure 2, we illustrate the set

Q(t n k+1 ) ⊂ R d , intersection of sets Q i (t n k+1
) whose boundaries are plotted in solid line. The set Q(t n k+1 , q n k ) is delimited by the dashed line. The theoretical and numerical projections, respectively qn

k+1 := P Q(t n k+1 ) (q n k + hf (t n k , q n k )) and q n k+1 are represented (for two examples of f (t n k , q n k )). Indeed, as Q(t n k+1
) is uniformly prox-regular, the projection onto

Q(t n k+1 ) of q n k + hf (t n k , q n k ) is well-defined for h small enough. The replacement of Q(t n k+1 ) by the convex set Q(t n k+1 , q n k
) is convenient because it allows us to use classical numerical methods to compute this projection. However, it raises some difficulties to prove the convergence of the scheme. On the one hand, we have to check that this approximation is sufficiently accurate (see Lemma 3.2). On the other hand, the set Q(t, q) does not vary smoothly enough to directly apply results about sweeping process (this point will be specified in Remark 3.7). Lemma 3.2. For all t ∈ [0, T ] and q ∈ Q(t), N(Q(t), q) = N( Q(t, q), q). Proof. We need the well-known following result (see e.g. [START_REF] Ciarlet | Introduction à l'analyse numérique matricielle et à l'optimisation[END_REF]) which rests on the reformulation of the minimization problem under constraints (associated to the projection), in terms of a saddle-point problem.

Lemma 3.3. Let t ∈ [0, T ], q ∈ Q(t) and q ∈ R d . Then, p = P Q(t,q) (q) is equivalent to the existence of λ ∈ (R + ) p such that (p, λ) satisfies:

         p = q + λ i ∇ q g i (t, q),
∀i , g i (t, q) + ∇ q g i (t, q), pq ≥ 0, λ i (g i (t, q) + ∇ q g i (t, q), pq ) = 0.

(P t,q,q )

First we check that N(Q(t), q) ⊂ N( Q(t, q), q). Let w ∈ N(Q(t), q). According to Proposition 2.8, this vector can be written

w = - i∈I(t,q) µ i ∇ q g i (t, q)
with nonnegative reals µ i . So by definition of Q(t, q), we get for all p ∈ Q(t, q), w, pq ≤ 0. Applying Lemma 2.4 (b), we deduce that w ∈ N( Q(t, q), q). It remains to prove N( Q(t, q), q) ⊂ N(Q(t), q). Letting w ∈ N( Q(t, q), q), we can write by Lemma 2.4 (a), q = P Q(t,q) (q + w) which implies by Lemma 3.3, q = (q + w) + µ i ∇ q g i (t, q), where (q, µ) ∈ Q(t) × (R + ) p satisfies System (P t,q,q+w ).

Consequently, w =µ i ∇ q g i (t, q) , µ i ≥ 0. The third relation in (P t,q,q+w ) is µ i g i (t, q) = 0, so µ i = 0 if g i (t, q) > 0, which concludes the proof.

Convergence

Before stating the result of convergence, we introduce some notations. We define the piecewise constant function f n as follows,

f n (t) = f (t n k , q n k ) if t ∈ [t n k , t n k+1 [, k < n and f n (T ) = f (t n n-1 , q n n-1 ).
We denote by q n the continuous, piecewise linear function satisfying for k ∈ {0, . . . , n}, q n (t n k ) = q n k . To finish, we introduce the functions ρ and θ defined by

ρ n (t) = t n k and θ n (t) = t n k+1 if t ∈ [t n k , t n k+1 [, ρ n (T ) = T and θ n (T ) = T.
Theorem 3.4. With the assumptions of Theorem 2.12, q n tends to q in C 0 ([0, T ], R d ), where t → q(t) is the unique solution of [START_REF] Maury | Un modèle de mouvement de foule[END_REF].

Proof. First we prove the boundedness of the sequence (q n ) and to do this, we use the same arguments as developed in [START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF] with the following lemma.

Lemma 3.5. There exists D > 0 such that for h small enough,

d Q(t n k+1 ,q n k ) (q n k ) ≤ Dh.
Without loss of generality, we can assume that T < 1/(2D + 4L) and h small enough to apply Lemma 3.5. It follows from ( 15) that

|q n k+1 -(q n k + hf (t n k , q n k ))| = d Q(t n k+1 ,q n k ) (q n k + hf (t n k , q n k )) ≤ d Q(t n k+1 ,q n k ) (q n k ) + h|f (t n k , q n k )| ≤ h(D + L)(1 + |q n k |). (24)
We deduce that

|q n k+1 -q n k | ≤ h(D + 2L)(1 + |q n k |) (25)
which implies that

|q n k+1 | ≤ |q n k | + h(D + 2L)(1 + |q n k |). Hence |q n n | ≤ |q 0 | + h(D + 2L) n-1 k=0 (1 + |q n k |) ≤ |q 0 | + nh(D + 2L)(1 + max 0≤k≤n |q n k |).
Since T = nh, we deduce that

max 0≤k≤n |q n k | ≤ |q 0 | + T (D + 2L) 1 -T (D + 2L) ≤ 2(|q 0 | + T (D + 2L)).
As a consequence, we obtain by letting C := 2(|q 0 | + T (D + 2L))

q n ∞ ≤ C. (26)
By ( 25) and (26), we have

dq n dt ∞ ≤ (D + 2L)(1 + C). (27)
By applying Arzela-Ascoli's Theorem, it can be easily shown that there exists a function q ∈ W 1,∞ ([0, T ], Q) and a subsequence (still denoted by q n ) satisfying

     dq n dt ⋆ ⇀ dq dt in L ∞ ([0, T ], R d ) q n ---→ n→∞ q uniformly in [0, T ].
By (26), it can be specified that

q ∞ ≤ C. ( 28
)
Since the time-interval is bounded, we get

dq n dt ⇀ dq dt in L 1 ([0, T ], R d ).
Now let us check that the limit function q satisfies the differential inclusion [START_REF] Maury | Un modèle de mouvement de foule[END_REF]. The beginning of the proof uses classical tools developped e.g. in [START_REF] Edmond | BV solutions of nonconvex sweeping process differential inclusion with perturbation[END_REF]. We want to show that

dq dt (t) -f (t, q(t)) ∈ -N(Q(t), q(t)) for a.e. t ∈ [0, T ]
which is equivalent to dq dt (t)f (t, q(t)) ∈ -N( Q(t, q(t)), q(t)) for a.e. t ∈ [0, T ] by Lemma 3.2.

Lemma 3.6.

f n ⇀ f (•, q(•)) in L 1 ([0, T ], R d ).
This lemma (later proved) implies that

dq n dt -f n ⇀ dq dt -f (•, q(•)) in L 1 ([0, T ], R d ).
Consequently, by Mazur's Lemma, there exists a sequence

z n ∈ L 1 ([0, T ], R d ) satisfying z n ∈ Conv dq k dt -f k , k ≥ n (29) and z n ---→ n→∞ dq dt -f (•, q(•)) in L 1 ([0, T ], R d ).
Extracting a subsequence, we may suppose that

z n ---→ n→∞ z = dq dt -f (•, q(•)) a.e. in [0, T ]. (30)
Furthermore, Inclusion ( 23) can be rewritten for almost every t ∈ [0, T ],

dq n dt (t) -f n (t) ∈ -N( Q(θ n (t), q n (ρ n (t))), q n (θ n (t))). ( 31
)
Let t ∈ [0, T ] such that z n (t) tends to z(t) and the above differential inclusion holds. By (29), it yields

∀ξ ∈ R d , z n (t), ξ ≤ sup k≥n dq k dt (t) -f k (t), ξ .
Passing to the limit, we obtain

∀ξ ∈ R d , z(t), ξ ≤ lim sup n dq n dt (t) -f n (t), ξ . (32)
From (31) and from Lemma 2.4 (c), we get

∀n, ∀ξ ∈ R d , dq n dt (t) -f n (t), ξ ≤ dq n dt (t) -f n (t) d Q(θ n (t),q n (ρ n (t))) (q n (θ n (t)) -ξ).
Hence, by ( 24) and (26), this inequality can be specified

∀n, ∀ξ ∈ R d , dq n dt (t) -f n (t), ξ ≤ (D + L)(1 + C) d Q(θ n (t),q n (ρ n (t))) (q n (θ n (t)) -ξ). ( 33
)
Proposition 3.8 with q = q(t), q n = q n (ρ n (t)) and t n = θ n (t) implies there exists ν > 0 such that for all ξ ∈ R d , |ξ| < ν,

d Q(θ n (t),q n (ρ n (t))) (q(t) -ξ) -d Q(t,q(t)) (q(t) -ξ) ---→ n→∞ 0. ( 34
)
Finally, by passing to the limit in (33), it comes from (32):

∀ξ ∈ R d , |ξ| < ν, z(t), ξ ≤ (D + L)(1 + C)d Q(t,q(t)) (q(t) -ξ),
which is equivalent to z(t) ∈ -N( Q(t, q(t)), q(t)),

by Lemma 2.4 (d). The required result follows from (30) provided we prove Lemmas 3.5, 3.6 and Proposition 3.8.

Remark 3.7. To estimate the upper limit in (32), we give a different method than the one written in [START_REF] Edmond | BV solutions of nonconvex sweeping process differential inclusion with perturbation[END_REF][START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF]. However the main argument is the same. Indeed in [START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF]

Ω := {(t, q), t ∈ [0, T ], q ∈ U (t)}
and

Ω c := {(t, q), t ∈ [0, T ], q ∈ Q(t)} , the map φ : Ω × R d -→ R (t, q, q) -→ d Q(t,q) (q)
is continuous on a neighbourhood of the set {(t, q, q), q = q} in Ω c × R d . In fact, the next proposition ensures that the map

Ω × R d -→ R d (t, q, q) -→ P Q(t,q) (q)
is continuous on the same neighbourhood (which is sufficient).

