Lledos Benjamin
email: benjamin.lledos@math.univ-toulouse.fr

A UNIQUENESS RESULT FOR A TWO DIMENSIONAL VARIATIONAL PROBLEM

We investigate the uniqueness of the solutions for a non-strictly convex problem in the Calculus of Variations of the form φ(∇v) -λv. Here, φ : R 2 → R is a convex function and λ is Lipschitz continuous. We prove the uniqueness when ∇λ is small and give some counterexamples when that is not the case. The proof is based on the global Lipschitz regularity of the minmizers and on the study of their level sets.

Mathematics Subject Classification: 35A02, 49J45, 49N99.

f (t) =      1 2 |t| 2 if |t| ≤ 1, |t| -1 2 if 1 < |t| < 2, 1 4 |t| 2 + 1 2 if 2 ≤ |t|.
For this functional, the admissible functions u belong to the Sobolev space W 1,2 (Ω) with a prescribed trace ψ : R 2 → R on the boundary ∂Ω of Ω. Our goal is to prove the uniqueness of solutions to the following minimization problem:

P λ : min u∈W 1,2 ψ (Ω)
I λ (u).

When λ ≡ λ 0 ∈ R + , this problem studied by Kawohl, Stara and Wittum in [START_REF] Kawohl | Analysis and numerical studies of a problem of shape design[END_REF] arises as the convexification a non convex problem of shape optimization in the theory of elasticity. In this example, f is the convexification of the minimum of two parabolas: t → 1 2 |t| 2 and t → 1 4 |t| 2 + 1 2 . Observe in particular that f is affine on the interval (1, 2). Since f is convex but not strictly convex, there is no obvious reason for P λ to have a unique solution.

In fact, the authors of [START_REF] Kawohl | Analysis and numerical studies of a problem of shape design[END_REF] rely on the assumption that the level sets of one minimizer u are star-shaped. Furthermore, they suppose that the boundary of the set in Ω where f ′ (|∇u|) = 1 is piecewise C 1 . In this paper, we do not require such additional assumptions.

There is no general answer to the question of uniqueness for non strictly convex problems in the Calculus of Variations, especially when λ is not constant. Hence we restrict our attention to the framework (1.1), where f can be replaced by more general convex functions provided that they are strictly convex around the origin and at infinity. 1.2. Main results. More precisely, let φ : R → R be an even C 1 convex function with φ(t) > φ(0) = 0 for all t ̸ = 0. Moreover, we assume that φ ∈ C 1,1 loc (R\{0}) We suppose that φ has p-growth for p > 1, namely, there exist C 1 > 0 and C 2 > 0 such that:

(1.3) C 1 |t| p ≤ φ(t) ≤ C 2 (1 + |t| p ) for all t ∈ R.
We introduce the following set of strict convexity of φ: SC = {x ∈ R, ∀y ∈ R\{x}, ∀t ∈]0, 1[, φ(tx + (1 -t)y) < tφ(x) + (1 -t)φ(y)}. For instance, SC = (-∞, -2) ∪ (-1, 1) ∪ (2, +∞) when φ is equal to the function f in (1.2). We make some structural assumptions on SC: • The set SC has finitely many connected components, in particular SC is open and

SC ∩ R + = N n=0
SC n with SC 0 := [0, b 0 ), SC n := (a n , b n ) for every n ∈ N * , n < N and SC N := (a N , +∞). For every n ∈ N, n < N we introduce d n := φ ′ (b n ) = φ ′ (a n+1 ).

• We assume that φ is C 2 and φ ′′ > 0 on SC\{0} and that φ is strongly convex at +∞ in the following sense:

lim inf t→+∞ tφ ′′ (t) φ ′ (t) > 0.
We define Φ(•) := φ(| • |) and for λ ∈ L ∞ (Ω) we introduce the following functional:

I λ : u → Ω Φ(∇u(x)) -λ(x)u(x)dx on W 1,p ψ (Ω),
where Ω is an open simply connected bounded set of R 2 with a Lipschitz continuous boundary ∂Ω and ψ is a Lipschitz-continuous function on ∂Ω. Here, W 1,p ψ (Ω) is the subset of those functions in W 1,p (Ω) that are equal to ψ on ∂Ω.

We introduce the minimization problem:

P λ : min u∈W 1,p ψ (Ω) I λ (u).
The main result of the paper is the following:

Theorem 1.1.
Let Ω be a simply connected bounded open set of R 2 . We assume that Ω has a C 1,1 boundary, ψ ∈ C 1,1 (R 2 ), λ is Lipschitz continuous on Ω, min where κ is the maximum curvature of Ω such that if ∥∇λ∥ L ∞ (Ω) ≤ C then P λ admits a unique solution on W 1,p ψ (Ω).

Remark 1.2. When λ is constant, a more general result, true in any dimension, can be found in [START_REF] Lledos | A uniqueness result for a translation invariant problem in the calculus of variations[END_REF].

Moreover, the boundedness condition on ∇λ is optimal since: Proposition 1.3. There exists λ ∈ C ∞ (B 1 (0)) with min

B 1 (0)
λ > 0 such that P λ has more than one solution on W 1,p 0 (B 1 (0)). 1.3. Ideas of the proof. We want to prove the uniqueness of the solution u for the variational problem P λ . We know by classical theory (see [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF] and [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]) that the problem P λ admits at least one minimizer u, this function u is bounded, globally Hölder continuous by [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Theorem 7.8] and locally Lipschitz continuous by [START_REF] Celada | Existence and regularity of minimizers of nonconvex integrals with p -q growth[END_REF]Theorem 1.1]. When λ = 0, the proof is substantially simplified. In this case, the strategy has been developed by Marcellini in [START_REF] Marcellini | A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations[END_REF] under additional assumptions and the proof itself in a general framework is due to Lussardi and Mascolo in [START_REF] Lussardi | A uniqueness result for a class of non-strictly convex variational problems[END_REF]. In those two papers the proof is divided into two parts:

• Part 1 If u and v are two solutions of the same problem, then v is constant on the level sets of u.

• Part 2 The level sets of u intersect the boundary ∂Ω of Ω. Since u and v are equal on ∂Ω they are equal on Ω. As observed in Remark 1.2, when λ ≡ λ 0 ∈ R + , a shorter proof can be found in [START_REF] Lledos | A uniqueness result for a translation invariant problem in the calculus of variations[END_REF] but when λ ∈ W 1,∞ (Ω) the proof requires new ideas and turns out to be fairly intricate. Part 1 remains true but Part 2 fails to be true. In fact, the term u → Ω λu changes the geometry of the level lines: they do not necessarily intersect the boundary ∂Ω of Ω. It is even possible that only one level set intersects the boundary, see Proposition 2.10.

A very important subset of Ω is the following: When λ ̸ = 0, the set U ∪ ∂Ω plays the same role as the one played by ∂Ω in Part 2 when there is no lower order term. However, the fact that u = v on U is far from being obvious. Nevertheless, if u and v are two minimizers of the same problem, we can easily see that ∇u and ∇v are equal on U and we can even prove that this is also the case on the level sets that intersect U . The aim of the proof is to show that u = v or ∇u = ∇v on the level sets of u and v. Hence, for a.e. x ∈ Ω, the Lipschitz map w(x)

:= u(x) -v(x) is equal to 0 or ∇w(x) = 0, thus u -v = w = 0.
This idea of using U ∪ ∂Ω comes from a paper by Bouchitté and Bousquet [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF] but in their case the fact that SC is of the form (1, +∞) implies that the boundary of every connected component of U intersects ∂Ω. Since ∇u = ∇v on U and u = v on ∂Ω we readily obtain that u = v on U and this part of the proof is easier. We warn the reader that this paper is not a generalization of [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF] because φ has no singularity at the origin unlike in [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF]. This singularity of φ in [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF] and in its generalization [START_REF] Lledos | A uniqueness result for a non-strictly convex problem in the calculus of variations[END_REF] creates some regularity issues that will not appear here.

For instance in our situation, we exploit the fact that thanks to [START_REF] Silva | Minimizers of convex functionals arising in random surfaces[END_REF], max(α, φ ′ (|∇u|)) ∈ W 1,2 loc (Ω) for any α > 0. Then we prove that max(d 0 , φ ′ (|∇u|)) has a representative that is absolutely continuous on almost every level sets.

The other major difference between this paper and [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF][START_REF] Lledos | A uniqueness result for a non-strictly convex problem in the calculus of variations[END_REF] is that in these references, max(1, φ ′ (|∇u|) is continuous on Ω which allows to prove their results for any dimension larger or equal to two. Here, we heavily rely on two results which are only valid in dimension two: a general regularity result for Lipschitz continuous functions, see Theorem 2.8 below and the Jordan curve theorem. The latter is the reason why we assume that Ω simply connected: this prevents the existence of holes inside the connected components of the upper level set

E s := [u > s] for s ∈ R.
When λ is small we can prove that almost every level set intersects a connected component of U where |∇u| < b 0 . This allows us to prove the following theorem: Theorem 1.5. When 0 ≤ λ(x) ≤ d 0 h Ω for a.e. x ∈ Ω, the problem P λ admits a unique minimizer.

