Tanguy Briançon
email: briancontanguy@yahoo.fr

Jimmy Lamboley
email: jimmy.lamboley@bretagne.ens-cachan.fr

Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints

Keywords: Shape optimization, eigenvalues of the Laplace operator, regularity of free boundaries

We consider the well-known following shape optimization problem:

where λ1 denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition, and D is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume a if such a ball exists in the box D (Faber-Krahn's theorem). In this paper, we prove regularity properties of the boundary of the optimal shapes Ω * in any case and in any dimension. Full regularity is obtained in dimension 2.

Introduction and main results

Let D be a bounded open subset of R d . For all open subset Ω of D, we denote by λ 1 (Ω) the first eigenvalue of the Laplace operator in Ω, with homogeneous boundary conditions, and by u Ω a normalized eigenfunction, that is

   -∆u Ω = λ 1 (Ω)u Ω in Ω, u Ω = 0 on ∂Ω, Ω u 2 Ω = 1.
We are interested here in the regularity of the optimal shapes of the following shape optimization problem, where a ∈ (0, |D|) (|D| denotes the Lebesgue measure of D)

:

Ω * open, Ω * ⊂ D, |Ω * | = a, λ 1 (Ω * ) = min{λ 1 (Ω); Ω * open, Ω ⊂ D, |Ω| = a}. (1)
By a well-known theorem of Faber and Krahn, if there is a ball B ⊂ D with |B| = a, then this ball is an optimal shape and it is unique, up to translations (and up to sets of zero capacity).

Here we adress the question of existence of a regular optimal set in all cases.

Existence of a quasi-open optimal set Ω * may be deduced from a general existence result by G. Buttazzo and G. Dal Maso (see [START_REF] Buttazzo | An existence result for a class of Shape Optimization Problems[END_REF]) for an extended version of [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF], where the variable sets Ω are not necessarily open. An optimal shape Ω * may not be more than a quasi-open set if D is not connected (we reproduce in the appendix the example mentioned in [START_REF] Brianc ¸on -M | Lipschitz continuity of state functions in some optimal shaping[END_REF]). On the other hand, it is proved in [START_REF] Brianc ¸on -M | Lipschitz continuity of state functions in some optimal shaping[END_REF] or [START_REF] Hayouni | Sur la minimisation de la première valeur propre du laplacien[END_REF] that such an open optimal set Ω * always exists for [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF] and, if moreover D is connected, then all optimal shapes Ω * are open. More precisely, it is proved in [START_REF] Brianc ¸on -M | Lipschitz continuity of state functions in some optimal shaping[END_REF] that, for any D, u Ω * is locally Lipschitz continuous in D. If moreover D is connected, then Ω * coincides with the support of u Ω * (and is therefore open). Let us summarize this as follows (see also [START_REF] Wagner | Optimal shape problems for eigenvalues[END_REF]): ∆u

Ω * + λ 1 (Ω * )u Ω * ≥ 0 in D, (2)
which means that ∆u

Ω * + λ 1 (Ω * )u Ω * is a positive Radon measure.
Here, we are interested in the regularity of ∂Ω * itself, and we prove the following theorem:

Theorem 1.2
Assume D is open, bounded and connected. Then any solution of (1) satisfies:

1. Ω * has locally finite perimeter in D and

H d-1 ((∂Ω * \ ∂ * Ω * ) ∩ D) = 0, (3)
where H d-1 is the Hausdorff measure of dimension d -1, and ∂ * Ω * is the reduced boundary (in the sense of sets with finite perimeter, see [START_REF] Evans -R | Measure theory and fine properties of functions[END_REF] or [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]).

2. There exists Λ > 0 such that We use the same strategy as in [START_REF] Brianc ¸on | Regularity of optimal shapes for the Dirichlet's energy with volume constraint[END_REF] (where the regularity is studied for another shape optimization problem). Theorem 1.2 essentially relies on the proof of the equivalence of (1) with a penalized version for the constraint |Ω| = a, as stated in Theorem 1.5 below. Once we have this penalized version, we can use techniques and results from [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF] (see also [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF] and [START_REF] Brianc ¸on | Regularity of optimal shapes for the Dirichlet's energy with volume constraint[END_REF]).