Proof of the claimed results

Proof of Lemma 3.5. This proof is similar to the one developed for Proposition 2.11. Let u given by Lemma 2.10 with t = t n k and q = q n k ∈ Q(t n k ), we introduce z(s) := q n k + su with s > 0. We claim that for s ≥ 2hβ δ (where c is introduced in (A0)),

∀i ∈ I ρ (t n k , q n k ), ∆ i := g i (t n k+1 , q n k ) + ∇ q g i (t n k+1 , q n k ) , z(s) -q n k ≥ 0.
Indeed, by Assumption (A2) and Proposition 3.1, g i (t n k+1 , q n k ) ≥ g i (t n k , q n k )βh ≥ -βh. As in Proposition 3.1, it can be proved that for all s ∈ [t n k , t n k+1 ], q n k ∈ U (s). Then it follows from Assumption (A3) that

∇ q g i (t n k+1 , q n k ) , z(s) -q n k ≥ ∇ q g i (t n k , q n k ) , z(s) -q n k -M hs.
For h ≤ δ 2M and by definition of u,

∆ i ≥ -βh -M hs + sδ ≥ s δ 2 -hβ ≥ 0.
Moreover, for i / ∈ I ρ (t n k , q n k ), Assumption (A2) yields

g i (t n k+1 , q n k ) ≥ g i (t n k , q n k ) -βh ≥ ρ -βh.
Consequently for i / ∈ I ρ (t n k , q n k ) and h ≤ ρ 2β ,

g i (t n k+1 , q n k ) + ∇ q g i (t n k+1 , q n k ) , z(s) -q n k ≥ ρ -βh -βs ≥ ρ 2 -βs.
For h small enough, for s ∈ [ 2hβ δ , ρ 2β ], z(s) belongs to Q(t n k+1 , q n k ). So we have proved that

d Q(t n k+1 ,q n k ) (q n k ) ≤ inf sδ≥2hβ |q n k -z(s)| ≤ 2β δ h.
Proof of Lemma 3.6. We want to show that f n weakly converges to

f (•, q(•)) in L 1 ([0, T ], R d ).
This property rests on the boundedness of the sequence

(f n ) n in L ∞ ([0, T ], R d ). Let us consider Ψ ∈ L ∞ ([0, T ], R d ) supposed to be continuous. T 0 f n (t), Ψ(t) dt = n-1 k=0 t n k+1 t n k f (t n k , q(t n k )), Ψ(t) dt + n-1 k=0 t n k+1 t n k f (t n k , q n (t n k )) -f (t n k , q(t n k )), Ψ(t) dt := I 1 n + I 2 n .
By ( 14),

|I 2 n | ≤ KT Ψ ∞ q n -q ∞ -----→ n→+∞ 0.
Furthermore

I 1 n = n-1 k=0 h f (t n k , q(t n k )), Ψ(t n k ) + n-1 k=0 t n k+1 t n k f (t n k , q(t n k )), Ψ(t) -Ψ(t n k ) dt := J 1 n + J 2 n .
The map f (•, q(•)) is bounded by L(1 + C) according to ( 15) and (28). Let us check that

J 1 n -----→ n→+∞ T 0 f (t, q(t)), Ψ(t) dt. ( 36
)
For all η > 0, there exists m ∈ N such that for all t, s ∈

[0, T ], |q(t) -q(s)| < η if |t -s| < δ := T /m since q ∈ W 1,∞ ([0, T ], Q). For all t, s ∈ [0, T ] satisfying |t -s| < δ, (14) yields |f (t, q(t)) -f (t, q(s))| < Kη. For n > m, J 1 n = n-1 k=0 h f (t n k , q(t n k )) -f (t n k , q(χ(t n k ))), Ψ(t n k ) + n-1 k=0 h f (t n k , q(χ(t n k ))), Ψ(t n k ) ,
where

χ(t) := δ t δ . Obviously, n-1 k=0 h f (t n k , q(t n k )) -f (t n k , q(χ(t n k ))), Ψ(t n k ) ≤ T Kη Ψ ∞ . Moreover n-1 k=0 h f (t n k , q(χ(t n k ))), Ψ(t n k ) = m-1 i=0 k t n k ∈[iδ,(i+1)δ[ h f (t n k , q(iδ)), Ψ(t n k ) .
Since for all i ∈ {0, ..., m -1}, f (•, q(iδ)) and Ψ are bounded and Riemann integrable by ( 13), we have

k t n k ∈[iδ,(i+1)δ[ h f (t n k , q(iδ)), Ψ(t n k ) ---→ n→∞ (i+1)δ iδ f (t, q(iδ)), Ψ(t) dt.
Thus, as previously,

lim sup n→∞ n-1 k=0 h f (t n k , q(χ(t n k ))), Ψ(t n k ) - T 0 f (t, q(t)), Ψ(t) dt ≤ T Kη Ψ ∞ .
This concludes the proof of (36). Moreover

|J 2 n | ≤ T L(1 + C) sup |t-s|≤T /n |Ψ(t) -Ψ(s)| -----→ n→+∞ 0.
Then, we deduce that for all continuous map

Ψ ∈ L ∞ ([0, T ], R d ), T 0 f n (t), Ψ(t) dt -----→ n→+∞ T 0 f (t, q(t)), Ψ(t) dt. ( 37
) Now let Φ ∈ L ∞ ([0, T ], R d ),
there exists a sequence of continuous maps (Ψ p ) p satisfying

Ψ p -Φ 1 ----→ p→+∞ 0. Let ε > 0, there exists p ∈ N such that L(1 + C) Ψ p -Φ 1 ≤ ε.
For that integer p, we write

T 0 f n (t), Φ(t) dt - T 0 f (t, q(t)), Φ(t) dt ≤ T 0 f n (t) -f (t, q(t)), Ψ p (t) dt + T 0 f (t, q(t)), Ψ p (t) -Φ(t) dt + T 0 f n (t), Φ(t) -Ψ p (t) dt ≤ T 0 | f n (t) -f (t, q(t)), Ψ p (t) | dt + T 0 | f (t, q(t)), Ψ p (t) -Φ(t) | dt + T 0 | f n (t), Φ(t) -Ψ p (t) | dt ≤ K 1 + K 2 + K 3 .
For i = 2 and 3, K i ≤ L(1 + C) Ψ p -Φ 1 ≤ ε and by (37), we know that for n large enough,

K 1 ≤ ε. So we deduce that for all Φ ∈ L ∞ ([0, T ], R d ), T 0 f n (t), Φ(t) dt -----→ n→+∞ T 0 f (t, q(t)), Φ(t) dt,
which concludes the proof.

Proposition 3.8. Let t ∈ [0, T ], q ∈ Q(t), a sequence (t n ) ∈ [0, T ] N converging to t and a sequence (q n ) ∈ (U (t n )) n converging to q such that d Q(tn,qn) (q n ) tends to zero. For all q ∈ R d ,
we denote p (respectively pn ) the projection of q onto Q(t, q) (respectively onto Q(t n , q n )). Then there exists ν > 0 so that for all q ∈ B(q, ν), the sequence ( pn ) converges to p.

Proof. By Lemma 3.3, the vector p satisfies System (P t,q,q ). We obtain similar systems denoted by (P tn,qn,q ) for all pn in substituting t, q and λ i by t n , q n and λ n i . The following lemma (which will be later proved) claims that the nonzero Kuhn-Tucker multipliers λ i , λ n i are associated to an index i belonging to I(t, q) (defined by ( 6)). Lemma 3.9. There exist ν > 0 and M 0 ∈ N such that for all n ≥ M 0 and all q ∈ B(q, ν), we have :

| pn -q| ≤ 2ν, and λ n i = λ i = 0, if g i (t, q) > 0, where (p, λ) and ( pn , λ n ) are respectively solutions of (P t,q,q ) and (P tn,qn,q ). There exists M 1 ∈ N such that for n ≥ M 1 , |qq n | < c. For all s ∈ [0, T ] and q, x ∈ Q(t), we denote by A t,q (s, x) the d × |I(t, q)| matrix defined as follows:

A t,q (s, x) := (∇ q g i (s, x)) i∈I(t,q) . Let M 0 and ν be fixed by Lemma 3.9. With the previous notation, for n ≥ max(M 0 , M 1 ) and q ∈ B(q, ν), the first equation of (P tn,qn,q ) can be written as

A t,q (t n , q n )[λ n ] = pn -q.
By (A1) and (A5'), we have

|λ n | ≤ γ α |A t,q (t, q)[λ n ]| ,
where |µ| represents the Euclidean norm of µ ∈ R |I(t,q)| . Moreover

|A t,q (t, q)[λ n ]| ≤ | pn -q| + |(A t,q (t n , q n ) -A t,q (t, q))[λ n ]|.
Thus by Lemma 3.9,

|λ n | ≤ γ α (2ν + ε n |λ n |) ,
with ε n := (A t,q (t n , q n ) -A t,q (t, q) . Hence

|λ n | 1 - ε n γ α ≤ 2νγ α .
Furthermore, ε n ≤ √ p max j∈I(t,q) |∇ q g j (t n , q n )-∇ q g j (t, q)|. Since q n ∈ U (t), it can be written

|∇ q g j (t n , q n ) -∇ q g j (t, q)| ≤ |∇ q g j (t n , q n ) -∇ q g j (t, q n )| + |∇ q g j (t, q n ) -∇ q g j (t, q)| ≤ M (|t -t n | + |q -q n |),
by Assumptions (A3) and (A4). Finally

ε n ≤ √ pM (|t -t n | + |q -q n |) ---→ n→∞ 0
and so we deduce that the sequence (λ n ) is bounded. A convergent subsequence can be also extracted and by passing to the limit in System (P tn,qn,q ), we obtain that the corresponding subsequence of ( pn ) converges to a point p ∞ which satisfies System (P t,q,q ). Lemma 3.3 implies that p ∞ = p and we can conclude by compacity arguments.

It remains to prove Lemma 3.9.

Proof of Lemma 3.9. This result is a consequence of the third equation of problems (P t,q,q ) and (P tn,qn,q ). By definition of I(t, q), ∃ε > 0, ∀ i / ∈ I(t, q), g i (t, q) > 2ε.