Here, d 0 = φ ′ (b 0 ) and h Ω is the Cheeger constant of Ω: Definition 1.6. The Cheeger constant of Ω is defined as:

h Ω = inf D⊂Ω Per(D, R 2 ) |D| where Per(D, R 2 ) = sup D div g g ∈ C 1 c (R 2 ; R 2 ), |g(x)| ≤ 1 , ∀x ∈ R 2 is called the Perimeter of the set D. A set D ⊂ Ω of finite perimeter is said to be a Cheeger set if Per(D, R 2 ) = h Ω |D|.
The proof of the main theorem will be based on an induction argument related to the family {d n , n ∈ N, 0 ≤ n < N } with the previous theorem as the initialization step.

We study the connected components l s (u) of L s (u) := u -1 (s) ⊂ R 2 such that l s (u) is a closed simple curve. The case l s (u) ∩ ∂Ω ̸ = ∅ is easy because u -v is constant on l s (u) and u -v = 0 on ∂Ω so that u -v on l s (u). Hence, we can assume that l s (u) ⋐ Ω and by the Jordan curve theorem, we can define F s as the bounded connected component of R 2 \l s (u). If l s (u) ∩ U = ∅ then we use the following proposition: Proposition 1.7. There exists a representative f 0 of max(d 0 , φ ′ (|∇u|)) such that for a.e. s ∈ R,

if l s (u) ∩ U = ∅ then f 0 is equal to a constant C(l s (u)) ∈ {d i , 0 ≤ i < N } on l s (u).
Another important result is a maximum principle proved in section 5 for smooth approximations of our problem P λ . We first regularize the problem to obtain a sequence (u n ) n∈N of smooth minimizers of smooth problems P λn , with (φ n ) n∈N and (λ n ) n∈N smooth approximations of φ and λ. In section 4, we use the fact that the sequence

(∇u n ) n∈N generates Young measures (ν x ) x∈Ω to prove that φ ′ n (∇u n ) → φ ′ (∇u) a.e.
in Ω when n → +∞. For such approximations we have:

Proposition 1.8. For a.e. s ∈ R, if l s (u) is a connected component of L s (u) which is a closed simple curve and such that l s (u) ⋐ Ω then sup ls(u) max(d 0 , φ ′ n (|∇u n |)) = sup Fs φ ′ n (|∇u n |).
1.4. Plan of the paper. In the next section, we recall some classical results and we introduce the notations and notions that are useful throughout the article. In Section 3, we study the regularity properties of the level sets of the minimizers. In the subsequent Section 4, we prove that max(α, φ ′ (|∇u|) ∈ W 1,2 loc (Ω). The maximum principle for max(d 0 , |σ n |) is proved in Section 5. Section 6 is dedicated to the proof of Theorem 1.1 and Theorem 1.5. In the last section, we state a possible extension to the main theorem.

Preliminary results

In this section we introduce some known results related to this problem.

2.1. Direct methods. We know by the direct method in the calculus of variations (see [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF] and [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]) that the problem P λ admits at least one minimizer. We recall that every minimizer u is bounded, globally Hölder continuous by [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Theorem 7.8] and locally Lipschitz continuous by [START_REF] Celada | Existence and regularity of minimizers of nonconvex integrals with p -q growth[END_REF]Theorem 1.1].

We begin this subsection by observing that the minimum of a minimizer is attained on the boundary of Ω.

0 = [u≤c] φ(|∇w|) < [u≤c] φ(|∇u|) and - [u≤c] λw = - [u≤c] λc ≤ - [u≤c] λu.
Hence, I λ (w) < I λ (u) on Ω, which contradicts the fact that u is a minimizer. Thus, u ≥ c on Ω. □ We now prove that the gradients of two minimizers of the same problem are collinear. This property is used in many subsequent proofs. Proof. Since u is a solution of P λ ,

I λ (u) ≤ I λ u + v 2 .
By the fact that φ is non decreasing and the convexity of φ and of the Euclidean norm,

I λ u + v 2 = Ω φ ∇u + ∇v 2 -λ u + v 2 ≤ 1 2 Ω (φ(|∇u|) -λu) + 1 2 Ω (φ(|∇v|) -λv) = 1 2 I λ (u) + 1 2 I λ (v).
Since v is another solution,

I λ (u) = 1 2 I λ (u) + 1 2 I λ (v).
This implies that

Ω φ ∇u + ∇v 2 = Ω 1 2 (φ(|∇u|) + φ(|∇v|)).
Hence for a.e.

U i of U , u -v is constant on U i .
To conclude this section, we introduce the weak Euler-Lagrange equation associated to P λ :

(2.2) div ∇Φ(∇u) = -λ on Ω.

Remark 2.6. By Lemma 2.2 the function ∇Φ(∇u) = φ ′ (|∇u|) ∇u |∇u| is independent of the choice of the minimizer of P λ and will be denoted by σ.

2.2.

Lipschitz regularity of a minimizer u and its level lines. In this subsection, we recall some Lipschitz regularity results for u and its level lines.

We use the following result from [START_REF] Lledos | A uniqueness result for a non-strictly convex problem in the calculus of variations[END_REF]Theorem 1.6] to show that our minimizers are globally Lipschitz-continuous on Ω: Proposition 2.7. We assume that Ω has a C 1,1 boundary and ψ ∈ C 1,1 (R 2 ). Then any minimizer u of P λ is globally Lipschitz-continuous on Ω. Moreover, there exists L > 0 such that

||∇u|| L ∞ (Ω) ≤ L(p, C 1 , |Ω|, ||λ|| L ∞ (Ω) , ||ψ|| C 1,1 (R 2 ) , κ)
where κ is the maximum of the curvatures of Ω and C 1 is introduced in (1.3).

For a function f : R 2 → R and for every s ∈ R, we introduce the following notation :

L * s (f ) is the union of all connected components l s (f ) of L s (f ) = f -1 (s) ⊂ R 2 such that H 1 (l s (f )) > 0.
Here, H 1 is the one-dimensional Hausdorff measure.

We will apply the following theorem from [1, Theorem 2.5] to prove Proposition 2.9.

Theorem 2.8. Let f : R 2 → R be a Lipschitz continuous function with compact support. For a.e. s ∈ R, we have:

• H 1 (L s (f )\L * s (f )) = 0. • Every connected component l s (f ) of L s (f ) that is not a point is a closed simple curve with a Lipschitz parametrization γ s . • L * s (f )
has a countable number of connected components. It follows that the level lines of a minimizer u have a Lipschitz-continuous parametrization: Remark 2.9. Let u be a globally Lipschitz-continuous minimizer of P λ with λ ∈ L ∞ (Ω). We extend it outside Ω by ψ that can be assumed compactly supported. For a.e. s ∈ R every connected component of L * s (u) ⊂ R 2 has a Lipschitz-continuous parametrization. 2.3. Explicit solution on the ball and counter-example. The application of [6, Theorem 1] to our problem P λ gives an explicit form for the unique solution on W 1,p 0 (B r (x 0 )) in dimension two when λ ≡ λ 0 ∈ R + : Proposition 2.10. When Ω = B r (x 0 ) and λ is constant, the problem P λ admits a unique minimizer on W 1,p 0 (B r (x 0 )). We can compute it explicitly:

u(x) := C - 2 λ φ * λ 2 |x -x 0 | with φ * (x) := sup y∈R ⟨x, y⟩ -φ(y), and C the constant such that u ∈ W 1,p 0 (B r (x 0 )).
The following proposition uses the Euler-Lagrange equation (2.2) to prove that a function is a minimizer.

Proposition 2.11. Let u be in W 1,p (Ω) and Φ a convex function. If there exist σ ∈ ∂Φ(∇u

) ∈ L p ′ (Ω; R 2 ) and λ ∈ L ∞ (Ω) such that div σ = -λ then u is a minimizer of P λ on W 1,p u (Ω).
Here ∂Φ is the convex subdifferential of Φ.

Proof. Since σ ∈ ∂Φ(∇u), for every w ∈ W 1,p u (Ω) we have:

Ω Φ(∇w) ≥ Ω Φ(∇u) + ⟨σ, ∇w -∇u⟩.
Since div σ = -λ we get:

Ω ⟨σ, ∇w -∇u⟩ = Ω λ(w -u).
Hence, I λ (w) ≥ I λ (u) for every w ∈ W 1,p u (Ω). Thus, u is a minimizer of P λ on W 1,p u (Ω). □

We apply this result to show that when λ is not constant, we can have more than one solution.

Proposition 2.12. Let φ : R + → R be a non-strictly convex function such that φ(0) < φ(t) for every t > 0. We assume that φ ∈ C 1 (R + ). Then, there exists

λ ∞ ∈ C ∞ (B 1 (0)), λ ∞ > 0 such that P λ∞ has an infinite number of solutions on W 1,p 0 (B 1 ) with Φ(•) = φ(| • |) and I λ∞ (u) := B 1 Φ(∇u(x)) -λ ∞ (x)u(x)dx.
Proof. We construct two different radial solutions u and v of the same problem. For every x ∈ B 1 (0), we set u(x) := ũ(|x|) and v(x) := ṽ(|x|). Our goal is to define ũ′ and ṽ′ on (0, 1). Since φ is not strictly convex on R + , there exist

a, b ∈ R + such that φ ′ is constant on (a, b) and φ ′ (t) ̸ = φ ′ (a) for every t / ∈ [a, b]. a)
We assume that the smallest possible a is strictly positive. Let us introduce a smooth increasing function f : R + → R + such that f (t) = t for every t ≥ 0 small and f (t) = φ ′ (a) for every t ≥ 1 2 . We use the fact that for every x > 0 if x ∈ ∂φ * (y) then φ ′ (x) = y. Hence for every t > 0, φ ′ (∂φ * (f (t))) = {f (t)}. For every t ∈ (0, 1), we set ũ′ (t) = -x t with x t ∈ ∂φ * (f (t)) such that ũ′ is measurable. Such a choice is possible by [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]Theorem 5.3,page 151]. In order to define ṽ′ , we set ṽ′ (t) = ũ′ (t) on (0, 1 2 ) and ṽ′ (t

) = -b for every t > 1 2 . Hence, φ ′ (|ṽ ′ |) = φ ′ (|ũ ′ |) = f is a smooth function. b)
We assume that a = 0. We set ũ′ = 0 on (0, 1), ṽ′ (t) = 0 on (0, 1 2 ) and ṽ′ (t) = -b on ( 1 2 , 1).