∆u Ω * + λ 1 (Ω * )u Ω * = √ ΛH d-1 ⌊∂Ω * ,
Remark 1.3 According to the results in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF], the third point in Theorem 1.2 is a direct consequence of the second one which says that u Ω * is a " weak solution" in the sense of [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF]. To obtain the full regularity of the boundary for d = 2, the fact that u Ω * is a weak solution is not sufficient, and more information has to be deduced from the variational problem. The approach is essentially the same as in Theorem 6.6 and Corollary 6.7 in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF]. The necessary adjustments are given at the end of this paper.

Remark 1.4 According to the result of [START_REF] Weiss | Partial regularity for weak solutions of an elliptic free boundary problem[END_REF][START_REF] Weiss | Partial regularity for a minimum problem with free boundary[END_REF][START_REF] Caffarelli -D. Jerison -C | Global energy minimizers for free boundary problems and full regularity in three dimensions[END_REF][START_REF] Silva | A singular energy minimizing free boundary[END_REF], it is likely that full regularity of the boundary may be extended to higher dimension (d ≤ 6 ?), and therefore that the estimate (3) can be improved. But this needs quite more work and is under study.

By a classical variational principle, we know that, for all Ω ⊂ D open,

λ 1 (Ω) = Ω |∇u Ω | 2 = min Ω |∇u| 2 , u ∈ H 1 0 (Ω), Ω u 2 = 1 . (4)
Here,

λ 1 (Ω * ) ≤ λ 1 (Ω) for all open set Ω ⊂ D with |Ω| = a. Since Ω ⊂ Ω ⇒ λ 1 (Ω) ≥ λ 1 ( Ω) , it follows that λ 1 (Ω * ) ≤ λ 1 (Ω)
for all open set Ω ⊂ D with |Ω| ≤ a. Coupled with (4), this leads to the following variation property of Ω * and u Ω * (see [START_REF] Brianc ¸on -M | Lipschitz continuity of state functions in some optimal shaping[END_REF] for more details), where we denote u = u Ω * , λ a = λ 1 (Ω * ), and Ω v = {x ∈ D; v(x) = 0}:

λ a = D |∇u| 2 = min D |∇v| 2 ; v ∈ H 1 0 (D), D v 2 = 1, |Ω v | ≤ a . (5)
Let us rewrite this as follows. For w ∈ H 1 0 (D), we denote

J(w) = D |∇w| 2 -λ a D w 2 .
Then applying [START_REF] Buttazzo | An existence result for a class of Shape Optimization Problems[END_REF] with v = w/( D w 2 ) 1/2 , we obtain that u is a solution of the following optimization problem:

J(u) ≤ J(w), for all w ∈ H 1 0 (D), with |Ω w | ≤ a. (6)
One of the main ingredient in the proof of Theorem 1.2 is to improve the variational property [START_REF] Caffarelli -D. Jerison -C | Global energy minimizers for free boundary problems and full regularity in three dimensions[END_REF] in two directions, as stated in Theorem 1.5 below. The approach is local.

Let B R be a ball included in D and centered on ∂Ω u ∩ D. We define

F = {v ∈ H 1 0 (D), u -v ∈ H 1 0 (B R )}. For h > 0, we denote by µ -(h) the biggest µ -≥ 0 such that, ∀ v ∈ F such that a -h ≤ |Ω v | ≤ a, J(u) + µ -|Ω u | ≤ J(v) + µ -|Ω v |. (7)
We also define µ + (h) as the smallest µ + ≥ 0 such that,

∀ v ∈ F such that a ≤ |Ω v | ≤ a + h, J(u) + µ + |Ω u | ≤ J(v) + µ + |Ω v |. (8)
The following theorem is a main step in the proof of Theorem 1.2:

Theorem 1.5 Let u, B R and F as above. Then for R small enough (depending only on u, a and D), there exists Λ > 0 and h 0 > 0 such that,