Setting ν = ε 8β , as (q n ) n and (t n ) n respectively converge to q and t, it yields

∃M 0 > 0, ∀n ≥ M 0 , g i (t n , q n ) ≥ ε, ∀ i / ∈ I(t, q) , d Q(tn,qn) (q n ) ≤ ν 2 and |q n -q| ≤ ν 2 .
Let q ∈ B(q, ν) , since p = P Q(t,q) (q) and q ∈ Q(t, q) (because q ∈ Q(t)), we obtain

|p -q| ≤ |q -q| ≤ ν and consequently, |p -q| ≤ |p -q| + |q -q| ≤ 2|q -q| ≤ 2ν. (38)
Moreover, as pn = P Q(tn,qn) (q),we get

| pn -q| ≤ |q n -q| + d Q(tn,qn) (q n ) ≤ |q -q| + |q -q n | + d Q(tn,qn) (q n ) ≤ ν + ν 2 + ν 2 = 2ν. Hence, | pn -q n | ≤ | pn -q| + |q -q n | ≤ 2|q -q n | ≤ 4ν. ( 39
)
For i / ∈ I(t, q), it follows from (A1), ( 38) and (39)

g i (t, q)+ ∇ q g i (t, q), p-q ≥ 2ε-2νβ > 0 and g i (t n , q n )+ ∇ q g i (t n , q n ), pn -q n ≥ ε-4νβ > 0.
The third equation of problems (P t,q,q ) , (P tn,qn,q ) and the nonnegativity of Kuhn-Tucker multipliers λ i , λ n i permit us to conclude.

Applications 4.1 A case in point

Here we deal with a simple case associated to a well-known game: the labyrinth tabletop game. The aim of this game is to maneuver a steel ball through a wooden labyrinth by tilting the surface in order to move the ball to the target. The angle of the labyrinth need to be carefully controlled so that the ball will not fall into the holes. We consider a particular labyrinth represented in Figure 3, with several holes (corresponding to the black disks). Initially the ball is located at the starting point S and it has to be rolled to the target T. The quasi-static evolution of this ball can be described by the following first order differential inclusion:

dq dt (t) + N(Q, q(t)) ∋ f (t),
where q ∈ R 2 is the position of the ball in the labyrinth surface, Q is the complementary set of the grey obstacles and the holes and f (t) is the perturbation caused by the leaning of the labyrinth plane. In Figure 3, we plotted the trajectory obtained with the following field f : beginning with f = f 1 , the ball arrives at the point P 1 , then with f = f 2 , the ball goes to the point P 2 and finally submitted to f = f 3 it reaches the target.

Obviously, the set Q is uniformly prox-regular and Assumptions (A1), (A4) and (A5') are satisfied. The evolution problem is well-posed by Theorem 2.12 and the trajectory of the ball can be estimated by the numerical scheme presented in Section 3.

An application to a crowd motion model

The aim of this subsection is to apply the previous results to a model of crowd motion in emergency evacuation. We refer the reader to [START_REF] Maury | Un modèle de mouvement de foule[END_REF][START_REF] Maury | A microscopic model of crowd motion[END_REF][START_REF] Venel | Modélisation mathématique et numérique des mouvements de foule[END_REF][START_REF] Maury | A discrete contact model for crowd motion[END_REF] for a complete and detailed description of this model.

Presentation of the model

We quickly recall the model. Its asset is to directly handle numerous contacts which are characteristic of emergency situations. This model allows us to deal with local interactions between people in order to describe the whole dynamics of the pedestrian traffic. This microscopic model for crowd motion rests on two principles. On the one hand, each individual has a spontaneous velocity that he would like to have in the absence of other people. On the other hand, the actual velocity must take into account congestion. Those two principles lead to define the actual velocity field as the projection of the spontaneous velocity over the set of admissible velocities (regarding the non-overlapping constraints).

We consider N persons identified to rigid disks. For convenience, the disks are supposed here to have the same radius r. The center of the i-th disk is denoted by q i ∈ R 2 . Since overlapping is forbidden, the vector of positions q = (q 1 , .., q N ) ∈ R 2N has to belong to the "set of feasible configurations", defined by

Q := q ∈ R 2N , D ij (q) ≥ 0 ∀ i = j , (40)
where D ij (q) = |q iq j | -2r is the signed distance between disks i and j.

We denote by U(q) = (U 1 (q 1 ), .., U N (q N )) ∈ R 2N the global spontaneous velocity of the crowd. We introduce the "set of feasible velocities" defined by:

C q = v ∈ R 2N , ∀i < j D ij (q) = 0 ⇒ G ij (q), v ≥ 0 ,
with G ij (q) = ∇D ij (q) = (0, . . . , 0, -e ij (q), 0, . . . , 0, e ij (q), 0, . . . , 0) ∈ R 2N

and e ij (q) = q j -q i |q j -q i | . The actual velocity field is defined as the feasible field which is the closest to U in the least square sense, which writes

dq dt = P Cq (U(q)) , (41)
where P Cq denotes the Euclidean projection onto the closed convex cone C q . We deduce from Farkas Lemma (see e.g. [START_REF] Ciarlet | Introduction à l'analyse numérique matricielle et à l'optimisation[END_REF]) and Proposition 2.8 the following results:

Proposition 4.1. The negative polar cone N q of C q defined by

N q := C • q := w ∈ R 2N , w, v ≤ 0 ∀v ∈ C q ,
is equal to the proximal normal cone N(Q, q). More precisely,

N q = N(Q, q) = - λ ij G ij (q), λ ij ≥ 0, D ij (q) > 0 =⇒ λ ij = 0 .
Using the classical orthogonal decomposition with two mutually polar cones (see [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF]), the main equation (41) becomes dq dt

+ P N(Q,q) (U(q)) = U(q). (42)
According to Proposition 3.3 in [START_REF] Bernicot | Existence of sweeping process in Banach spaces under directional prox-regularity[END_REF], we know that for a Lipschitz map U, this differential equation is equivalent to the following differential inclusion:

dq dt + N(Q, q) ∋ U(q). (43)
For all (i, j), D ij is a convex function and belongs to C 2 (U ij ), with

U ij := {q ∈ R 2N , |q i -q j | -r > 0}
and satisfies Assumption (A0) with c = r/ √ 2. Moreover it is obvious that

∀q ∈ U ij , |G ij (q)| = √ 2 and |D 2 D ij (q)| ≤ 2 r .
As a consequence, Assumptions (A1) and (A4) are satisfied. It remains to check the inverse triangle inequality (Assumption (A5')), which is the aim of the following proposition.

Proposition 4.2 (Inverse triangle inequality).

There exists γ > 1 such that for all q ∈ Q,

(i,j)∈I(q) α ij |G ij (q)| ≤ γ (i,j)∈I(q) α ij G ij (q) ,
where I(q) = {(i, j), i < j, D ij (q) = 0} and α ij are nonnegative reals.

Constant γ can be fixed as follows

γ = 3 √ 2N 3 sin 2π N N .
The next subsection is devoted to its proof. Remark 4.3. Note the sign of coefficients α ij . From a general point of view, this inequality is obviously wrong if these coefficients are just assumed real. Indeed, for N ≥ 6, the cardinal of the set I(q) is strictly larger than 2N , which involves a relation between vectors G ij (q). Remark 4.4. In [START_REF] Maury | A discrete contact model for crowd motion[END_REF], we have already proved such a result. The proof we propose here gives a smaller constant γ and above all, it allows to better understand how the non-uniqueness of the Kuhn-Tucker multipliers appears. Proposition 2.9, Theorems 2.13 and 3.4 imply the next results. 42) is wellposed in the set of the absolutely continuous functions and the related numerical scheme ( 22) is convergent.

Other numerical simulations based on this scheme, we refer the reader to [START_REF] Maury | Un modèle de mouvement de foule[END_REF][START_REF] Maury | Handling of contacts on crowd motion simulations[END_REF][START_REF] Venel | Integrating strategies in numerical modelling of crowd motion[END_REF][START_REF] Maury | A discrete contact model for crowd motion[END_REF].

Proof of the inverse triangle inequality

In order to prove Proposition 4.2, we are firstly going to show that the terms λ ij appearing in the following equation :

i<j λ ij G ij (q) = F are bounded.
Proposition 4.7. For all q ∈ Q, for all F ∈ R 2N , the following set

Λ q,F :=    λ ∈ R N(N-1) 2 , i<j λ ij G ij (q) = F, λ ij ≥ 0, λ ij = 0 if D ij (q) > 0   
is uniformly bounded with respect to q. More precisely,

∀λ ∈ Λ q,F , ∀i < j, λ ij ≤ |F| a N with a = 3 sin( 2π N )
.

First we check that Proposition 4.7 implies Proposition 4.2. Indeed, by Proposition 4.7, we have for all (k, l) ∈ I(q),

α kl ≤ a N (i,j)∈I(q) α ij G ij (q) , with a = 3 sin 2π N . Since |G kl (q)| = √ 2
, by summing coefficients α kl for all (k, l) ∈ I(q) , we obtain

(k,l)∈I(q) α kl |G kl (q)| ≤ √ 2 |I(q)| a N (i,j)∈I(q) α ij G ij (q) .
In the monodisperse case, each disk has at most 6 neighbours. As a consequence, |I(q)| ≤ 3N, and thus,

(k,l)∈I(q) α kl |G kl (q)| ≤ 3 √ 2N a N (i,j)∈I(q) α ij G ij (q) .
It now remains to prove Proposition 4.7.

Proof of Proposition 4.7. Suppose that set Λ q,F is not empty, we want to estimate the solutions λ of the following system containing 2N equations:

i < j Dij (q) = 0 λ ij G ij (q) = F , λ ij ≥ 0. (P )
By specifying the expression of vectors G ij (q), we can write the system concerning individual i 0 ,

n j = 1 j = i0 j neighbour of i0 λ ji 0 e ji 0 = F i 0 , (P i 0 )
where

λ ji 0 = λ ji 0 if j < i 0 λ i 0 j if j > i 0
and e ji 0 = e ji 0 (q).

We assume here that the configuration q ∈ Q is a cluster in the sense that the set N i=1 B(q i , r) is connected by arcs. Otherwise, for the other configurations, it suffices to deal with the different clusters one by one, as the associated problems are independent from each other. We denote by A = {q 1 , .., q N } the set of all positions. We are going to solve the problems (P i ) one by one. After having solved a problem (P i 0 ), q i 0 is removed from A and problem (P i 0 ) is deleted from the whole problem (P ), and the values of λ ji 0 are taken into account in terms F j , for each person j neighbour of individual i 0 . The goal is to reduce A to a singleton. The order of this algorithm is important. Less terms there are in problem (P i 0 ), more easily solvable it is. First and foremost, we deal with people having only one neighbour (Case 1) because in that case the solution is trivial. When there is no single contact, we consider people with more neighbours. Since we want to have an upper bound for the terms λ ji 0 , we will take care of controlling the angles between vectors e i 0 j 1 and e i 0 j 2 if individuals j 1 and j 2 are in contact with person i 0 (see Case 2).