Now, we can set u(x) := 1 |x| -ũ ′ (t)dt and v(x) := 1 |x| -ṽ ′ (t)
dt that are Lipschitz-continuous on B 1 (0) and vanish at the boundary. It remains to find σ and λ ∞ .

In the case a), we set for every

x ∈ B 1 (0), σ(x) = -f (|x|) x
|x| that is smooth by assumptions on f .

In the case b), we set σ(x) = -φ ′ (0) x |x| for every x ∈ B 1 (0) such that |x| > 1 2 . On B 1 2 (0), we set

σ(x) = -f (x)
x |x| with f smooth satisfying the same assumptions as f . Finally, we set

λ ∞ (x) = -div σ ∈ C ∞ (B 1 (0)).
Hence, by Proposition 2.11, u and v are solutions of the same problem P λ∞ . Moreover, a direct computation shows that λ ∞ (x) > 0 on B 1 (0). □ Now, we give an explicit counter-example where P λ has more than one solution with λ > 0 and U ̸ = ∅ with φ is as in (1.2), namely:

φ(t) =      1 2 |t| 2 if |t| ≤ 1, |t| -1 2 if 1 < |t| < 2, 1 4 |t| 2 + 1 2 if 2 ≤ |t|. Proposition 2.13. There exists λ ∈ C ∞ (B 1 (0)), min B 1 (0)
λ > 0 such that P λ has more than one solution on W 1,2 0 (B 1 (0)) and U ̸ = ∅.

Proof. We take the same notations as in the previous proof. In this case, we have

φ * (t) = 1 2 |t| 2 if |t| ≤ 1, |t| 2 -1 2 if 1 < |t|.
For t ∈ R, we define:

g(t) =      t if t ≤ 1 4 , 3t -1 2 if 1 4 < t < 1 2 , 1 if 1 2 ≤ t.
Hence, we can take f as the convolution of g with a smooth standard mollifier. For instance, 5 8 ) and ṽ′ = -2 on ( 5 8 , 1]. Thus, u(x) :=

f (t) := C t+ 1 8 t-1 8 g(s) exp - 1 1 64 -|t -s| 2 ds with C -1 = 1 8 -1 8 exp - 1 1 64 -|s| 2 ds. Then ũ′ = -f on [0, 1], ṽ′ = -f on [0,
1 |x| -ũ ′ (t)dt and v(x) := 1 |x| -ṽ ′ (t)dt are two solutions on W 1,2 0 (B 1 (0)) with λ(x) := N -1 |x| f (|x|) + f ′ (|x|). Moreover, U ̸ = ∅ since |∇u| < 1 on B 1 2 (0). □ 2.
4. BV functions. We start by giving the definitions of functions of bounded variations and sets of finite perimeter:

Definition 2.14. A function f ∈ L 1 (Ω) has bounded variations in Ω if Ω |Df | := sup Ω f div g dx g ∈ C 1 c (Ω; R 2 ), |g(x)| ≤ 1 , ∀x ∈ Ω < ∞.
We denote by BV (Ω) the set of functions in L 1 (Ω) having bounded variations in Ω.

If f ∈ BV (Ω), the distributional gradient of f is a vector valued Radon measure that we denote by Df and |Df | is the total variation of Df . Definition 2.15. Let E be a Borel set. We say that E has finite perimeter in Ω if the characteristic function χ E of E belongs to BV (Ω). The perimeter Per(E, Ω) is defined as: Remark 2.17. The reduced boundary ∂ * E is a subset of ∂E.

Per(E, Ω) = Ω |Dχ E | = sup E div g g ∈ C 1 c (Ω; R 2 ), |g(x)| ≤ 1 , ∀x ∈ Ω .
We recall the coarea formula for Lipschitz continuous functions from [11, Theorem 3.4.2.1, page 112] that will be useful throughout the article. Proposition 2.18. (Coarea formula) Let u be a Lipschitz continuous function with compact support and f be a nonnegative measurable function. Then

R 2 f |∇u|dx = R Ls(u) f (x)dH 1 (x)ds where L s (u) := u -1 (s) ⊂ R 2 .
Remark 2. [START_REF] Marcellini | A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations[END_REF]. By replacing f by the indicator function

1 A (x) = 1 if x ∈ A, 0 if x / ∈ A, we observe that if |A| = 0 then for a.e. s ∈ R, H 1 (L s (u) ∩ A) = 0.
Proposition 2.20. Let v be a Lipschitz continuous function with compact support in R 2 . For a.e. s ∈ R we have By density, for every g ∈ L ∞ (supp v; R N ), we get

∇v(x) |∇v(x)| = D1 [v>s] (x) |D1 [v>s] |(x) for H 1 a.e. x ∈ L s (v).
R 2 ⟨g, ∇v⟩ = R R 2 ⟨g, D1 [v>s] ⟩ds.
We fix g := ∇v |∇v| when ∇v ̸ = 0, g := 0 when ∇v = 0 and obtain

(2.3) R 2 |∇v| = R R 2 ⟨ ∇v |∇v| , D1 [v>s] ⟩ds.
But, by [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]Theorem 4.4] we have for a.e. s ∈ R that

|D1 [v>s] | = H 1 ¬ ∂ * [v > s]. Hence, (2.4) R 2 ⟨g, D1 [v>s] ⟩ = R 2 ⟨g, D1 [v>s] |D1 [v>s] | ⟩d|D1 [v>s] | = ∂ * [v>s] ⟨g, D1 [v>s] |D1 [v>s] | ⟩dH 1 .
Since ⟨g,

D1 [v>s] |D1 [v>s] | ⟩ ≤ 1 for a.e. s ∈ R, with (2.
3) and (2.4) we get

R 2 |∇v| ≤ R H 1 (∂ * [v > s])ds.
By remark 2.17, we have

R 2 |∇v| ≤ R H 1 (∂ * [v > s])ds ≤ R H 1 (L s (v))ds.
By the coarea formula given in Proposition 2.18, the following equalities hold true:

R 2 |∇v| = R H 1 (∂ * [v > s])ds = R H 1 (L s (v))ds.
Hence, we get ⟨g,

D1 [v>s] |D1 [v>s] | ⟩ = 1 for H 1 a.e. x ∈ ∂ * [v > s] and H 1 (L s (v)\∂ * [v > s]) = 0 for a.e. s ∈ R. Thus, g = D1 [v>s]
|D1 [v>s] | H 1 a.e. on L s (v) for a.e s ∈ R, as desired. □

3.

Relation between the level lines and U.

In this section we use the Lipschitz regularity of the level lines of a minimizer u to prove that they are, in a generic sense, level sets for the other minimizers. We then study the case when a level line intersects the set U , which implies that the gradient of another solution is equal to ∇u on that particular level line.

3.1. Equality on level lines. We first prove that the difference between two minimizers is constant on every connected component of almost every level sets. Proposition 3.1. Let u and v be two minimizers of the same problem P λ . There exists a negligible subset N 0 of R such that for every s ∈ S 0 := R\N 0 , for every connected component l s (u) of L s (u), the map u -v is constant on l s (u).

Proof. We consider that u and v are extended by ψ outside of Ω. By Proposition 2.9 there is a negligible set N ∞ such that for every s ∈ R\N ∞ , every connected component l s (u) of L s (u) ⊂ R 2 that is not a point has a parametrization that is Lipschitz continuous.

Since ∇u and ∇v are defined and collinear a.e. on R 2 , by the coarea formula we obtain that there exists a negligible set N ′ ∞ such that for every s ∈ R\N ′ ∞ , ∇u and ∇v are defined and collinear H 1 a.e. on L s (u). We set N 0 := N ∞ ∪ N ′ ∞ . Hence, for every s ∈ R\N 0 we have that ∇v is orthogonal to each Lipschitz connected curve l s (u). We introduce γ s : [0, length(l s (u))) → l s (u) a Lipschitz-continuous parametrization of l s (u). Then, by the chain rule we have (v • γ s ) ′ = ⟨∇v(γ s ), γ ′ s ⟩ a.e. on [0, length(l s (u))). By orthogonality of ∇v to l s (u), we have that v is constant on l s (u). □

The following proposition is the first step to prove the uniqueness result.