∀ h ∈ (0, h 0 ), 0 < µ -(h) ≤ Λ ≤ µ + (h) < +∞,
and, moreover,

lim h→0 µ + (h) = lim h→0 µ -(h) = Λ. ( 9
)
Remark 1.6 We can compare the existence of µ + (h) with Theorem 2.9 in [START_REF] Brianc ¸on -M | Lipschitz continuity of state functions in some optimal shaping[END_REF]. This theorem shows that there exists µ + such that

D |∇u| 2 ≤ D |∇v| 2 + λ a 1 - D v 2 + + µ + (|Ω v | -a), for v ∈ H 1 0 (D) and |Ω v | ≥ a.
The difference with [START_REF] Brianc ¸on -M | Lipschitz continuity of state functions in some optimal shaping[END_REF] is that, in (8), we have the term λ a [1 -D v 2 ] (not only the positive part), but we allowed only perturbations in B R . We cannot expect to have something like [START_REF] Silva | A singular energy minimizing free boundary[END_REF] for perturbations in all D (because we may find v with |Ω v | > a and J(v) < 0, so lim t→+∞ J(tv) = -∞).

In the next section, we will prove Theorem 1.5. In the third section, we will prove Theorem 1.2. In the appendix, we discuss the case D non-connected.

2 Proof of Theorem 1.5

In the next lemma, we give an Euler-Lagrange equation for our problem. The proof follows the steps of the Euler-Lagrange equation in [START_REF] Crouzeix | Variational approach of a magnetic shaping problem[END_REF].

Lemma 2.1 (Euler-Lagrange equation) Let u be a solution of [START_REF] Caffarelli -D. Jerison -C | Global energy minimizers for free boundary problems and full regularity in three dimensions[END_REF]. Then there exists Λ ≥ 0 such that, for all

Φ ∈ C ∞ 0 (D, R d ), D 2(DΦ∇u, ∇u) - D |∇u| 2 ∇ • Φ + λ a D u 2 ∇ • Φ = Λ Ωu ∇ • Φ .
Proof. We start by a general remark that will be useful in the rest of the paper.

If v ∈ H 1 0 (D) and if Φ ∈ C ∞ 0 (D, R d ), we define v t (x) = v(x + tΦ(x))
; therefore, for t small enough, v t ∈ H 1 0 (D). A simple calculus gives (when t goes to 0),

|Ω vt | = |Ω v | -t Ωv ∇ • Φ + o(t), J(v t ) = J(v) + t D 2(DΦ∇v.∇v) - D |∇v| 2 ∇ • Φ + λ a D v 2 ∇ • Φ + o(t).
Now we apply this with v = u and Φ such that Ωu ∇ • Φ > 0. Such a Φ exists, otherwise we would get, using that D is connected, Ω u = D or ∅ a.e. We have |Ω ut | < |Ω u | for t ≥ 0 small enough and, by minimality,

J(u) ≤ J(u t ) = J(u) + t D 2(DΦ∇u, ∇u) - D |∇u| 2 ∇ • Φ + λ a D u 2 ∇ • Φ + o(t),
and so,

D 2(DΦ∇u, ∇u) - D |∇u| 2 ∇ • Φ + λ a D u 2 ∇ • Φ ≥ 0. ( 10
)
Now, we take Φ with Ωu ∇ • Φ = 0. Let Φ 1 be such that Ωu ∇ • Φ 1 = 1. Writing [START_REF] Gilbarg -N | Elliptic Partial Differential Equations of Second order[END_REF] with Φ + ηΦ 1 and letting η goes to 0, we get [START_REF] Gilbarg -N | Elliptic Partial Differential Equations of Second order[END_REF] with this Φ and, using -Φ, we get [START_REF] Gilbarg -N | Elliptic Partial Differential Equations of Second order[END_REF] with an equality instead of the inequality. For a general Φ, we use this equality with Φ -

Φ 1 ( Ωu ∇ • Φ) (we have Ωu ∇ • Φ -Φ 1 Ωu ∇ • Φ = 0
), and we get the result with

Λ = D 2(DΦ 1 ∇u, ∇u) - D |∇u| 2 ∇ • Φ 1 + λ a D u 2 ∇ • Φ 1 ≥ 0, using (10)
.