Case 1: elimination of single contacts Suppose that there exists q i ∈ A such that person i has only one neighbour j. Problem (P i ) becomes λ ji e ji = F i .

Consequently, λ ji = |F i |. We remove q i from A and (P i ) from (P ). Then, we substitute F j by F jλ ji e ij , which implies that the term |F j | is replaced with |F j | + |F i |. In this way, all single contacts are taken off, which allows us for example to completely deal with the case illustrated in Figure 4. If A is not reduced to a singleton after these eliminations, we consider the following case.

Case 2: there are no single contacts We define C the convex hull of A. The set of its extremal points is denoted by E and the cardinal of E by p. We denote by P the boundary of C, which is a convex polygon whose vertices are the points of E. This polygon has p ≥ 3 corners at least, otherwise we would rather consider Case 1.

The angle sum equals to π(p -2). Therefore there exists an angle of P lower than π(1 -2 p ) < π. Suppose that q i is the corresponding vertex. Because of the angle condition, individual i has two or three neighbours. Indeed, if individuals j and k are in contact with person i, the minimum angle between vectors e ij and e ik is equal to π 3 .

Case 2 (a): individual i has 2 neighbours Using a rotation around q i if needed, we can assume that the concerned configuration is represented in Figure 5. In view of the choice of q i , θ ≥ 2π p ≥ 2π N and θ ≤ 2π 3 . Note that N ≥ p ≥ 3. By writing the expression of e ji and e ki , we can solve Problem (P i ) after a simple computation:

Note that N ≥ p ≥ 6. In fact, for N < 7 persons, we can see that the cluster A can be reduced to a singleton only by considering Case 1.

q i q j q l q k θ φ In this case, the terms λ ji , λ ki and λ li are not unique (3 unknowns and 2 equations). We define a particular solution of Problem (P i ) (λ 0 ji , 0, λ 0 li ), where

  λ 0 ji λ 0 li   = 1 sin θ
sin θ cos θ 0 1

F x i F y i . ( 46
)
To obtain all solutions, it remains to describe the kernel of System (P i ). We can easily show the next lemma. Since we are only looking for the nonnegative solutions, the sign of sin θ implies t ≥ 0. Furthermore, the signs of sin(φθ) and of sin φ involve t ≤ t max where

t max = min λ 0 ji -sin(φ -θ) , λ 0 li -sin φ . ( 47
)
Moreover, the following inequalities are satisfied:

λ ji ≤ √ 2

Conclusion

By iterating this process, we solve Problem (P ) with at most (N -1) steps. We can conclude with the following lemma, whose the proof is a straightforward computation. Lemma 4.9. At every step, the norm |F| is replaced at worst with 3 sin( 2π N ) |F|.

This lemma ends the proof of Proposition 4.7.

Remark 4.10. A similar result can be proved in the polydisperse situation (where the radii are not assumed equal). Case 2 presents also more possibilities because the maximum number of neighbours that a person can have, depends on the radii r i . By denoting n v this number, a simple computation shows that n v ≤ π arcsin r min r max + r min , where r min = min r i et r max = max r i . With these notations, it can be proved that in the polydisperse case (see [START_REF] Venel | Modélisation mathématique et numérique des mouvements de foule[END_REF])

∀λ ∈ Λ q,F , ∀i < j, λ ij ≤ |F| b N with b = 2 √ n v min sin π n v + 1
, sin 2π N Indeed Proposition 4.2 still holds with I ρ (t, q), ρ < r by replacing γ by another constant.

Definition 2 . 1 .

 21 Let S be a closed subset of R d . The proximal normal cone to S at x is defined by: N(S, x) := v ∈ R d , ∃α > 0, x ∈ P S (x + αv) , where P S (y) := {z ∈ S, d S (y) = |y -z|}, with d S (y) := inf z∈S |y -z| corresponds to the Euclidean projection onto S.

Lemma 2 . 4 .

 24 Let S be a closed convex set included in R d . Then for all x ∈ S and w ∈ R d : w ∈ N(S, x) def ⇔ x = P S (x + w) (a) ⇔ ∀y ∈ S, w, yx ≤ 0 (b)

3 Figure 3 :

 33 Figure 3: Labyrinth game.

Proposition 4 . 5 .Theorem 4 . 6 .

 4546 The set Q ⊂ R 2N , defined by (40) is η-prox-regular with a constant Assume that U is Lipschitz. The Cauchy problem associated to (

Figure 4 :Figure 5 :

 45 Figure 4: Case of a chain.q i q j

  |F i | and λ ki ≤ 1 sin θ |F i |.We remove q i from A and we replace|F j | (respectively |F k |) with |F j | + √ 2|F i |/ sin(2π/N ) (respectively |F k | + |F i |/ sin(2π/N )) because sin θ ≥ sin( 2π N ). Problem (P i ) is also eliminated.Case 2 (b): individual i has 3 neighbours Using a rotation around q i if needed, we can assume that the concerned configuration is represented in Figure6. Because of the non-overlapping constraint, angles θ and φ satisfy the following

Figure 6 :

 6 Figure 6: Case 2.

Lemma 4 . 8 .

 48 The kernel of System (P i ) is generated by: In addition, the signs of the coordinates can be specified sin(φθ) ≤sin 2π N < 0, sin θ ≥ sin 2π N > 0 and sin φ ≤ -

Remark 4 . 11 .

 411 The radii can be allowed to depend on time in a Lipschitz way. This new problem fits into the framework of sweeping process considered in this paper provided that r := inf t∈[0,T ] inf i∈{1,...,N } r i (t) > 0.

  sin θ |F i |, λ li ≤ 1 sin θ |F i | and λ ki ≤ t max sin θ ≤ 2 √ 3 |F i |.So we remove q i from A and we replace|F j | (respectively |F k | and |F l |) with |F j |+ √ 2|F i |/ sin(2π/N ) (respectively |F k | + 2|F i |/ √ 3 and |F l | + |F i |/ sin(2π/N )). Problem (P i ) is also eliminated. Now we have removed one person from set A. If this set is not reduced to a singleton, we return to Case 1 or Case 2.


Yves Colin De Verdière
email: yves.colin-de-verdiere@ujf-grenoble.fr

Françoise Truc
email: francoise.truc@ujf-grenoble.fr

Confining quantum particles with a purely magnetic field

We consider a Schrödinger operator with a magnetic field (and no electric field) on a domain in the Euclidean space with a compact boundary. We give sufficient conditions on the behaviour of the magnetic field near the boundary which guarantees essential self-adjointness of this operator. From the physical point of view, it means that the quantum particle is confined in the domain by the magnetic field. We construct examples in the case where the boundary is smooth as well as for polytopes; these examples are highly simplified models of what is done for nuclear fusion in tokamacs. We also present some open problems.

1 Introduction

The problem

Let us consider a particle in a domain Ω in R d (d ≥ 2) in the presence of a magnetic field B. We will always assume that the topological boundary ∂Ω := Ω \ Ω of Ω is compact. At the classical level, if the strength of the field tends to infinity as x approaches the boundary ∂Ω, we expect that the charged particle is confined and never visits the boundary: the Hamiltonian dynamics is complete. At the quantum level the fact that the particle never feels the boundary amounts to saying that the magnetic field completely determines the motion, so there is no need for boundary conditions. At the mathematical level, the problem is to find conditions on the behaviour of B(x) as x tends to ∂Ω which ensure that the magnetic operator H A is essentially self-adjoint (see Section 2.6) on C ∞ 0 (Ω) (the space of compactly supported smooth functions). These conditions will not depend on the gauge A, but only on the field B. One could have called such pairs (Ω, A) "magnetic bottles", but this denomination is already introduced in the important paper [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF] for Schrödinger operators with magnetic fields in the whole of R d having compact resolvents. This question may be of technological interest in the construction of tokamacs for the nuclear fusion [30]. The ionised plasma which is heated is confined thanks to magnetic fields.

Previous works

The same problem, concerning scalar (electric) potentials, has been intensively studied. In the many-dimensional case the basic result appears in a paper by B. Simon [START_REF] Simon | Essential Self-adjointness for Schrödinger operators with singular potentials: A generalized Kalf-Walter-Schminke theorem[END_REF] which generalises results of H. Kalf, J. Walter and U.-V. Schminke (see [START_REF] Kalf | On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials[END_REF] for a general review). Concerning the magnetic potential, the first general result is by Ikebe and Kato: in [START_REF] Ikebe | Uniqueness of the self-adjoint extension of singular elliptic differential operators[END_REF], they prove self-adjointness in the case of Ω = R d for any regular enough magnetic potential. This result was then improved in [START_REF] Simon | Schrödinger operators with singular magnetic vector potentials[END_REF][START_REF] Shubin | The essential Self-adjointness for Semi-bounded Magnetic Schrödinger operators on Non-compact Manifolds[END_REF].

Concerning domains with boundary, we have not seen results in the purely magnetic case. A regularity condition on the direction of the magnetic field was introduced in the important paper [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF] (Corollary 2.10 p. 853) in order to construct "magnetic bottles" in R d . It was used later in many papers like [START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques[END_REF][START_REF] Dufresnoy | Un exemple de champ magnétique dans R ν[END_REF][START_REF] Truc | Trajectoires bornées d'une particule soumise à un champ magnétique linéaire[END_REF][START_REF] Truc | Semi-classical asymptotics for magnetic bottles[END_REF][START_REF] Erdoes | Semiclassical eigenvalue estimates for the Pauli operator with strong nonhomogeneous magnetic fields[END_REF][START_REF] Erdoes | Magnetic Lieb-Thirring inequalities with optimal dependence on the field strength[END_REF][START_REF] Erdoes | Magnetic Lieb-Thirring inequalities for the 3dimensional Pauli operator with a strong nonhomogeneous magnetic fields[END_REF][START_REF] Balinsky | Generalized Hardy Inequality for the Magnetic Dirichlet Forms[END_REF]].

In the recent paper [START_REF] Nenciu | On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R n[END_REF], G. Nenciu and I. Nenciu give an optimal condition for essential self-adjointness on the electric potential near the boundary of a bounded smooth domain; they use Agmon-type results on exponential decay of eigenfunctions combined with multidimensional Hardy inequalities.