Proposition 3.2. For s ∈ S 0 , if l s (u) ∩ (R 2 \Ω) ̸ = ∅ then u = v on l s (u).
Proof. Thanks to the previous proposition we know that u -v is constant on l s (u). Since u and v are extended by ψ outside Ω, we have u ≡ v on R 2 \Ω. By assumption, we have

l s (u) ∩ R 2 \Ω ̸ = ∅ then u = v on l s (u). □ 3.2.
Relation between U and the level lines. In this section we consider two minimizers u and v of I λ with the same boundary condition. We know by Proposition 2.4 that ∇u = ∇v on U . We will extend this result to the level lines that intersect U .

Notation. We denote by S ⊂ R the set of these s that satisfy the following conditions:

• s ∈ S 0 with S 0 defined in Proposition 3.1,

• ∇u ̸ = 0 H 1 a.e. on L s (u). We introduce the following set:

Γ := {l i s (u), s ∈ S and i ∈ I s } where the index set I s corresponds to the non-constant curves l i s (u) among the connected components of L s (u) inside Ω which do not intersect ∂Ω. Proposition 3.3. Given s ∈ S and i ∈ I s , let F i s be the bounded connected component of R 2 \l i s (u) given by the Jordan curve theorem. Then for every i ∈ I s , u > s on F i s . Proof. We fix s ∈ S and we call E s the set

[u > s] ⊂ R 2 . By continuity of u, E s is an open set. Let l i s (u) a connected component of L s (u) such that l i s (u) ⋐ Ω.
Since Ω is simply connected, F i s ⊂ Ω and by Proposition 2.1, we have that u ≥ s on

F i s . If l ′ s (u) is a connected component of L * s (u) that is inside F i s then for H 1 a.e. y ∈ l ′ s (u)
, ∇u(y) is defined and ∇u(y) ̸ = 0. Since u ≥ s on F i s , every point y in l ′ s (u) is a local minimum on F i s and hence either ∇u(y) = 0 or ∇u(y) is not defined. Thus there is no such l ′ s (u) in F i s . By assumptions on s, we have

H 1 (L s \L * s ) = 0. Hence, H 1 ([u = s] ∩ F i s ) = 0.
Let us assume that there exists x ∈ F i s such that u(x) = s. Then by Proposition 2.1 for every ϵ < dist(x, l i s (u)) there exists y ϵ ∈ ∂B ϵ (x) such that u(y ϵ ) = s and we define the following set

Y := {y ϵ , 0 < ϵ < dist(x, l i s (u))}. We have H 1 (Y ) := lim δ→0 H 1 δ (Y )
with

H 1 δ (Y ) := inf{ n∈N diam(V n )}
where the infimum is taken over the families of sets

(V n ) n∈N such that Y ⊂ n V n and diam(V n ) < δ
for every n ∈ N.

For every admissible (V n ) n∈N , we define E n as the set of those ϵ such that y ϵ ∈ V n ∩ Y . We define e m := inf{ϵ ∈ E n }, e M := sup{ϵ ∈ E n } and V n := [e m , e M ]. We have diam(

V n ) = e M -e m ≤ diam(V n ) < δ and ]0, dist(x, l i s (u))[⊂ V n . Hence, ( V n ) n∈N is admissible for H 1 δ ((0, dist(x, l i s (u))))
and

H 1 δ (Y ) ≥ H 1 δ (0, dist(x, l i s (u)))
. By taking the limit when δ goes to 0, we obtain:

H 1 (Y ) ≥ H 1 ((0, dist(x, l i s (u))) = dist(x, l i s (u)). Thus H 1 ([u = s] ∩ F i s ) ≥ dist(x, l i s (u)) > 0.
That is a contradiction. Hence there is no such x, thus u > s on F i s . □

A direct consequence of that result is the following: The main result of this subsection is the following: Proposition 3.6. For a.e. s ∈ S, for every i ∈ I s , if

l i s (u)∩ U ̸ = ∅ then ∇(u -v) = 0 H 1 a.e. on l i s (u).
In order to prove this result, we state two technical lemmata:

Lemma 3.7. For a.e. s ∈ S, for every i ∈ I s there exists a decreasing sequence (s n ) n∈N converging to s such that:

• There exists a simple connected curve l sn (u) in L sn with Lipschitz parametrization that is inside F i s .

• (F sn ) n∈N is an increasing sequence with n∈N

F sn = F i s .
Here, F t is the bounded connected component of R 2 \l t (u) given by the Jordan curve theorem and F i s is the bounded connected component of R 2 \l i s (u). Proof of Lemma 3.7. By Proposition 3.3 we have that u > s on F i s . By the coarea formula 2.18 there exists s 0 > s, s 0 ∈ S such that H 1 (L s 0 (u) ∩ F i s ) > 0. Moreover, by Theorem 2.8, H 1 (L s 0 (u)\L * s 0 (u)) = 0. Hence, there exists l s 0 (u) in L s 0 satisfying the assumptions of Lemma 3.7. We next select s < s 1 < s 0 with s 1 ∈ S such that H 1 (L s 1 (u) ∩ F i s ) > 0. We have that that F s 0 ⊂ E s 1 . Hence, F s 0 is in one connected component of E s 1 , we call l s 1 (u) the boundary of that connected component. By Proposition 3.5 we have that l s 1 (u) is a simple connected curve with Lipschitz parametrization. We repeat this argument to find a sequence (s n ) n∈N that satisfies the first part of the lemma. By construction (F sn ) n∈N is an increasing sequence, it remains to prove that n∈N

F sn = F i s .
We introduce F ∞ := n∈N F sn a subset of F i s . If y ∈ ∂F ∞ there exists a sequence (y n ) n∈N such that y n ∈ ∂F sn and y n → y. By continuity of u and the fact that y n converges to y we obtain that u(y) = s.

By Proposition 3.3 we have that ∂F ∞ ⊂ ∂F i s . We claim that F ∞ = F i s . Indeed, if those two sets are not equal there exists x ∈ F i s \F ∞ . Since F i s is a connected open set, for every y ∈ F ∞ there exists a continuous path from x to y included in F i s . By continuity this path must intersect ∂F ∞ ⊂ ∂F i s , contradicting the fact that the path is in F i s . Hence, F ∞ = F i s . □ Lemma 3.8. Let u be a minimizer. For a.e. s ∈ S and every i ∈ I s , we consider a sequence (l sn ) n∈N as in the previous lemma. Then, we have that lim n→+∞ D(l sn , l i s (u)) = 0 where D(E, F ) := sup e∈E inf f ∈F |e-f |. Moreover, let v be another minimizer, if there exists a constant C such that u-v ≡ C on every l sn then ∇(u -v) = 0 H 1 a.e on l i s (u).

Proof of Lemma 3.8. For every ϵ > 0, we claim that there exists N ∈ N such that for every n ≥ N , D(l sn , l i s (u)) < ϵ. Indeed, assume by contradiction that there exists ϵ > 0 such that ∀N , there exist n ≥ N and y n ∈ l sn (u) such that ∀x ∈ l i s (u), d(y n , x) ≥ ϵ. Since all these y n are in Ω, there exists a sequence (y g(n) ) n∈N converging towards some y ∈ F i s . We have that d(y, x) ≥ ϵ for every x ∈ l i s (u). By continuity of u, we have that u(y) = s. Thus, by Proposition 3.3, y ∈ ∂F i s = l i s (u). That is a contradiction.

For H 1 a.e. x ∈ l i s (u) we have that ∇u(x) ̸ = 0 and ∇v(x) exist. Moreover, for H 1 a.e. x ∈ l i s (u) we have that ∇u(x) and ∇v(x) are orthogonal to l i s (u) at x in the sense that ⟨∇u(x), γ ′ s (γ -1 s (x))⟩ = 0 where γ s is a Lipschitz parametrization of l i s (u). We consider d x := x + R∇u(x). Let us call H - and H + the two half-planes of R 2 \d x . For every r > 0, H ± ∩ l i s (u) ∩ B r (x) ̸ = ∅, otherwise it would contradict the fact that ∇u(x) is orthogonal to l i s (u). A direct consequence is the fact that

H -∩ F i s ̸ = ∅ and H + ∩ F i s ̸ = ∅.
We assume that for every N ∈ N, there exists ñ ≥ N such that l s ñ (u) ∩ d x = ∅. Thus, we can assume that l s ñ (u)∩H -= ∅. Hence, there exist y ∈ l i s (u) and ϵ 0 > 0 such that d(y, F s ñ ) > ϵ 0 . Since l s ñ(u) → l i s (u) in the sense of D that is absurd. Then, there exists N ∈ N such that for every n ≥ N , l sn (u) ∩ d x ̸ = ∅. For every such n we take x n as a point that minimizes d(x, y) on l sn (u) ∩ d x . Since this sequence (x n ) n∈N is bounded, it converges, up to a subsequence, to a point x ′ ∈ F i s such that u(x ′ ) = s. By Proposition 3.3, x ′ ∈ l i s (u). If x ′ ̸ = x then there exists r > 0 such that d x ∩ B r (x) ∩ l sn (u) = ∅ for every n ∈ N large enough. Hence, d x ∩ B r (x) ∩ F sn = ∅ for every n ∈ N large enough which contradicts the fact that n∈N F sn = F i s . Thus, x = x ′ and we can find a sequence (x n ) n∈N such that x n ∈ l sn (u) ∩ d x and x n → x. By assumption, u -v ≡ C on l sn (u) for n large enough. By continuity of u -v we obtain that u

-v(x) = C. Then, (u-v)(x)-(u-v)(xn) |x-xn| = C-C
|x-xn| = 0. Moreover, ∇(u -v)(x) is collinear to ∇u(x), hence, we obtain ∇(u -v)(x) = 0. Since that is the case for H 1 a.e. x ∈ l i s (u), we have the desired conclusion. □ Proof of Proposition 3.6. For every s ∈ S, l i s (u) is a Lipschitz continuous closed curve such that ∇u and ∇v are defined and collinear H 1 a.e. on l i s (u) and ∇u ̸ = 0 H 1 a.e. on l i s (u).