Remark 2.2 We will have to prove that, in fact, Λ > 0.

Let us remind our notations: let u be a solution of [START_REF] Caffarelli -D. Jerison -C | Global energy minimizers for free boundary problems and full regularity in three dimensions[END_REF], and let B R be a ball included in D and centered on ∂Ω u ∩ D. We define

F = {v ∈ H 1 0 (D), u -v ∈ H 1 0 (B R )}.
Before proving Theorem 1.5, we give the following useful lemma:

Lemma 2.3 Let u, B R and F as above. Then there exists a constant C such that, for R small enough,

∀v ∈ F, J(v) ≥ 1 2 BR |∇v| 2 -C.
Proof. We know that λ 1 (B R ) = λ 1 (B 1 )/(R 2 ) (we just use the change of variable x → x/R). If R is small enough we have:

λ 1 (B R ) ≥ 1, 4λ a λ 1 (B R ) ≤ 1/2. ( 11
) Let v ∈ F; so u -v ∈ H 1 0 (B R )
, and using the variational formulation of λ 1 (B R ), we get

u -v 2 L 2 (BR) ≤ ∇(u -v) 2 L 2 (BR) λ 1 (B R ) .
We deduce that,

v 2 L 2 (BR) ≤ 2 ∇(u -v) 2 L 2 (BR) λ 1 (B R ) + 2 u 2 L 2 (BR) ≤ 4 ∇v 2 L 2 (BR) λ 1 (B R ) + C λ a ,
(we use [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]) where C depends only on the L 2 norms of u and his gradient. Now we have

J(v) ≥ D |∇v| 2 -λ a 4 ∇v 2 L 2 (BR) λ 1 (B R ) + C λ a ,
and we get the result using [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF].

Remark 2.4 This lemma is interesting for two reasons. The first one is that J is bounded from below on F . The second one is that, if

v n ∈ F is a sequence such that J(v n ) is bounded, then ∇v n L 2 (BR) is also bounded. Since v n = u outside B R we deduce that v n is bounded in H 1 0 (
D) (and so weakly converges up to a sub-sequence...).

Proof of Theorem 1.5: We divide our proof into four parts. Let Λ ≥ 0 be as in Lemma 2.1.

First part: Λ ≤ µ + (h) < +∞.
We start the proof by showing that µ + (h) is finite. Since B R is centered on the boundary on ∂Ω u , we first show:

0 < |Ω u ∩ B R | < |B R |.
The first inequality comes from the fact that Ω u is open. The second one comes from the following lemma:

Lemma 2.5 Let ω be an open subset of D, and let u be a solution of [START_REF] Caffarelli -D. Jerison -C | Global energy minimizers for free boundary problems and full regularity in three dimensions[END_REF].

If |Ω u ∩ ω| = |ω|, then -∆u = λ a u in ω,
and therefore ω ⊂ Ω u .

Proof of Lemma 2.5. Since u > 0 a.e. on ω, we define v ∈ H 1 0 (D) by v = u outside ω and -∆v = λ a u in ω. From the strict maximum principle, we get v > 0 on ω and

|Ω v | = |Ω u |. By minimality (J(u) ≤ J(v)) we have, ω (∇u -∇v).(∇u -∇v + 2∇v) -λ a ω (u -v)(u + v) ≤ 0 ω |∇u -∇v| 2 + λ a ω (u -v)(2u -u -v) ≤ 0, (we use that u -v ∈ H 1 0 (ω) and -∆v = λ a u in ω).
We get that u = v a.e. in ω and by continuity u = v > 0 everywhere in ω.

If |Ω u ∩ B R | = |B R |, applying this lemma to ω = B R , we would get Ω u ∩ B R = B R , which is impossible since B R is centered on ∂Ω u . If R is small enough we can also suppose, 0 < |Ω u \ B R | < |D \ B R |.
For the first inequality, we need that |B R | < a, and for the second one we need a < |D| -|B R |.