Rough description of our results

As we will see, in the case of a magnetic potential the Agmon-type estimates still hold, whereas the Hardy inequalities cannot be used because there is no separation between kinetic and potential energy. Actually the point is that we need, to apply the strategy of [START_REF] Nenciu | On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R n[END_REF], some lower bound on the magnetic quadratic form h A associated with the magnetic potential A. Our main result is as follows: under some continuity assumption on the direction of B(x) at the boundary, for any ǫ > 0 and R > 0, there exists a constant C ǫ,R ∈ R such that the quadratic form h A satisfies the quite optimal bound

∀u ∈ C ∞ 0 (Ω), h A (u) ≥ (1 -ǫ) Ω∩{x| |x|≤R} |B| sp |u| 2 |dx| -C ǫ,R u 2 . (1.1)
Here |B(x)| sp is a suitable norm on the space of bi-linear antisymmetric forms on R d , called the spectral norm. This implies that H A is essentially self-adjoint if there exists η > 0 so that

|B(x)| sp ≥ (1 + η)D(x) -2
where D is the distance to the boundary of Ω.

We study then examples in the following cases:

• The domain Ω is a polytope

• The boundary ∂Ω is smooth and the Euler characteristic χ(∂Ω) vanishes (toroidal domain)

• The boundary ∂Ω is smooth and the Euler characteristic χ(∂Ω) does not vanish (non toroidal domain)

• The domain Ω is R 3 \ 0 and the field is a monopole or a dipole

• The domain Ω is the unit disk: for any ǫ > 0 and d = 2, we construct an example of a non essentially self-adjoint operator

H A with |B(x)| sp ∼ ( √ 3/2 -ǫ)D(x) -2
showing that our bound is rather sharp.

Open problems

The following questions seem to be quite interesting:

• What are the properties of a classical charged particle in a confining magnetic box? Are almost all trajectories not hitting the boundary?

• What is the optimal constant C in the estimates |B(x)| sp ≥ CD(x) -2 of our main result 3.2? From our main results and the example in the unit disk given in Section 5.4, we see that the optimal constant lies in the interval [ √ 3/2, 1].

Definitions and background results

In this section, we will give precise definitions and related notations. We will also review some known results with references to the literature.

The domain Ω

In what follows, we will keep the following definitions: Ω is an open set in the Euclidean space R d (d ≥ 2) with a compact topological boundary ∂Ω = Ω \ Ω, so that either Ω or R d \ Ω is bounded.

Definition 2.1 We will denote by d R the distance defined on Ω by the Riemannian metric induced by the Euclidean metric:

d R (x, y) = inf γ∈Γx,y length(γ)
where Γ x,y is the set of smooth curves γ : [0, 1] → Ω with γ(0) = x, γ(1) = y.

We will denote by Ω the metric completion of (Ω, d R ) and by Ω ∞ = Ω \ Ω the metric boundary of Ω.

We say that

Ω is regular if Ω ∞ is compact.
If Ω is regular, ∂Ω is compact. In fact the identity map of Ω extends to a continuous map π from Ω onto Ω and π(Ω

∞ ) = ∂Ω. ( Ω, π) is a "desingularisation" of Ω. If X = ∂Ω is a compact C 1 sub-manifold or a compact simplicial complex embedded in a piecewise C 1 way, Ω is regular. If X = ∪ n∈N [0, 1]e n with e n a sequence of unit vectors in R 2 converging to e 0 , then R 2 \ X is not regular, even if ∂Ω = X is compact. ∂Ω π Ω ∞ e 2
e n e 0 e 1 The Lebesgue measure will be |dx| and we will denote by u, v := Ω uv|dx| the L 2 scalar product and by u the L 2 norm of u. We will denote by C ∞ 0 (Ω) the space of complex-valued smooth functions with compact support in Ω. The inequality |D(x) -D(x ′ )| ≤ d R (x, x ′ ) follows from the triangle inequality for dR . The almost everywhere differentiability of Lipschitz functions is the celebrated Theorem of Hans Rademacher [START_REF] Rademacher | Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale[END_REF]; see also [START_REF] Morrey | Multiple Integrals in the Calculus of Variations[END_REF] p. 65 and [START_REF] Heinonen | Lectures on Lipschitz analysis[END_REF].

The distance to the boundary

Adapted charts for smooth boundaries

Assuming that the boundary is smooth, we can find, for each point

x 0 ∈ ∂Ω, a diffeomorphism from an open neighbourhood U of x 0 in R d onto an open neigh- bourhood V of 0 in R d x1,x ′ satisfying: • x 1 (F (x)) = D(x) • The differential F ′ (x 0 ) of F is an isometry • F (U ∩ Ω) = V ∩ {x 1 > 0}.
We will call such a chart an adapted chart at the point x 0 . Such a chart is an ǫ-quasi-isometry (see the definition in Section 4.2) with ǫ as small as one wants by choosing U small enough.

Antisymmetric forms

Let us denote by ∧ k R d the space of real-valued k-linear antisymmetric forms on the Euclidean space R d . The space ∧ 1 R d is the dual of R d , and it is equipped with the natural Euclidean norm:

| d j=1 a j dx j | 2 = d j=1 a 2 j . The space ∧ 2 R d is equipped with the spectral norm: if B ∈ ∧ 2 R d ,

Magnetic fields

Let us give the basic definitions and notations concerning magnetic fields in a domain Ω. The magnetic potential is a smooth real 1-form A on Ω ⊂ R d , given by A = d j=1 a j dx j , and the associated magnetic field is the 2-form B = dA; more explicitly, we have

B(x) = 1≤j<k≤d b jk (x)dx j ∧dx k with b jk (x) = ∂ j a k (x)-∂ k a j (x) .
Let us define now the Schrödinger operator with magnetic field B = dA:

Definition 2.7 The magnetic connection ∇ = (∇ j ) is the differential operator defined by

∇ j = ∂ ∂x j -ia j .
The magnetic Schrödinger operator H A is defined by

H A = - d j=1 ∇ 2 j .

The magnetic Dirichlet integral h

A = H A .|. is given, for u ∈ C ∞ 0 (Ω), by h A (u) = Ω d j=1 |∇ j u| 2 |dx| .
Let us note the commutator formula [∇ j , ∇ k ] = -ib jk which will be very important. From the previous definitions and the fact that the formal adjoint of

∇ j is -∇ j , it is clear that the operator H A is symmetric on C ∞ 0 (Ω). Definition 2.8 We will say that B = dA is a confining field in Ω if H A is essen- tially self-adjoint (see Section 2.6).

The Riemannian context

"Regular" Riemannian manifolds

The context of an Euclidean domain is not the most natural one for our problem.

In particular, the "regularity assumption" of Definition 2.1 can easily be extended to the Riemannian context. Let (Ω, g) be a smooth Riemannian manifold. We are interested in cases where (Ω, g) is not complete. Let us recall that g induces on Ω a distance d g defined by d g (x, y) = inf γ∈Γx,y length(γ) where Γ x,y is the set of smooth paths γ : [0, 1] → Ω so that γ(0) = x, γ(1) = y. We will denote by Ω the metric completion of Ω and by Ω ∞ = Ω \ Ω the metric boundary. In the case where Ω ⊂ R d is equipped with the Euclidean Riemannian metric, Ω ∞ is in general not equal to the boundary ∂Ω. Definition 2.1 is now replaced by: Definition 2.9 The Riemannian manifold (Ω, g) is regular if Definition 4.4) to an open set of R d with an Euclidean metric.

1. Ω ∞ is compact 2. For any ǫ > 0, every x 0 ∈ Ω ∞ has a neighbourhood U so that so that U ∩ Ω is ǫ-quasi-isometric (see
A function f : Ω → C is regular at the boundary if it extends by continuity to Ω.

Magnetic fields on Riemannian manifolds

The magnetic potential is a smooth real valued 1-form A on Ω, the magnetic field is the 2-form B = dA. The norm |B(x)| sp is calculated with respect to the Euclidean metric g x0 . The magnetic potential defines a connection ∇ on the trivial line bundle

Ω × C → Ω by ∇ X f = df (X) -iAf . The magnetic Dirichlet integral is h A (f ) = Ω ∇f 2 g |dx| g where the norm of the 1-form ∇f (x) is calculated with the dual Riemannian norm: ∇f 2 g = ij g ij ∇ ∂i f ∇ ∂j f and |dx| g = θ|dx 1 • • • dx d | is the Riemannian volume.
The magnetic Schrödinger operator is then defined by:

H A f = -θ -1 ij ∇ ∂i θg ij ∇ ∂j f .

Essential self-adjointness

In this section, we will review what is an essentially self-adjoint operator and give some easy propositions which we were not able to point in the literature.

Essentially self-adjoint operators

Let us recall the following Definition 2.10 A differential operator P :

C ∞ 0 (Ω) → C ∞ 0 (Ω) is essentially self- adjoint in L 2 (Ω, |dx|) if P is formally symmetric (for any u, v ∈ C ∞ 0 (Ω), P u|v = u|P v ) and the closure of P is self-adjoint.
A basic criterion for essential self-adjointness is the following (see criterion (4) of Theorem X.1 and Corollaries in [START_REF] Reed | Methods of Modern mathematical Physics II[END_REF] ): Proposition 2.11 Let P be as before and formally symmetric. Let us assume either that [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations[END_REF] there exists E ∈ R so that any solution v ∈ L 2 (Ω) of (P -E)v = 0 (in the weak sense of Schwartz distributions) vanishes, or that (2) there exists λ ± ∈ C with ±ℑ(λ ± ) > 0 so that any solution v ∈ L 2 (Ω) of (Pλ ± )v = 0 (in the weak sense of Schwartz distributions) vanishes.

Then P is essentially self-adjoint.

Essential self-adjointness depends only on the boundary behaviour

Proposition 2.12 Let X be a smooth manifold with a smooth density |dx|. Let L j , j = 1, 2 be two formally symmetric elliptic differential operators of degree m on L 2 (X, |dx|) and let us assume that L 1 is essentially self-adjoint and

L 2 -L 1 = M is compactly supported. Then L 2 is essentially self-adjoint.
Proof.-It is enough to show that L 2 -ci is invertible for c real and large enough. We have

L 2 -ci = Id + M (L 1 -ci) -1 (L 1 -ci).
Moreover the domain of L 1 contains H m 0 (the space of compactly supported H m functions). So that M (L 1ci) -1 = O(c -1 ).