If l i s (u) ∩ U ̸ = ∅ then there exists U i a connected component of U such that l i s (u) ∩ U i ̸ = ∅.
By Proposition 2.5 and Proposition 3.1 we have u -v ≡ C i on l i s (u). We consider the sequences (s n ) n∈N , (l sn (u)) n∈N and (F sn ) n∈N from Lemma 3.7.

Hence, by the first part of Lemma 3.8, there exists N ∈ N such that for every n ≥ N , l sn (u)∩U i ̸ = ∅. Thus, by Proposition 2.5 and Proposition 3.1, u -v ≡ C i on l sn (u) for every n ≥ N . By the second part of Lemma 3.8, we have that ∇(u -v) = 0 H 1 a.e on l i s (u). □ 4. W 1,2 regularity of |σ|.

In this section, we prove the following proposition:

Proposition 4.1. For every α > 0, the function

f := max(α, |σ|) is in W 1,2 (Ω ′ ) for any Ω ′ ⋐ Ω.
We prove this result in four parts. In Step 1 , we regularize our problem in order to work with smooth solutions (u n ) n∈N . Then in Step 2 , we prove that || max(α,

|∇Φ n (∇u n )|)|| W 1,2 (Ω ′ ) is uniformly bounded in n ∈ N. In the subsequent Step 3 , we show that max(α, |∇Φ n (∇u n )|) → f a.e. on Ω. We conclude that f is in W 1,2 (Ω ′ ) in Step 4 .
Step 1 For every n ∈ N, we introduce (ρ n ) n∈N a standard mollifying sequence with supp ρ n ⊂ B 1 n (0). If we set φ n := φ * ρ n and λ n := λ * ρ n then (φ n ) n∈N and (λ n ) n∈N are sequences of smooth approximations of φ and λ. We consider

Φ n := φ n (|•|)+ 1 n g(|•|).
The function g is smooth quadratic around the origin such that 0 < g ′′ (x) for every x ∈ R and

C -|x| p ≤ g(z) ≤ C + (|x| p + 1)
for all |x| ≥ 1 with 0 < C -< C + . Let u n be the minimizer of:

I n : v → Ω Φ n (∇v(x)) -λ n v(x)dx on W 1,p ψ (Ω). Proposition 4.2. The sequence (u n ) n∈N is uniformly bounded in W 1,p (Ω).
There exists a subsequence still denoted by (u n ) n∈N that weakly converges in W 1,p (Ω) towards u. Moreover, u is a minimizer of P λ on W 1,p ψ (Ω). Proof. By Proposition 2.7 we have that the sequence (u n ) n∈N is uniformly bounded in W 1,∞ (Ω). Hence, we can extract a subsequence, still denoted by (u n ) n∈N , that converges strongly in L p (Ω) and weakly in W 1,p ψ (Ω) towards u. It remains to prove that u is a minimizer of P λ on W 1,p ψ (Ω). By Jensen's inequality, we have Φ n ≥ Φ. Hence (4.1)

lim inf n→+∞ Ω Φ n (∇u n ) ≥ lim inf n→+∞ Ω Φ(∇u n ).
By weak lower semi-continuity of I λ , (4.1) and the fact that u n is the minimizer for I n we have

(4.2) Ω Φ(∇ u) -λ u ≤ lim inf n→+∞ Ω Φ(∇u n ) -λu n ≤ lim inf n→+∞ Ω Φ n (∇u n ) -λ n u n ≤ lim n→+∞ Ω Φ n (∇u) -λ n u.
By the dominated convergence theorem applied to the last quantity we obtain

Ω Φ(∇ u) -λ u ≤ Ω Φ(∇u) -λu.
Hence, u is a minimizer on W 1,p ψ (Ω). □

Step 2 For every n ∈ N, we introduce σ n := ∇Φ n (∇u n ). In this part, we prove the following result on f n := max(α, |σ n |): Proposition 4.3. For every α > 0 and every Ω ′ ⋐ Ω, the functions f n := max(α, |σ n |) are uniformly bounded in W 1,2 (Ω ′ ).

Proof. By [10, Proposition 2.4], we have for every b > 0 and k ∈ {1, 2} that:

Ω ′ ∩{∂ k un≥b} |∇σ n | 2 ≤ C 1 b, ||∇u n || L ∞ (Ω) , sup b≤t≤||∇un|| L ∞ (Ω) φ ′′ n (t) + g ′′ (t) n .
Thus,

Ω ′ ∩{|∇un|≥b} |∇σ n | 2 ≤ C 2 b, ||∇u n || L ∞ (Ω) , sup b≤t≤||∇un|| L ∞ (Ω) φ ′′ n (t) + g ′′ (t) n .
Finally,

Ω ′ ∩{|σn|≥φ ′ n (b)} |∇σ n | 2 ≤ C 2 b, ||∇u n || L ∞ (Ω) , sup b≤t≤||∇un|| L ∞ (Ω) φ ′′ n (t) + g ′′ (t) n .
By Proposition 2.7 we have that ||∇u n || L ∞ (Ω) can be bounded by L uniformly in n ∈ N. Moreover,

φ ∈ C 1,1 loc (R\{0}) and φ n is a convolution of φ. Hence, sup b≤t≤||∇un|| L ∞ (Ω) φ ′′ n (t) + g ′′ (t) n can be bounded by sup b 2 ≤t≤L
φ ′′ (t) + 1 for every n ∈ N such that φ ′′ n is close enough to φ ′′ on ( b 2 , +∞) and larger than sup b≤t≤L g ′′ (t). Namely, every n ∈ N larger than max{ 1 2b , sup b≤t≤L g ′′ (t)}. Thus, we get

Ω ′ ∩{|σn|≥φ ′ n (b)} |∇σ n | 2 ≤ C 2 (b, L, sup b 2 ≤t≤L φ ′′ (t) + 1)
for every n ∈ N larger than max{ 1 2b , sup b≤t≤L g ′′ (t)}.

By growing assumptions on φ, for every α > 0 we can find b > 0 such that φ ′ n (b) ≤ α for n ∈ N large enough. Hence, for every α > 0 and every n ∈ N large enough we have:

Ω ′ ∩[|σn|>α] |∇σ n | 2 ≤ C(α, φ ′′ , L).
Thus, the sequence (f n ) n∈N is uniformly bounded in W 1,2 (Ω ′ ). □

Since the functions f n := max(α, |σ n |) are uniformly bounded in W 1,2 (Ω ′ ), we can extract a subsequence which converges weakly.

Step 3 We prove that σ n → σ a.e. Ω up to a subsequence. To do so we use the Young measures associated to (∇u n ) n∈N . Proposition 4.4. We have the following equality

(4.3) lim inf n→+∞ Ω Φ(∇u n ) = Ω Φ(∇ u).
Proof. If we replace u by u in the last term of (4.2) we obtain that

(4.4) Ω Φ(∇ u) -λ u ≤ lim inf n→+∞ Ω Φ(∇u n ) -λu n ≤ lim inf n→+∞ Ω Φ n (∇u n ) -λ n u n ≤ lim inf n→+∞ Ω Φ n (∇ũ) -λ n ũ.
By Fatou's lemma with the lim sup the last term is equal to the first term. Hence all those inequalities are equalities, in particular:

lim inf n→+∞ Ω Φ(∇u n ) -λu n = Ω Φ(∇ u) -λ u.
Since u n → u in L p (Ω) we have that

lim n→+∞ Ω λu n = Ω λ u.
Hence,

lim inf n→+∞ Ω Φ(∇u n ) = Ω Φ(∇ u).

□

We consider (u ψ(n) ) n∈N a subsequence such that

(4.5) lim inf n→+∞ Ω Φ(∇u n ) = lim n→+∞ Ω Φ(∇u ψ(n) ).
In order to simplify the notations, we still denote (u ψ(n) ) n∈N by (u n ) n∈N .

Proposition 4.5. For a.e. x ∈ Ω we have

Φ(∇ũ(x)) = Φ(x) := R 2 Φ(y)dν x (y)
where ν x is a probability measure that depends on x and on the weak convergence of (∇u n ) n∈N towards ∇ u. Moreover, supp ν x ⊂ {y ∈ R 2 , ∇Φ(y) = ∇Φ(∇ũ(x))} for a.e. x ∈ Ω.