Let h > 0 be such that

h < |B R | -|Ω u ∩ B R | (and so, if v ∈ F with |Ω v | ≤ a + h, then |Ω v ∩ B R | < |B R |). Let (µ n ) an increasing sequence to +∞. There exists v n ∈ F such that |Ω vn | ≤ a + h and, J(v n ) + µ n (|Ω vn | -a) + = min v∈F ,|Ωv|≤a+h J(v) + µ n (|Ω v | -a) + . ( 12
)
For this we use remark 2.4, and so the functional J(v) + µ n (|Ω v |a) + is bounded by below for v ∈ F. Moreover, a minimizing sequence for this functional is bounded in H 1 0 (D) and so weakly converges in H 1 0 (D), strongly in L 2 (D) and almost everywhere (up to a sub-sequence) to some v n . Using the lower semi-continuity of v → D |∇v| 2 for the weak convergence, the strong convergence in L 2 (D) and the lower semi-continuity of v → |Ω v | for the convergence almost everywhere we see that v n is such that ( 12) is true.

If |Ω vn | ≤ a then ( 8) is true with µ n , so we will suppose to the contrary that |Ω vn | > a for all n.

Step 1: Euler-Lagrange equation for v n . If we set b n = |Ω vn |, then v n is also solution of

J(v n ) = min v∈F ,|Ωv|≤bn J(v).
With the same proof as in lemma 2.1, we can write an Euler-Lagrange equation for

v n in B R . That is, there exists Λ n ≥ 0 such that, for Φ ∈ C ∞ 0 (B R , R d ), D 2(DΦ∇v n .∇v n ) - D |∇v n | 2 ∇ • Φ + λ a D v 2 n ∇ • Φ = Λ n Ωv n ∇ • Φ. ( 13
)
Step 2:

Λ n ≥ µ n . There exists Φ ∈ C ∞ 0 (B R ) such that Ωv n ∇ • Φ = 1. Let v t n (x) = v n (x + tΦ(x)).
We have v t n ∈ F for t ≥ 0 small enough, and using derivation results recalled in the proof of lemma 2.1 and |Ω vn | > a, we get

a < |Ω v t n | = |Ω vn | -t + o(t) ≤ a + h, J(v t n ) = J(v n ) + tΛ n + o(t). Now we use (12) with v = v t
n in order to get,

J(v n ) + µ n (|Ω vn | -a) ≤ J(v n ) + tΛ n + o(t) + µ n (|Ω vn | -t -a),
and dividing by t > 0 and letting t goes to 0, we finally get Λ n ≥ µ n .

Step 3: v n strongly converges to some v. Using [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF] with v = u, we get

J(v n ) + µ n (|Ω vn | -a) ≤ J(u) (14)
and so, using Remark 2.4, we can deduce that v n weakly converge in H 1 0 (up to a sub-sequence) to some v ∈ F with |Ω v | ≤ a + h. We also have the strong convergence in L 2 (D) and the convergence almost everywhere. Since J is bounded from below on F , we see from ( 14) that µ n (|Ω vn |a) is bounded and we get lim n→∞ |Ω vn | = a, and so |Ω v | ≤ a. From J(v n ) ≤ J(u), we get J(v) ≤ lim inf J(v n ) ≤ J(u) and so v is a solution of ( 6). Finally we can write, using [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF], that J(v n ) ≤ J(v) and we get, using the strong convergence of

v n in L 2 , lim sup n→∞ D |∇v n | 2 ≤ D |∇v| 2 .
We also have, with weak convergence in

H 1 0 (D) that D |∇v| 2 ≤ lim inf n→∞ D |∇v n | 2 .
We deduce that lim n→∞ ∇v n L 2 (D) = ∇v L 2 (D) . With the weak-convergence, this gives the strong convergence of v n to v in H 1 0 (D).