2.6.3 Essential self-adjointness is independent of the choice of a gauge Proposition 2.13 Let X be a smooth manifold with a smooth density |dx|. Let us consider a Schrödinger operator H A1 and

A 2 = A 1 + dF with F ∈ C ∞ (X, R). Then, if H A1 is essentially self-adjoint, H A2 is also essentially self-adjoint.
Proof.-We have formally (as differential operators)

H A2 = e iF H A1 e -iF .
Hence,

H A2 -ci = e iF (H A1 -ci) e -iF . The domain D 2 of the closure of H A2 (defined on C ∞ 0 (X)
) is e iF times the domain D 1 of the closure of H A1 . The result follows from the fact that e ±iF is invertible in L 2 and an isomorphism of the domains.

Main results

Let us take H A with domain D(H

A ) = C ∞ 0 (Ω).
As explained in the introduction, we are looking for growth assumptions on |B| sp close to ∂Ω ensuring essential selfadjointness of H A . We formulate now our main results: Theorem 3.1 Let us take d = 2. Assume that ∂Ω is compact with a finite number of connected components and that B(x) satisfies near ∂Ω

|B(x)| sp ≥ (D(x)) -2 , (3.1)
then the Schrödinger operator H A is essentially self-adjoint. This still holds true for any gauge A ′ such that dA ′ = dA = B.

Theorem 3.2 Let us take d > 2. Assume that Ω is regular and that there exists η > 0 such that B(x) satisfies near ∂Ω

|B(x)| sp ≥ (1 + η) (D(x)) -2 , (3.2)
and that the functions

n jk (x) = b jk (x) |B(x)| sp (3.
Ω := {x ∈ R d | f (x) > 0} with f : R d → R smooth, df (y) = 0 for y ∈ ∂Ω, then f (x) ∼ |df (x)|D(x)
B sp ≥ (1 + ǫ)D -2
near Ω ∞ and that, for each x 0 ∈ Ω ∞ , the direction n(x) of B, calculated with the metric g x0 (i.e. using the trivialisation of the tangent bundle associated with g x0 ), has a limit as x → x 0 , then H A is essentially self-adjoint on C ∞ 0 (Ω). The exponent 2 of the leading term in Equations (3.1) and (3.2) is optimal, as shown in the following Proposition 3.5 For any 0 < α < √ 3/2, there exists a magnetic field B such that H A (with dA = B) is not essentially self-adjoint and such that the growth of |B| sp near the boundary ∂Ω satisfies

|B(x)| sp ≥ α (D(x)) 2 .
We prove this proposition in Section 5.4 in the case d = 2, but the proof can be easily generalised to larger dimensions.

As a consequence of this proposition, together with Theorem 3.1 (respectively 3.2 ), we get that the optimal constant in front of the leading term (D(x)

) -2 is in [ √ 3/2, 1].
Hence we see that the situation for confining magnetic fields is not the same as for confining potentials (for which the optimal constant is 3/4, hence is smaller than √ 3/2). Indeed this is due to the difference between the Hardy inequalities in the two situations: the term 1/(4D 2 ) does not appear in the magnetic case, as it does in the case of a scalar potential, where it plays the role of an "additional barrier".

Proof of the main results

In this Section, we prove Theorems 3.1, 3.2 and 3.4 using the method of [START_REF] Nenciu | On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R n[END_REF] which we first review.

Agmon estimates

The following statement is proved, using Agmon estimates [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations[END_REF], in [START_REF] Nenciu | On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R n[END_REF]: Theorem 4.1 Assume that ∂Ω is compact, and that there exists

c ∈ R such that, for all u ∈ C ∞ 0 (Ω), h A (u) -Ω D(x) -2 |u(x)| 2 |dx| ≥ c u 2 . Then H A is essentially self-adjoint.
Reading the proof in [START_REF] Nenciu | On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R n[END_REF], one sees that the only property of Ω which is used is that the function D(x) is smooth near the boundary and satisfies |dD(x)| ≤ 1. One can extend the proof to the case where ∂Ω is not a smooth manifold by using the properties of the function D described in Lemma 2.4. The fact that Ω is bounded does not play an important role, only the compactness of ∂Ω is important. The essential self-adjointness of H A results from the Proposition 2.11 and the following Theorem 4.2 Let v ∈ L 2 (Ω) be a weak solution of (H A -E)v = 0. Let us assume that there exists a constant c > 0 such that, for all u ∈ C ∞ 0 (Ω),

u|(H A -E)u - {x∈Ω | D(x)≤1} |u(x)| 2 D(x) 2 |dx| g ≥ c u 2 . (4.1)
Then v ≡ 0.

For the reader's convenience, we give here the proof of Theorem 4.2 following the strategy of [START_REF] Nenciu | On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R n[END_REF] in a slightly simplified way.

Proof.-The proof is based on the following simple identity ( [START_REF] Nenciu | On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R n[END_REF])

Lemma 4.3 Let v be a weak solution of (H A -E)v = 0, and let f be a real-valued Lipschitz function with compact support. Then

f v|(H A -E)(f v) = v | |df (x)| 2 v . (4.2)
Let us give two numbers ρ and R satisfying respectively 0 < ρ < 1 2 and 1 < R < +∞. We will apply identity (4.2) with f = F (D) where F (u) the piecewise smooth function defined by 2 almost everywhere. From the inequality (4.1) applied to f v, we get:

F (u) =            0 for u ≤ ρ and for u ≥ R + 1 2(u -ρ) for ρ ≤ u ≤ 2ρ u for 2ρ ≤ u ≤ 1 1 for 1 ≤ u ≤ R R + 1 -u for R ≤ u ≤ R + 1 ρ R R + 1 2ρ 1 F (u) u 1 0 Figure 2: The function F We have |df | 2 = F ′ (D)
(H A -E)(f v) | f v ≥ 2ρ≤D(x)≤1 |v| 2 |dx| g + c f v 2 . (4.3)
On the other hand, using the explicit values of df and Equation (4.2), we get:

(H A -E)(f v) | f v ≤ 4 ρ≤D(x)≤2ρ |v| 2 |dx| g + • • • • • • 2ρ≤D(x)≤1 |v| 2 |dx| g + R≤D(x)≤R+1 |v| 2 |dx| g . (4.4)
Putting together the inequalities (4.3) and (4.4), we get

c f v 2 ≤ 4 ρ≤D(x)≤2ρ |v| 2 |dx| g + R≤D(x)≤R+1 |v| 2 |dx| g . (4.5)
Taking ρ → 0 and R → +∞ in the inequalities (4.5), we get that the L 2 norm of v vanishes.

Quasi-isometries

In section 5 we give examples which have smooth boundaries (excepting the convex polyhedra (section 5.1)). In order to build new examples, like non convex polyhedra, one can use quasi-isometries.

Definition 4.4 Given 0 < c ≤ C, a (c, C)-quasi-isometry of Ω 1 onto Ω 2 is an homeomorphism of F : Ω 1 onto Ω 2 whose restriction to Ω 1 is a smooth diffeomor- phism onto Ω 2 and such that ∀x, y ∈ Ω 1 , cd R (x, y) ≤ d R (F (x), F (y)) ≤ Cd R (x, y) .
An ǫ-quasi-isometry is an (1ǫ, 1 + ǫ) quasi-isometry.

Lemma 4.5 We have the bounds

F ′ ≤ C, (F -1 ) ′ ≤ c -1 , |det(F ′ )| ≤ C d , cD 1 (x) ≤ D 2 (F (x)) ≤ CD 1 (x)
, where, for i = 1, 2, D i (x) denotes, for any x ∈ Ω i , the distances to the boundary (Ω i ) ∞ .

We will start with a magnetic potential A 2 in Ω 2 and define A 1 = F ⋆ (A 2 ). We want to compare the magnetic quadratic forms h A2 (u) and h A1 (u • F ) as well as the L 2 norms. We get:

Theorem 4.6 Assuming that, for any u ∈ C ∞ 0 (Ω 2 ), h A2 (u) ≥ K Ω2 |u| 2 D 2 2 |dx 2 | -L u 2 , we have, for any v ∈ C ∞ 0 (Ω 1 ), h A1 (v) ≥ K c C d+2 Ω1 |v| 2 D 2 1 |dx 1 | -Lc 2 v 2 .
In other words, we can check that H A1 is essentially self-adjoint from an estimate for h A2 using Theorem 4.1.

Proof.-Let us start making the change of variables

x 2 = F (x 1 ) in the integral h A2 (u). Putting v = u • F , we get h A2 (u) = Ω1 ∇ A1 v(x 1 ) 2 g |det(F ′ (x 1 ))||dx 1 |
where g is the inverse of the pull-back of the Euclidean metric by F . Using Lemma 4.5, we get the estimate.

Lower bounds for the magnetic Dirichlet integrals

+ • • • + | b 2 d-1,2 du|u | . Proof.-We have | b 12 u|u | = | [∇ 1 , ∇ 2 ]u|u | ≤ 2| ∇ 1 u|∇ 2 u | ≤ Ω (|∇ 1 u| 2 + |∇ 2 u| 2 )|dx| .
We take the sum of similar inequalities replacing the indices (1, 2) by [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF][START_REF] Balinsky | Generalized Hardy Inequality for the Magnetic Dirichlet Forms[END_REF]

, • • • , (2 d- 1, 2 d).
Lemma 4.8 Let Ω be a regular open set in R d . Let x 0 ∈ Ω ∞ and assume that B(x) does not vanish near the point x 0 and that the direction of B is regular near x 0 . Let A be a local potential for B near x 0 , then, for any ǫ > 0, there exists a neighbourhood U of x 0 in R d so that, for any φ ∈ C ∞ 0 (U ∩ Ω),

h A (φ) ≥ (1 -ǫ) U |B(x)| sp |φ(x)| 2 |dx| , (4.6)
where |B(x)| sp is defined in Definition 2.5.