Proof. Let (ν x ) x∈Ω be the Young measures associated to a subsequence of (∇u n ) n∈N given by [4, Theorem 2]. We have for every Carathéodory function F such that F •, ∇u n (•) n∈N is uniformly integrable:

(4.6) lim n→+∞ Ω F (x, ∇u n (x))dx = Ω F (x)dx with F (x) = R 2 F (x, y)dν x (y). Moreover, for a.e. x ∈ Ω, (4.7)
∇ u(x) = R 2 ydν x (y). Since u n is uniformly bounded in W 1,∞ (Ω), (4.8)
lim n→+∞ Ω Φ(∇u n (x))dx = Ω Φ(x)dx where Φ(x) = R 2 Φ(y)dν x (y).
If we combine this last equation with (4.5) we get

Ω Φ(∇ u) = lim n→+∞ Ω Φ(∇u n (x))dx = Ω Φ(x)dx.
If we apply the triangle inequality and Jensen's inequality to (4.7) we obtain for a.e. x ∈ Ω,

(4.9) Φ(∇ u(x)) ≤ R 2 Φ(y)dν x (y)dx = Φ(x).
If we combine the two last equations we obtain for a.e. x ∈ Ω (4.10)

Φ(∇ u(x)) = Φ(x).
By Jensen's inequality, Φ is affine on supp ν x and thus, for a.e. x ∈ Ω we have that supp ν x ⊂ {y ∈ R 2 , ∇Φ(y) = ∇Φ(∇ũ(x))}. □ Now, we can prove the following convergence result: Proposition 4.6. We have that σ n → σ in L 1 (Ω) when n → +∞. Here, σ n = ∇Φ n (∇u n ).

Proof. If we set F (x, y) = |∇Φ(y) -σ(x)| in (4.6), we obtain that

lim n→+∞ Ω |∇Φ(∇u n (x)) -σ(x)|dx = Ω R 2 |∇Φ(y) -σ(x)|dν x (y)dx = 0.
Since ∇u n is uniformly bounded in L ∞ (Ω), we have that

lim n→+∞ Ω |∇Φ n (∇u n ) -∇Φ(∇u n )| = 0.
Hence, by the triangle inequality,

lim n→+∞ Ω |σ n (x) -σ(x)|dx = 0. Hence, σ n → σ in L 1 (Ω) when n → +∞. □
Thanks to the previous Proposition, we can extract a subsequence, we do not relabel, such that σ n → σ a.e. on Ω when n → +∞.

Step 4 Since σ n → σ a.e. on Ω when n → +∞, we have that f n → max(α, |σ|) a.e. on Ω. By Proposition 4.3, we have that max(α, |σ|) ∈ W 1,2 (Ω ′ ).

Continuity of |σ| on the level lines and a maximum principle

In this section, we prove that generically, max(d 0 , |σ|) is continuous on the level lines of u and satisfies a maximum principle.

For Ω ′ ⋐ Ω, we introduce Γ ′ := {l i s (u), s ∈ S and i ∈ I ′ s } where S ⊂ R is the set of those s that satisfy the conclusion of Theorem 2.8 and such that ∇u, ∇v are defined, ∇u ̸ = 0, ∇u and ∇v are collinear H 1 a.e. on L s (u). The index set I ′ s corresponds to the non-constant curves l i s (u) among the connected components of L s (u) such that l i s (u) ⋐ Ω ′ . Proposition 5.1. There exists a representative f 0 of max(d 0 , |σ|) that is absolutely continuous on l i s (u) for a.e. s ∈ S and every i ∈ I s . Proof. We consider the sequence (σ n ) n∈N introduced in the previous section. We have that σ n → σ a.e. on Ω when n → +∞. By Proposition 4.1, we have ||∇ max(d 0 , |σ n |)|| L 2 (Ω 1 ) ≤ C 1 with Ω 1 ⋐ Ω and C 1 independent of n ∈ N. Thus, there exists a constant C 2 independent of n such that

C 2 ≥ Ω 1 |∇ max(d 0 , |σ n |)| 2 |∇u| = R Ls(u)∩Ω 1 |∇ max(d 0 , |σ n |)| 2 dH 1 ds
where the equality is given by Proposition 2.18. With Fatou's lemma we obtain that for a.e. s ∈ S,

(5.1) lim inf n→+∞ Ls(u)∩Ω 1 |∇ max(d 0 , |σ n |)| 2 dH 1 ≤ C 3 (s).
Here, the index I 1 s corresponds to the non-constant curves l i s (u) among the connected components of L s (u) such that l i s (u) ⋐ Ω 1 . We define S 1 as the subset of S such that (5.1) holds. We have

|S\S 1 | = 0.
Now, we fix s ∈ S 1 and i ∈ I 1 s . We can extract a subsequence such that for every n ∈ N:

l i s (u) |∇ max(d 0 , |σ n |)| 2 dH 1 ≤ 2C 3 (s).
Let us call γ i s : 0, length(l i s (u)) → l i s (u) a Lipschitz continuous parametrization of l i s (u). We have that max(d 0 ,

|σ n |) • γ i s is bounded in W 1,2 [0 , length(l i s (u)))
. By the Arzelà-Ascoli theorem there exists a subsequence of max(d 0 , |σ n |) • γ i s converging uniformly to v ∈ C 0 [0, length(l i s (u))) . Since max(d 0 , |σ n |) → max(d 0 , |σ|) H 1 a.e. on l j t (u) for a.e. t ∈ R and every j ∈ I t , we choose f 0 as a representative of max(d 0 , |σ|) such that f 0 = v • (γ i s ) -1 on l i s (u). Now, we introduce an increasing sequence of open sets

(Ω k ) k∈N such that χ Ω k → χ Ω in L 1 (R 2 ).
For a.e. s ∈ S 1 and for every i ∈ I 2 s \I 1 s , we can define f 0 as we did on Ω 1 . Hence, there exists S 2 ⊂ S 1 such that |S\S 2 | = 0 and for every s ∈ S 2 and every i ∈ I 2 s , we have f 0 absolutely continuous on l i s (u). Thus, we can select by induction a representative of max(d 0 , |σ|) that is absolutely continuous on l i s (u) for a.e. s ∈ S and every i ∈ I s . □ For a.e. s ∈ S, if l i s (u) ∩ U = ∅ we have some additional information that will be useful in the final proof. Proposition 5.2. For a.e. s ∈ S, for every i ∈ I s , if

l i s (u) ∩ U = ∅ then f 0 = C i s is constant on l i s (u) with C i s ∈ {d n , n ∈ N, 0 ≤ n < N }.
Proof. For a.e. x ∈ Ω\U we have |σ(x)| ∈ {0} ∪ φ ′ (R\SC). By the coarea formula for a.e. s ∈ R, for

H 1 a.e. x ∈ (Ω ∩ L s (u))\U we have |σ(x)| ∈ φ ′ (R\SC). Hence, for a.e. s ∈ R if l i s (u) ∩ U = ∅ then f 0 (l i s (u)) ⊂ φ ′ (R\SC) ∪ f 0 (X)
for some X ⊂ l i s (u) with H 1 (X) = 0. Moreover for a.e. s ∈ R and every i ∈ I s , l i s (u) is a Lipschitz continuous curve such that f 0 is absolutely continuous on l i s (u). Since φ ′ (R\SC) is finite and f 0 (X) is the image of a negligible set by an absolutely continuous function we have |φ ′ (R\SC) ∪ f 0 (X)| = 0. The continuity of f 0 on l i s (u) implies that f 0 is constant on l i s (u). Since l i s (u) ∩ U = ∅ we obtain

f 0 = C i s ∈ {d n , n ∈ N, 0 ≤ n < N }. □
We use the notations of Section 4, where (σ n ) n∈N is a smooth approximation that converges a.e. on Ω to σ. We prove the following maximum principle on max(d 0 , |σ n |): Proposition 5.3. We assume that Ω has a C 1,1 boundary, ψ ∈ C 1,1 (R 2 ), λ is globally Lipschitz continuous on Ω and λ > 0. There exists

Υ := Υ |Ω|, max Ω λ, min Ω λ, ||ψ|| C 1,1 (R 2 ) , κ > 0
with κ the maximum of the principal curvatures of ∂Ω such that if ||∇λ|| L ∞ (Ω) ≤ Υ then for n ∈ N large enough, for a.e. every s ∈ S, for every i ∈ I s , if l i s (u) ∩ U = ∅ we have sup

F i s |σ n | ≤ sup l i s (u) max(d 0 , |σ n |)
where F i s is the bounded connected component of R 2 \l i s (u).

Remark 5.4. When λ is constant this result is true even if Ω and ψ are only Lipschitz continuous.

Proof. By the coarea formula in Proposition 2.18, σ n → σ H 1 a.e. on l i s (u) for a.e. s ∈ S and every i ∈ I s . We apply the maximum principle from [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 15.1] to |∇u n | on F i s . To do so we assume that

||∇λ|| L ∞ (Ω) ≤ min Ω λ 2 2 × L × sup x∈[ b 0 2 ,L] φ ′′ (x) + φ ′ (x)
x .