Step 4: lim Λ n = Λ. We see that v is a solution of ( 6), so we can apply Lemma 2.1 to get that there exists a Λ v such that

∀ Φ ∈ C ∞ 0 (D, R d ), D 2(DΦ∇v.∇v) - D |∇v| 2 ∇ • Φ + λ a D v 2 ∇ • Φ = Λ v Ωv ∇ • Φ.
We have u = v outside B R so, using this equation and the Euler-Lagrange equation for u we see that Λ v = Λ. Now, we write the Euler-Lagrange for

v n and Φ ∈ C ∞ 0 (D, R d ) such that Ωv ∇•Φ = 0, D 2(DΦ∇v n .∇v n ) - D |∇v n | 2 ∇ • Φ + λ a D v 2 n ∇ • Φ = Λ n Ωv n ∇ • Φ,
and, using the strong convergence of v n to v, we get that

lim n→∞ Λ n = lim n→∞ D 2(DΦ∇v n .∇v n ) -D |∇v n | 2 ∇ • Φ + λ a D v 2 n ∇ • Φ Ωv n ∇ • Φ = D 2(DΦ∇v.∇v) -D |∇v| 2 ∇ • Φ + λ a D v 2 ∇ • Φ Ωv ∇ • Φ = Λ.
Since lim µ n = +∞ we get the contradiction from Steps 2 and 4, and so µ + (h) is finite. To conclude this first part, we now have to see that Λ ≤ µ + (h). Let Φ ∈ C ∞ 0 be such that Ωu ∇ • Φ = -1, and let u t (x) = u(x + tΦ(x)). Using the calculus in the proof of Lemma 2.1 we have, for t ≥ 0 small enough,

a ≤ |Ω ut | = a + t + o(t) ≤ a + h, J(u t ) = J(u) -tΛ + o(t).
Now, using (8), we have

J(u) + µ + (h)a ≤ J(u) -tΛ + µ + (h)(a + t) + o(t),
and we get Λ ≤ µ + (h).

Second part: lim µ

+ (h) = Λ.
We first see that µ + (h) > 0 for h > 0. Indeed, if µ + (h) = 0 we write

for every ϕ ∈ C ∞ 0 (B R ) with |{ϕ = 0}| < h, J(u) ≤ J(u + tϕ), so -∆u = λ a u in B R , which contradicts 0 < |Ω u ∩ B R | < |B R |.
Let ε > 0 and h n > 0 a decreasing sequence tending to 0. Because h → µ + (h) is non-increasing, we just have to see that lim µ

+ (h n ) ≤ Λ + ε for a sub-sequence of h n . If Λ > 0, let ε ∈]0, Λ[ and 0 < α n := µ + (h n ) -ε < µ + (h n ); if Λ = 0, let 0 < α n = µ + (h n )/2 < µ + (h n ). There exists v n such that J(v n ) + α n (|Ω vn | -a) + = min v∈F ,|Ωv|≤a+hn J(v) + α n (|Ω v | -a) + . Since α n < µ + (h n ) we see that |Ω vn | > a (otherwise we write J(u) ≤ J(v n ) + α n (|Ω vn | -a) + ).
We now have 4 steps that are very similar to the 4 steps used in the previous part to show that µ + (h n ) is finite.

Step 1: Euler-Lagrange equation for

v n . If v ∈ F is such that |Ω v | ≤ |Ω vn |, we have J(v n ) ≤ J(v).
Then, as in Lemma 2.1 we can write the Euler-Lagrange equation ( 13) for v n in B R for some Λ n .

Step 2: Λ n ≥ α n . Since |Ω vn | > a the proof is the same as step 2 in the first part, with α n instead of µ n .