Proof.-Let us choose U so that, for all

x ∈ U ∩ Ω, |n(x) -n(x 0 )| Eucl ≤ ǫ 2 d(d-1) , where | i<j a ij dx i ∧ dx j | 2 Eucl = i<j a 2 ij , by applying Definition 2.2 to n(x) at the point x 0 . We choose orthonormal coordinates in R d so that n(x 0 ) = n 12 dx 1 ∧ dx 2 + n 34 dx 3 ∧ dx 4 + • • • with n 2k-1,2k ≥ 0 and k n 2k-1,2k = 1. From Lemma 4.7, we have, for φ ∈ C ∞ 0 (Ω ∩ U ), h A (φ) ≥ U |B(x)| sp (n 12 (x) + n 34 (x) + • • •)|φ(x)|
c R ∈ R so that, ∀u ∈ C ∞ 0 (Ω), h A (u) ≥ Ω∩B(O,R) |B||u| 2 |dx| -c R u 2 . (4.7)
Proof.-As B does not vanish near ∂Ω, the sign of B is constant near each connected component of ∂Ω. Let us write Ω ⊂ ∪ 3 l=1 Ω l with Ω l open sets such that Ω 1 ∩∂Ω = ∅, B > 0 on Ω 2 and B < 0 on Ω 3 . We can assume that Ω 2 and Ω 3 are bounded. Take a partition of unity φ j , j = 1, 2, 3, so that, for j = 2, 3, φ j ∈ C ∞ 0 (Ω j ), and φ 2 j ≡ 1. Now we use the IMS formula (see [START_REF]Geometrical methods in the Quantum Many Body Problem, Nonexistence of very negative ions[END_REF])

h A (u) = 2 l=0 h A (φ l u) - Ω 2 l=0 |dφ l | 2 |u| 2 |dx| . (4.8)
with the lower bound of Lemma 4.7 in Ω l ∩ Ω for l = 2, 3 and the lower bound 0 for Ω 1 .

The case d > 2

Theorem 4.11 Let us assume that ∂Ω ⊂ B(O, R). Assume that B = dA does not vanish near ∂Ω and that the functions n jk (x) are regular at the boundary ∂Ω, then, for any ǫ > 0, there exists C ǫ,R > 0 so that, ∀u ∈ C ∞ 0 (Ω),

h A (u) ≥ (1 -ǫ) Ω∩B(O,R) |B| sp |u| 2 |dx| -C ǫ,R Ω |u| 2 |dx| . (4.9) Proof.-We first choose a finite covering of Ω ∞ by open sets U l , l = 1, • • • N of R d
which satisfies the estimates of Lemma 4.8. We choose then a partition of unity

φ l , l = 0, • • • , N with • For l ≥ 1, φ l ∈ C ∞ 0 (U l ) • φ 0 is C ∞ 0 (Ω) • l φ 2 l ≡ 1 in Ω • sup l |dφ l | 2 = M .
Using the estimates given in Lemma 4.8 for l ≥ 1 and the fact that l |dφ l | 2 is bounded by M , we get, using IMS identity (4.8), the inequality (4.9).

End of the proof of the main theorems

Using Theorem 4.1, it is enough to show that there exists c ∈ R such that, for all

u ∈ C ∞ 0 (Ω), h A (u) ≥ Ω∩B(O,R) |D(x)| -2 |u(x)| 2 |dx| -c u 2 ,
under the assumptions of Theorems 3.1 and 3.2. This is a consequence of Theorem 4.10 for d = 2 and Theorem 4.11 for d > 2.

The proof of Theorem 3.4 is an adaptation of the case of an Euclidean domain. The partition of unity is constructed using only the distance function which has enough regularity. We use also the fact that near each point x 0 of the boundary the metric is quasi-isometrically close to the Euclidean metric g x0 .

Examples

Polytopes

A polytope is a convex compact polyhedron. Let Ω be a polytope given by

Ω = ∩ N i=1 {x | L i (x) < 0} ,
where the L i 's are the affine real-valued functions

L i (x) = d j=1 n ij x j + a i .
We will assume that,

for i = 1, • • • , d, d j=1 n 2 ij = 1 (normalisation)
and n i1 = 0 (this last condition can always be satisfied by moving Ω by a generic isometry). We have the

Theorem 5.1 The operator H A in Ω with

A = 1 n 11 L 1 + 1 n 21 L 2 + • • • dx 2 ,
is essentially self-adjoint.

Proof.-We have

B = 1 L 2 1 + 1 L 2 2 + • • • dx 1 ∧ dx 2 + d j=3 b j dx j ∧ dx 2 ,
and

D = min 1≤i≤N |L i |. So that B = b 12 dx 1 ∧ dx 2 + d j=3 b j2 dx j ∧ dx 2 with b 12 ≥ D -2 .
We then apply directly Lemma 4.7 and Theorem 4.1.

5.2

Examples in domains whose Euler characteristic of the boundary vanishes ("toroidal domains").

Let us assume that ∂Ω is a smooth compact manifold of co-dimension 1 and denote by j : ∂Ω → R d the injection of ∂Ω into R d . A famous theorem of H. Hopf (see [START_REF] Alexandroff | Topologie. Band I[END_REF][START_REF] Guillemin | Differential topology[END_REF]) asserts that there exists a nowhere vanishing tangent vector field to ∂Ω (or 1-form) if and only if the Euler characteristic of ∂Ω vanishes.

Theorem 5.2 Let us assume that the Euler characteristic of ∂Ω vanishes (we say that Ω is toroidal). Let A 0 be a smooth 1-form on Ω so that the 1-form on ∂Ω defined by ω = j ⋆ (A 0 ) does not vanish, and consider a 1-form A in Ω defined, near ∂Ω, by A = A 0 /D α . We assume that either α > 1, or α = 1 with the additional condition that for any y ∈ ∂Ω, |ω(y)| > 1. Then H A is essentially self-adjoint.

Remark 5.3

The existence of ω is provided by the topological assumption on ∂Ω. This works if Ω ⊂ R 3 is bounded by a 2-torus. It is the case for tokamacs.

Proof.-We will apply Theorem 3.2. We have to check:

• The uniform continuity of the direction of the magnetic field or the extension by continuity to Ω. It has to be checked locally near the boundary ∂Ω. We will use an adapted chart (see section 2.2.2).

In these local coordinates we write

A 0 = a 1 dx 1 +β with β = a 2 dx 2 +• • •+a d dx d and ω = a 2 (0, x ′ )dx 2 + • • • + a d (0, x ′ )dx d so we get B = d A 0 x α 1 = x 1 dA 0 -αdx 1 ∧ β x α+1 1 .
Thus we get that the direction of B is equivalent as x 1 → 0 + to that of dx 1 ∧ω which is non vanishing and continuous on Ω.

• The lower bound (3.2) |B| sp ≥ (1 + η)D -2 near ∂Ω. The norm of B near the boundary is given, as x → y by

|B(x)| sp ∼ α|ω(y)|/D α+1 .
Therefore we conclude that the hypotheses of Theorem 3.2 are fulfilled.

Remark 5.4

The asymptotic behaviour of B(x) as x → ∂Ω is

B(x) ∼ - αdx 1 ∧ ω(y) D α+1 .
It follows that ω and α depend only of B and are invariant by any gauge transform in Ω.

Remark

Non toroidal domains

Statement of results

We try to follow the same strategy than in Section 5.2, but now we will allow the 1-form ω on X = ∂Ω to have some zeroes. This is forced by the topology if the Euler characteristic of ∂Ω does not vanish. We need the Definition 5.6 A 1-form ω on a compact manifold X is generic if ω has a finite number of zeroes and dω does not vanish at the zeroes of ω.

We have the Theorem 5.7 Let Ω ⊂ R d with a smooth compact boundary X = ∂Ω. Let A 0 be a smooth 1-form in R d so that ω = j ⋆ X (A 0 ) is generic. We assume also that, at each zero m of ω,

|dω(m)| sp > 1 , (5.1)
where the norm |dω(m)| sp is calculated in the space of anti-symmetric bi-linear forms on the tangent space

T m ∂Ω. Then, if A is a 1-form in Ω such that near X, A = A 0 /D 2 , B = dA is confining in Ω.
We see that the field need to be more singular than in the toroidal case. We could have taken this highly singular part only near the zeroes of ω.

Local model

We will work in an adapted chart at a zero of ω. We take A = A 0 /x 1 2 with j ⋆ (A 0 ) = ω, we have: A 0 = a 1 dx 1 + β and β(0) = 0.

We have

B = dω x 2 1 + dx 1 ∧ ρ + 0(x -1 1 ).
Applying the basic estimates of Lemma 4.7 in some orthonormal coordinates in R d-1 so that dω(0) = b 23 dx 2 ∧ dx 3 + • • • , we see, using the assumption (5.1), that there exists a neighbourhood U of the origin and an η > 0 so that, for any u ∈ C ∞ 0 (U ),

h A (u) ≥ (1 + η) U |u| 2 x 2 1
|dx| .

Globalisation

Near each zero of ω, we take a local chart of R d where A is given by the local model. Such a chart is an ǫ-quasi-isometry (see 4. The corresponding magnetic field B writes B(x, y) = α(r-2) (r-1) 2 dx ∧ dy , and, near the boundary, |B(x)| ∼ α/(D(x)) 2 . We have, in polar coordinates (r, θ),

H A = - ∂ 2 ∂r 2 - 1 r ∂ ∂r - 2iαr r -1 ∂ ∂θ + α 2 r 2 (r -1) 2 .
Hence the operator H A splits as a sum m∈Z H A,m where H A,m acts on functions e imθ f (r). We will look at the m = 0 component: Theorem 5.8 follows from the Lemma 5.9 If 0 < α < √ 3/2, on the Hilbert space L 2 (]0, 1[, rdr), the operator

H = - d 2 dr 2 - 1 r d dr + α 2 r 2 (r -1) 2
is in the limit circle case near r = 1 and hence is not essentially self-adjoint.

Proof.-Let U be the unitary transform

U : u → r 1/2 u from L 2 (]0, 1[, rdr) onto L 2 (]0, 1[, dr). Then K = U HU -1 is given by - d 2 dr 2 - 1 4r 2 + α 2 r 2 (r -1) 2 .
K is known to be in the limit circle case at r = 1 (Theorem X.10 in [START_REF] Reed | Methods of Modern mathematical Physics II[END_REF].)