Here L is the Lipschitz constant introduced in Proposition 2.7. For every n ∈ N, there exists b n such that φ

′ n (b n ) = d 0 . Hence, for n ∈ N large enough, b n ≥ b 0 2 and ||∇λ n || L ∞ (Ω) ≤ min Ω λ 2 n ||∇u n || L ∞ (Ω) × sup x∈[ bn 2 ,L] φ ′′ n (x) + φ ′ n (x) x .
Thus, thanks to [14, Theorem 15.1, Equation (15.15)] for a.e. s ∈ R we have

sup Fs |∇u n | ≤ sup l i s (u) max( b 0 2 , |∇u n |) ≤ sup l i s (u) max(b n , |∇u n |) for n ∈ N large enough. Since φ ′ n is increasing we obtain sup Fs |σ n | ≤ sup l i s (u) max(d 0 , |σ n |).
□ Proposition 5.5. Let us consider s ∈ S and i ∈ I s such that l i s (u) ∩ U = ∅ and f 0 = C i s on l i s (u). Then for a.e. t > s, for every j ∈ I t such that l j t (u) ⋐ F i s and l j t (u) ∩ U = ∅, we have max(d 0 , |σ|) = C j t H 1 a.e. on l j t (u) with C j t ∈ {d n , n ∈ N, 0 ≤ n < N } not larger than C i s . Proof. By construction of f 0 in the proof of Proposition 5.1 and Proposition 5.2, we can construct a subsequence max(d 0 , |σ ψ(n) |) converging uniformly to C i s on l i s (u) that also converges uniformly to C j t on l j t (u). Hence, with the previous proposition we get:

C j t ≤ lim sup n→+∞ sup l j t (u) max(d 0 , |σ ψ(n) |) ≤ lim sup n→+∞ sup F i s (u) max(d 0 , |σ ψ(n) |) ≤ lim n→+∞ sup l i s (u) max(d 0 , |σ ψ(n) |) = C i s .
Thus, we have that C j t ≤ C i s . □ 6. Proof of the main theorem 6.1. Pseudo Cheeger problem. In this part we combine the maximum principle for |σ| and the Euler-Lagrange equation to prove that the level sets are almost Cheeger sets. We recall the definition of the Cheeger constant of a set: Definition 6.1. The Cheeger constant of Ω is defined as:

h Ω = inf D⊂Ω Per(D, R 2 ) |D| A set D ⊂ Ω of
F i s λ = C i s Per(F i s )
where F i s is the bounded connected component of R 2 \l i s (u) and C i s is the constant introduced Proposition in 5.2.

Proof. By [11, Section 5.11, Theorem 1], for a.e. s ∈ S and for every i ∈ I s , we have that D1 F i s ∈ BV (Ω).

We consider the sequence (σ n ) n∈N that converges a.e. on Ω towards σ introduced in Section 4. By [11, Section 5.8, Theorem 1] we obtain:

F i s div(σ n )dx = Ω 1 F i s (x)div(σ n )dx = - Ω ⟨σ n , D1 F i s |D1 F i s | ⟩d|D1 F i s | = - ∂ * F i s ⟨σ n , D1 F i s |D1 F i s | ⟩dH 1 .
The set ∂ * F i s is introduced in Definition 2.16. We can use Proposition 2.20 that gives:

- F i s div(σ n )dx = ∂ * F i s ⟨σ n , ∇u |∇u| ⟩dH 1 .
But by the coarea formula σ n → σ H 1 a.e. on ∂ * F i s ⊂ l i s (u) for a.e. s ∈ R and every i ∈ I s . By Proposition 5.2 and since σ is collinear to ∇u |∇u| H 1 a.e. on l i s (u), we get for such an s:

(6.1) lim n→+∞ - F i s div(σ n )dx = ∂ * F i s |σ|dH 1 = C i s Per(F i s ).
Moreover, (6.2) -

F i s div(σ n ) = F i s λ n → F i s λ
when n → +∞, where λ n := λ * ρ n . Hence, with (6.1) and (6.2), we have the desired result:

F i s λ = C i s Per(F i s )
for a.e. s ∈ S, for every i ∈ I s if l i s (u) ∩ U = ∅. □

We also have: Proposition 6.4. For every set F ⊂ F i s of finite perimeter we have

F λ ≤ C i s Per(F ).
Proof. We follow the same ideas developed in the previous proof. We have:

- F div(σ n ) = ∂ * F ⟨σ n , ν F ⟩dH 1 .
The term in the left hand side tends to F λ when n → +∞. For the term in the right hand side we get:

∂ * F ⟨σ n , ν F ⟩dH 1 ≤ ∂ * F |σ n |dH
F i s λ = C i s P er(F i s ).
We assume that such a l i s (u) exists. Since F i s ⋐ Ω by Remark 6.2 and the previous equality, we have

h Ω < P er(F i s ) |F i s | = 1 C i s |F i s | F i s λ. We have that ||λ|| L ∞ (Ω) ≤ d 0 h Ω . Thus, h Ω < d 0 h Ω C i s .
Hence, C i s < d 0 which is a contradiction. Thus, for a.e. s ∈ R we have l i s (u) ∩ (U ∪ ∂Ω) ̸ = ∅. Let v be another minimizer. By Proposition 3.2 and Proposition 3.6, for a.e. s ∈ R, on every connected component l i s (u) of L s (u) that is not a point we have u = v on l i s (u) or ∇(u -v) = 0 H 1 a.e. on l i s (u). By the coarea formula:

R 2 ∩[u̸ =v] |∇(u -v)||∇u| = R Ls(u)∩[u̸ =v]
|∇(u -v)|dH 1 ds. By Theorem 2.8, for a.e. s ∈ R, H 1 (L s \L * s ) = 0 and L * s is composed by a countable number of curves l i s (u). For every i ∈ I s , we have: Step 1 As an initialisation step, we assume that C i s = d 0 . By Proposition 3.3, u > s on F i s . By the coarea formula, for a.e. t > s, t belongs to S. We assume that there exists t > s and j ∈ I t such that l j t (u) ∩ U = ∅ and F t ⋐ F i s . By Proposition 5.3, |σ| = d 0 a.e. on l j t (u). Thus, by Proposition 6.3 we have that Ft λ = d 0 P er(F t ). For r > 1 close to 1 and x 0 ∈ Ω, we introduce F r t = r(F t -x 0 ) + x 0 ⋐ F s . Hence, by Proposition 5. , we have that rλ(r(x -x 0 ) + x 0 ) -λ(x) > 0 for every x ∈ F i s . That is a contradiction. Hence, for a.e. t > s and every j ∈ I t such that l j t (u) ⋐ F i s we have l j t (u) ∩ U ̸ = ∅. By Proposition 3.6, ∇(u -v) = 0 H 1 a.e. on l j t (u). By the coarea formula, ∇(u -v) = 0 a.e. in F i s . By Lemma 3.8, we have that ∇(u -v) = 0 H 1 a.e. on l i s (u).

Step 2 Now, we prove the induction part. We consider 1 ≤ n < N . Let us assume that for every k < n, for a.e. t ∈ R and every j ∈ I t if l j t (u) ∩ (U ∪ ∂Ω) = ∅ and C j t = d k then ∇(u -v) = 0 H 1 a.e. on l j t (u). If l i s (u) is such that l i s (u) ∩ (U ∪ ∂Ω) = ∅ and C i s = d n , we consider t > s such that l j t (u) ∩ U = ∅ and F t ⋐ F i s . Hence, by Proposition 5.5, either C j t = d n or C j t < d n . If C j t = d n , then as in Step 1 we construct F r t ⋐ F i s and we prove the ∇(u -v) = 0 H 1 a.e. on l j t (u). By induction we have ∇(u -v) = 0 H 1 a.e. on l j t (u) in the second case. Hence, ∇(u -v) = 0 H 1 a.e. on l j t (u). We can conclude as in Step 1 that ∇(u -v) = 0 H 1 a.e. on l i s (u).

Step 3 For a.e. s ∈ S, we consider l s (u) a connected component of L * (u). If l s (u) ∩ (R 2 \Ω) ̸ = 0 then by Proposition 3.2, u = v on l s (u). If l s (u) ⊂ Ω and l s (u) ∩ U ̸ = ∅ then by Proposition 3.6, ∇(u -v) = 0 H 1 a.e. on l s (u). Finally, thanks to Step 2 if l s (u) ⋐ Ω and l s (u) ∩ U = ∅ then we have ∇(u -v) = 0 H 1 a.e. on l s (u). Hence, we can prove with the coarea formula, as in the proof of Theorem 1.5, that u = v. □

Extensions

In this section we present an extension of the main theorem where SC has a countable number of connected components. We assume that φ is C 2 and φ ′′ > 0 on int(SC)\{0} and:

SC ∩ R + = SC ∞ ∪ n∈N SC n
with SC 0 := [0, b 0 ), SC n := (a n , b n ) for every n ∈ N * and SC ∞ is defined below. We assume that (a n ) n∈N * and (b n ) n∈N are strictly increasing sequences. Moreover, the sequence (a n ) n∈N * is bounded and lim Proof. With this new structural assumptions, the minimizers are still globally Lipschitz-continuous on Ω. We can define U as previously with int(SC) instead of SC. The function max(d 0 , φ ′ (∇u)) is still in H 1 loc (Ω). Since |φ ′ (R\SC)| = 0, Proposition 5.2 remains valid. Hence, the last crucial point is the end of the induction argument in Step 2 of the proof of Theorem 1.1. We assume that there exists l i s (u) ⋐ Ω such that l i s (u) ∩ U = ∅ and C i s = φ ′ (a ∞ ). Then for every l t ⋐ F i s , we either have that C t = φ ′ (a ∞ ) or ∇(u -v) = 0 H 1 a.e. on l t . Hence, we have that ∇(u -v) = 0 H 1 a.e. on l i s (u). Thus, u = v on Ω. □ Remark 7.2. The sets

[0, 1 2 ) ∪

1 .F

 1 Introduction 1.1. A model case. The motivation of this article is to study non strictly convex problems in the Calculus of Variations in dimension two as in the following model case: (1.1)I λ : u → Ω (∇u(x)) -λ(x)u(x)dxwhere Ω is an open bounded set in R 2 , λ ∈ L ∞ (Ω) and F (y) = f (|y|) with (1.2)

x∈Ω λ(x) > 0 .