Step 3: v n strongly converge to some v. As in step 3 above, we just write,

J(v n ) + α n (|Ω vn | -a) + ≤ J(u),
Now, using that ∇Ψ = ∇u(1ϕ) -(∇ϕ)u + ε∇ϕ and the L ∞ bounds for u and ∇u, we see that v εu is bounded in H 1 (D). Now, up to a subsequence, v ε weakly converges in H 1 0 (D) to v such that:

   -∆v = λ a u in (B r \ B r/2 ) ∩ Ω u v = u in D \ B r v = 0 in B r/2 ∪ (B r ∩ {u = 0}).
Using [START_REF] Crouzeix | Variational approach of a magnetic shaping problem[END_REF] with h = |B r/2 |, and u = v in D \ B r , we have:

Br |∇u| 2 -λ a Br u 2 + µ -(h)|Ω u ∩ B r | ≤ Br |∇v| 2 -λ a Br v 2 + µ -(h)|Ω v ∩ B r |,
and so,

B r/2 |∇u| 2 µ -(h)|Ω u ∩ B r/2 | ≤ Br \B r/2 ∇(v -u).∇(u -v + 2v) -λ a Br \B r/2 (v 2 -u 2 ) + λ a B r/2 u 2 ≤ lim inf ε→ 0 2 Dε ∇(v ε -u).∇v ε -λ a Dε (v 2 ε -u 2 ) + λ a B r/2 u 2 = lim inf ε→ 0 2 ∂B r/2 ∩{u>ε} (ε -u) ∂v ε ∂n + 2λ a Dε (v ε -u)u -λ a Dε (v 2 ε -u 2 ) + λ a B r/2 u 2 = lim inf ε→ 0 2 ∂B r/2 ∩{u>ε} (ε -u) ∂v ε ∂n + λ a Dε (2uv ε -u 2 -v 2 ε ) + λ a B r/2 u 2 ≤ lim inf ε→ 0 2 ∂B r/2 ∩{u>ε} (ε -u)∇v ε . -→ n + λ a B r/2 u 2 ,
where -→ n is the outward normal of D ε and so the inward normal of B r/2 . Let w ε be such that,

   -∆w ε = λ a u on B r \ B r/2 w ε = u on ∂B r ∩ {u > ε} w ε = ε on (∂B r ∩ {u ≤ ε}) ∪ ∂B r/2 .
Because w ε ≥ ε on ∂(B r \B r/2 ) and super-harmonic in B r \B r/2 , we get that w ε ≥ ε in B r \B r/2 . In particular

w ε ≥ v ε = ε in ∂D ε ∩(B r \B r/2
). Moreover, we also have w ε ≥ v ε on ∂D ε ∩(∂B r ∪∂B r/2 ), and since w εv ε is harmonic in D ε , we get

w ε ≥ v ε in D ε . Using w ε = v ε = ε on ∂B r/2 ∩ {u > ε},
we can now compare the gradients of w ε and v ε on this set,

0 ≤ -∇v ε . -→ n ≤ -∇w ε . -→ n on ∂B r/2 ∩ {u > ε}. (23)
Let now w 0 ε be defined by w 0 ε = w ε on ∂(B r \ B r/2 ) and harmonic in B r \ B r/2 . We use now the following estimate:

0 ≤ -∇w 0 ε . -→ n ≤ C r - ∂Br (u -ε) + ≤ Cγ on ∂B r/2 , (24)
where γ = 1 r -∂Br u (to get this estimate, we can first prove, using a comparison argument, that

|∇w 0 ε | ≤ C r w 0 ε -ε ∞,B 3r/4 \B r/2
, and then conclude using again maximum principle and Poisson formula for functions that are harmonic in a ball). Let w 1 ε = w εw 0 ε , we have w 1 ε = 0 on ∂(B r \ B r/2 ) and -∆w 1 ε = λ a u in B r \ B r/2 and so,

∇w 1 ε ∞,Br\B r/2 ≤ Cr u ∞ ≤ Cr. (25)
Now using ( 22), ( 23), ( 24) and (25) we get,