Singular points

Monopoles

We will first discuss the case of monopoles in R 3 . Here Ω is R 3 \ 0. Definition 5.10 The monopole of degree m, m ∈ Z \ 0, is the magnetic field B m = (m/2)p ⋆ (σ) where p : R 3 \ 0 → S 2 is the radial projection and σ the area form on S 2 . In coordinates

B m = m 2 xdy ∧ dz + ydz ∧ dx + zdx ∧ dy (x 2 + y 2 + z 2 ) 3/2 .
Remark 5.11 Let us note, for comparisons with the case where ∂Ω is of codimension 1, that |B m | sp = |m| 2 r -2 . The flux of B m through S 2 is equal to 2πm. This is a well-known quantisation condition which is needed in order to build a quantum monopole. In order to define the Schrödinger operator H m , we first introduce an Hermitian complex line bundle L m with an Hermitian connexion ∇ m on Ω with curvature B m . We first construct L m and ∇ m on S 2 and then take their pull-backs: ∇ m in a direction tangent to a sphere is the same and ∇ m vanishes on radial directions. We have, using spherical coordinates,

H m = - ∂ 2 ∂r 2 - 2 r ∂ ∂r + 1 r 2 K m
, where K m is the angular Schrödinger operator on S 2 (discussed for example in [START_REF] Torki-Hamza | Stabilité des valeurs propres et champ magnétique sur une variété riemannienne et sur un graphe[END_REF]). Let us denote by λ m 1 the lowest eigenvalue of K m . The self-adjointness of H m depends of the value of λ m 1 . As a consequence of Weyl's theory for Sturm-Liouville equations, H m is essentially self-adjoint if and only if λ m 1 ≥ 3/4. From [START_REF] Kuwabara | On spectra of the laplacian on vector bundles[END_REF][START_REF] Kuwabara | Spectrum of the Schrödinger operator on a line bundle over complex projective spaces[END_REF][START_REF] Torki-Hamza | Stabilité des valeurs propres et champ magnétique sur une variété riemannienne et sur un graphe[END_REF]] (sketched in Section 5.5.2), we know that λ m 1 = |m|/2 so that Theorem 5.12 The Schrödinger operator H m (monopole of degree m) is essentially self-adjoint if and only if |m| ≥ 2.

5.5.2

The spectra of the operators K m , the "spherical Landau levels"

These spectra are computed in [START_REF] Kuwabara | On spectra of the laplacian on vector bundles[END_REF][START_REF] Kuwabara | Spectrum of the Schrödinger operator on a line bundle over complex projective spaces[END_REF] and in the PhD thesis [START_REF] Torki-Hamza | Stabilité des valeurs propres et champ magnétique sur une variété riemannienne et sur un graphe[END_REF]. We sketch here the calculus. Recall that K m is the Schrödinger operator with magnetic field mσ/2 where σ is the area form on S 2 . The metric is the usual Riemannian metric on S 2 : Theorem 5. [START_REF] Heinonen | Lectures on Lipschitz analysis[END_REF] The spectrum of K m is the sequence

λ k = 1 4 k(k + 2) -m 2 , k = |m|, |m| + 2, • • • ,
with multiplicities k + 1. In particular, the ground state λ |m| of K m is |m|/2, with multiplicity |m| + 1. The ground state is exactly the norm of the magnetic field.

If m = 0, the reader will recognise the spectrum of the Laplace operator on S 2 . We start with the sphere S 3 with the canonical metric. Looking at S 3 ⊂ C 2 , we get an free isometric action of S 1 θ on S 3 : θ.(z 1 , z 2 ) = e iθ (z 1 , z 2 ). The quotient manifold is S 2 with 1/4 times the canonical metric; the volume 2π 2 of S 3 divided by 2π is π which is one forth of 4π.

The quotient map S 3 → S 2 is the Hopf fibration, a S 1 -principal bundle. The sections of L m over S 2 are identified with the functions on S 3 which satisfy f (θz) = e imθ f (z). With this identification of the sections of L m , we have

K m = 1 4 ∆ S 3 -m 2 ,
where 1/4 comes from the fact that the quotient metric is 1/4 of the canonical one and m 2 from the action of ∂ 2 θ which has to be removed. It is enough then to look at the spectral decomposition of ∆ S 3 using spherical harmonics: the kth eigenspace of ∆ S 3 is of dimension (k + 1) 2 and splits into k + 1 subspaces of dimension k + 1 corresponding to m = -k, -k + 2, • • • , k.

A general result for

Ω = R d \ 0
In this section Ω = R d \ 0 and B is singular at the origin. Theorem 5.14 If lim x→0 |x| 2 |B(x)| sp = +∞ and, for any x = 0, the direction n(tx) has a limit as t → 0 + , then M B is essentially self-adjoint Proof.-The proof is essentially the same as the proof of Theorem 3.2 except that in the application of IMS method, we have to take a conical partition of unity whose gradients can only be bounded by |x| -1 .

Multipoles

Let us denote, for x ∈ R 3 , B x the monopole with centre x: B x = τ ⋆ x (B 2 ) with τ x the translation by x and B 2 the monopole with m = 2. If P ∂ ∂x is a homogeneous linear differential operator of degree n on R 3 with constant coefficients, we define B P = P (B x ) x=0 . Then B P is called a multipole of degree n. All multipoles are exact! It is a consequence of the famous Cartan's formula: if P is of degree 1, hence a constant vector field,

B V = L V B 0 = d (ι(V )B 0 ) .
A multipole of degree 1 is called a dipole; viewed from very far away, the magnetic field of the earth looks like a dipole.

Theorem 5.15 If B V = dA V is a dipole , H AV is essentially self-adjoint.

Proof.-Because B V is homogeneous of degree -α = -3, it is enough, using 5.14, to show that B V does not vanish. V is a constant vector field, hence up to a dilatation, we can take V = ∂/∂z. We have

B

Figure 1 :Definition 2 . 2

 122 Figure 1: An example where ∂X is compact while X ∞ is not compact We will use the following regularity property: Definition 2.2 Let us assume that Ω is regular. A continuous function f : Ω → C is regular at the boundary if it extends by continuity to Ω.

2. 2 . 1

 21 The distance function Definition 2.3 Let us denote by dR the extension of d R by continuity to Ω. For x ∈ Ω, let D(x) be the distance to the boundary Ω ∞ , given by D(x) = min y∈Ω∞ dR (x, y).

Lemma 2 . 4

 24 The function D is 1-Lipschitz and almost everywhere differentiable in Ω. At any point x of differentiability of D, we have |dD(x)| ≤ 1.

  there exists an orthonormal basis of R d so that B = b 12 dx 1 ∧ dx 2 + b 34 dx 3 ∧ dx 4 + • • • + b 2 d-1,2 d with d = [d/2] and b 12 ≥ b 34 ≥ • • • ≥ 0; the sequence b 12 , b 34 , • • • is unique: the eigenvalues of the antisymmetric endomorphism B of R d associated with B(x) are ±ib 12 , ±ib 34 , • • • , ±ib 2 d-1,2 d and 0 if d is odd. Definition 2.5 We define the spectral norm of B by |B| sp := d j=1 b 2j-1,2j

  for x close to ∂Ω. And we can replace in the estimates (3.2) D(x) by f (x)/|df (x)|. Theorem 3.2 can be extended to Riemannian manifolds as follows: Theorem 3.4 Let (Ω, g) be a regular Riemannian manifold with a magnetic field B = dA. Let us assume that

4. 3 . 1

 31 Basic magnetic estimates Lemma 4.7 For any u ∈ C ∞ 0 (Ω), we have h A (u) ≥ | b 12 u|u | + | b 34 u|u |

  [START_REF] Balinsky | Generalized Hardy Inequality for the Magnetic Dirichlet Forms[END_REF] ) with ǫ as small as one wants. This gives the local estimate near the zeroes of ω. The local estimate outside the zeroes of ω is clear because we have then |B| sp ≥ C/D 3 with C > 0: this follows from the estimates in Section 5.2 with α = 2. We finish the proof of Theorem 5.7 with IMS formula and the local estimates needed in Theorem 4.1.5.4 An example of a non essentially self-adjoint Schrödinger operator with large magnetic field near the boundaryLet us consider the 1-form defined on Ω = {(x, y) ∈ R 2 | x 2 + y 2 = r 2 < 1} by A = α(xdyydx)/(r -1) where 0 < α < √ 3/2. The magnetic potential A is invariant by rotations. Then Theorem 5.8 The operator H A is not essentially self-adjoint.

.

  |B| sp is one half of the trace norm of B, hence|B| sp is a norm on ∧ 2 R d . If d = 2, |B| sp = |B|; if d = 3, |B| spis the Euclidean norm of the vector field B associated with B, defined by ι( B)dx ∧ dy ∧ dz = B where ι( B)ω is the inner product of the vector field B with the differential form ω.

	Remark 2.6 |B| sp is the infimum of the spectrum of the Schrödinger operator with constant magnetic field B in R d .

  5.5 If d = 3, the magnetic field B can be identified with a vector field B in Ω defined by ι B dx 1 ∧ dx 2 ∧ dx 3 = B as in Section 2.3. Using the induced Riemannian structure, we can identify any 1-form ω on ∂Ω with a vector field ω. Moreover ∂Ω is oriented by any 2-form Σ = ι(ν)dx 1 ∧ dx 2 ∧ dx 3 with ν any outgoing vector field near ∂Ω. Using the previous identifications, the asymptotic behaviour of B near ∂Ω is given by B ∼ αr ( ω) /D α+1 , where r is the rotation by +π/2 in the tangent space to ∂Ω. It means that B is very large near ∂Ω and parallel to ∂Ω. From the point of view of classical mechanics, the trajectories of the charged particle are spiralling around the field lines and do not cross the boundary. It would be nice to have a precise statement.

  dz + 3yzdz ∧ dx + (2z 2x 2y 2 )dx ∧ dy (x 2 + y 2 + z 2 ) 5/2 .The form B ∂/∂z does not vanish in Ω.Remark 5.[START_REF] Kostant | Quantization and unitary representations[END_REF] We do not know if all multipoles of degree ≥ 2 are essentially selfadjoint.

		3/2	,
	which gives
	B ∂/∂z =	3xzdy ∧

∂/∂z = d dt |t=0 xdy ∧ dz + ydz ∧ dx + (zt)dx ∧ dy (x 2 + y 2 + (zt) 2 )