 0 There exists a positive constant C := C(N, |Ω|, max Ω λ, min Ω λ, ||ψ|| C 1,1 (Ω) , κ)

Proposition 1 . 4 .

 14 There exists an open set U such that for every minimizer u, one has u ∈ C 1 (U ) and for every x ∈ U , |∇u(x)| ∈ SC\{0} while for a.e. x / ∈ U , |∇u(x)| / ∈ SC\{0}.

Proposition 2 . 1 .

 21 Let u be a minimizer of P λ on W 1,p ψ (Ω) with λ ∈ L ∞ (Ω) and λ(x) ≥ 0 for a.e. x ∈ Ω. Then min Ω u = min ∂Ω ψ. Proof. Since min Ω u ≤ c := min ∂Ω ψ, we have to prove that min Ω u ≥ min ∂Ω ψ. We introduce w := max(u, c). If there exists a point x ∈ Ω such that u(x) < c, then by continuity of u the set [u < c] has a positive measure. We have w = u and ∇w = ∇u on [u > c]. Moreover, since [u < c] has positive measure we have:

Lemma 2 . 2 .

 22 Let u and v be two minimizers of P λ with λ ∈ L ∞ (Ω). Then ∇u(x) and ∇v(x) are collinear and φ is affine on the interval [|∇u(x)|, |∇v(x)|] for a.e. x ∈ Ω.

Definition 2 . 16 .

 216 For a set E of finite perimeter in R 2 , we define the reduced boundary ∂ * E of E as the subset of supp |Dχ E | such that for every x ∈ ∂ * E, ν E (x) := lim r→0 Br(x) Dχ E Br(x) |Dχ E | exists and |ν E (x)| = 1.

Proof.

  By the vector valued coarea formula [2, Theorem 3.40] we have that A ∇v = R A D1 [v>s] ds for every Borel set A. By linearity, for every linear combination of indicator functions χ, we have supp v ⟨χ, ∇v⟩ = R supp v ⟨χ, D1 [v>s] ⟩ds.

R 2 ∩

 2 l i s (u)∩[u̸ =v] |∇(u -v)|dH 1 = 0.By Proposition 3.2, we getLs(u)∩[u̸ =v] |∇(u -v)|dH 1 = 0. Hence, [u̸ =v] |∇(u -v)||∇u|dx = 0.For the same reasons,R 2 ∩[u̸ =v] |∇(u -v)||∇v|dx = 0. Hence we have ∇(u -v) = 0 a.e. on [u ̸ = v]. This implies that the map u -v is constant on R 2 . Since u = v on ∂Ω, we have that u = v on Ω.□ Now, we are ready to prove the main theorem:Proof of Theorem 1.1. Let u and v be two minimizers of P λ . We assume that ||∇λ|| L ∞ (Ω) ≤ Υ < min Ω λ diamΩ up to decreasing the constant Υ from Proposition 5.3. For a.e. s ∈ R, every i ∈ I s , ifl i s (u) ∩ (U ∪ ∂Ω) = ∅ by Proposition 5.2, |σ| = C i s H 1 a.e. on l i s (u) with C i s ∈ {d n , 0 ≤ n < N }.We prove by induction on 0 ≤ n < N that if |σ| = d n on l i s (u) then ∇(u -v) = 0 H 1 a.e. on l i s .

2

 2 [START_REF] Anzellotti | Convex functionals and partial regularity[END_REF] we have |σ| ≤ d 0 on ∂F r t . Then, by Proposition 6.4,r Ft λ(r(x -x 0 ) + x 0 )dx = F r t λ(y)dy ≤ d 0 P er(F r t ) = rd 0 P er(F t ) = r Ft λ(x)dx. Thus, Ft rλ(r(x -x 0 ) + x 0 ) -λ(x) ≤ 0.Since ||∇λ|| L ∞ (Ω)

Proposition 7 . 1 .

 71 n→+∞ a n = α. For every n ∈ N, d n := φ ′ (b n ) = φ ′ (a n+1 ) is an increasing sequence. The connected component SC ∞ is exceptional because SC ∞ := (a ∞ , +∞) if α < a ∞ and SC ∞ := [α, +∞) if α = a ∞ .In that case Theorem 1.1 is still valid.

  The set U does not depend on the choice of a minimizer. Moreover, let u and v be two minimizers of P λ , then ∇u = ∇v on U .Proof. Let us consider two minimizers u and v of the same problem. We define respectively, U u and U v as the open sets of the previous proposition for u and v. By Lemma 2.2 and strict convexity of φ on SC, we have that ∇u = ∇v a.e. on U u . Hence, v ∈ C 1 (U u ) and for every x ∈ U u , |∇v(x)| ∈ SC\{0}. Thus, by definition of U v , we have that U u ⊂ U v . To prove the other inclusion we just have to exchange u and v. Hence, U u = U v = U and ∇u = ∇v on U .

	Hence for a.e. x ∈ Ω, φ • | • | is affine on the segment [∇u(x), ∇v(x)]. In view of the definition of φ
	and the strict convexity of the lower level sets of | • |, this means that ∇u(x) and ∇v(x) are collinear
	for a.e. x ∈ Ω but also that φ is affine on [|∇u(x)|, |∇v(x)|] for a.e. x ∈ Ω.
						□
	We use a result of [3] to introduce the following set where ∇u is continuous and |∇u| takes its
	values in SC\{0}, which is defined in Section 1.2.
	Proposition 2.3. When λ ∈ C 0 (Ω), there exists an open set U such that u ∈ C 1 (U ) and for every
	x ∈ U , |∇u(x)| ∈ SC\{0} while for a.e. x / ∈ U , |∇u(x)| / ∈ SC\{0}.
	Proof. By [3, Theorem 6.1], for a.e. x ∈ Ω such that |∇u(x)| ∈ SC\{0} there exists a neighborhood
	V of x such that u ∈ C 1,α (V). Since SC\{0} is open there exists ϵ > 0 such that for every
	x ′ ∈ B ϵ (x), |∇u(x ′ )| ∈ SC\{0}. Let U be the set of such x, then U is open and for a.e. x / ∈ U ,
	|∇u(x)| / ∈ SC\{0}.					□
	One of the interests of this set is the following:
	Proposition 2.4. □
	A direct consequence of this result is that:
	Remark 2.5. For every connected component
	x ∈ Ω,
	(2.1)	φ	∇u(x) + ∇v(x) 2	=	1 2	(φ(|∇u(x)|) + φ(|∇v(x)|)).

  Proposition 3.4. For every s ∈ S, I s is countable. Moreover, for every i ∈ I s , l i s (u) is the boundary of a connected componentF i s of E s = [u > s].Proof. By Theorem 2.8, I s is countable. Let us consider i ∈ I s . By the previous proposition, l i s (u) is the boundary of a connected component of E Proposition 3.5. For every s ∈ S, every connected component F s of E s , if F s ⋐ Ω then F s is simply connected and its boundary is a closed simple curve l s (u) with Lipschitz parametrization.Proof. Since F s is bounded, R 2 \F s has only one unbounded connected component. We call F s the complement of this unbounded set. We claim that F s = F s . We have that ∂ F s ⊂ ∂F s . Hence, u ≡ s on ∂ F s . Since F s is simply connected, ∂ F s is a connected set in L s (u) with H 1 (∂ F s ) > 0. By Theorem 2.8, ∂ F s is a closed subset of a closed simple curve with Lipschitz parametrization l s (u). Hence, F s is a bounded set such that ∂ F s ⊂ l s (u). We have that F s is an open set in R 2 \l s (u). Since, ∂ F s ⊂ l s (u), F s is also closed in R 2 \l s (u). The fact that F s is bounded and connected gives that F s is the bounded connected component of R 2 \l s (u) and by the Jordan curve theorem we have ∂ F s = l s (u). By Proposition 3.3, F s ⊂ E s . Since F s contains F s , we get that F s = F s . Moreover, we proved that F s is simply connected with l s (u) as boundary.□

s = [u > s]. □

We also have that:

  finite perimeter is said to be a Cheeger set if Per(D, R 2 ) = h Ω |D|. There is no Cheeger set D of Ω such that D ⋐ Ω because the function t → P er(tD,R 2 ) Proposition 6.3. For a.e. s ∈ S, for every i ∈ I s if l i s (u) ∩ U = ∅ we have

	Remark 6.2. |tD|
	is 1 t -homogeneous.
	The following equality is a consequence of Proposition 5.2:

  , |σ n |). By Proposition 5.1 and Proposition 5.2 we have max(d 0 , |σ n |) → C i s H 1 a.e. on l i s (u) when n → +∞. Hence, Proof of Theorem 1.5. For a.e. s ∈ R, for every i ∈ I s if l i s (u) ∩ (U ∪ ∂Ω) = ∅ then by Proposition 6.3,

	By Proposition 5.3, sup F i s	|σ n | ≤ sup l i s	max(d 0 F	λ ≤ C i s Per(F ).
				□
	6.2. Main proof. We first prove Theorem 1.5:

1

.