L := B r/2 |∇u| 2 + µ -(h)|Ω u ∩ B r/2 | ≤ C(γ + r) ∂B r/2 u + λ a B r/2 u 2 . ( 26
)
Our goal is now to bound from above the right-hand of this inequation with CL(γ + r): and so if γ and r are small enough we will get L = 0 and so u ≡ 0 in B r/2 . We now give an estimate of u ∞,B r/2 in term of γ. Let w = 0 on ∂B r and -∆w = λ a u in B r . We have (using (2)) ∆(uw) = ∆u + λ a u ≥ 0 in B r and uw = u on ∂B r so,

u -w ∞,B r/2 ≤ C - ∂Br u ≤ Cγr.
We also have that w ∞,Br ≤ Cr 2 u ∞,Br ≤ Cr 2 , and finally,

u ∞,B r/2 ≤ C(γr + r 2 ). ( 27
)
We now write (using ( 27)),

∂B r/2 u ≤ C B r/2 |∇u| + 1 r B r/2 u ≤ C 1 2 B r/2 |∇u| 2 + 1 2 |Ω u ∩ B r/2 | + 1 |Ω u ∩ B r/2 | u ∞,B r/2 .
Here we use Theorem 1.5 to see that there exists h 0 such that

Λ 2 ≤ µ -(h) ≤ Λ, 0 < h ≤ h 0 .
And so, we have

∂B r/2 u ≤ C B r/2 |∇u| 2 + µ -(h)|Ω u ∩ B r/2 | + C|Ω u ∩ B r/2 |(γ + r) ≤ CL(1 + γ + r), (28)
with C independent of r for every r small enough such that h = |B r/2 | ≤ h 0 . We also have (using

(27)) B r/2 u 2 ≤ C|Ω u ∩ B r/2 |(γr + r 2 ) ≤ CL(γr + r 2 ). (29)
We now get, from (26), ( 28) and (29), if γ ≤ 1 and r ≤ 1,

L ≤ C(γ + r)L(1 + γ + r) + CL(γr + r 2 ) ≤ CL(γ + r),
and, if we suppose r ≤ 1 2C we get,

L ≤ CLγ + L 2 ,
and so, if γ < 1 2C we get L = 0 and u ≡ 0 on B r/2 .

With the help of this lemma, we are now able to successively prove the three properties (a),(b) and (c) of (20).

Proof of (a). The proof is now, using (21) in lemma 3.1, the same as in [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF] or in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF]. Here are the main steps: we first show that there exists C 1 , C 2 and r 0 such that, for every B(x 0 , r) ⊂ B with r ≤ r 0 ,

0 < C 1 ≤ |B(x 0 , r) ∩ Ω u | |B(x 0 , r)| ≤ C 2 < 1,
and

C 1 r d-1 ≤ (∆u + λ a u)(B(x 0 , r)) ≤ C 2 r d-1 .
The proof is the same as in [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF] with λ a u instead of f . It gives directly (using the Geometrical measure theory, see section 5.8 in [START_REF] Evans -R | Measure theory and fine properties of functions[END_REF]) the first point of Theorem 1.2.

Proof of (b). For the second point, we see that ∆u + λ a u is absolutely continuous with respect to H d-1 ⌊∂Ω u which is a Radon-Measure (using the first point), so we can use Radon's Theorem. To compute the Radon's derivative, we argue as in Theorem 2.13 in [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF] or (4.7,5.5) in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF]. The main difference is that here, we have to use [START_REF] Evans -R | Measure theory and fine properties of functions[END_REF] in Theorem 1.5 to show that, if u 0 denotes a blow-up limit of u(x 0 + rx)/r (when r goes to 0), then u 0 is such that, for every v such that v = u 0 outside B(0, 1). To show this, in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF] or in [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF] the authors use only perturbations in B(x 0 , r) with r goes to 0, so using (9), we get the same result. We can compute the Radon's derivative and get (in B)

B
∆u + λ a u = √ ΛH d-1 ⌊∂Ω u .
Now, u is a weak-solution in the sense of [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF] and [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF] and we directly get the analytic regularity of ∂ * Ω u (this regularity is shown for weak-solutions).

Proof of (c). If d = 2, in order to have the regularity of the whole boundary, we have to show that Theorem 6.6 and Corollary 6.7 in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF] (which are for solutions and not weak-solutions) are still true for our problem. The Corollary directly comes from the Theorem. So we need to show that, if d = 2 and x 0 ∈ ∂Ω u , then The only difference now with [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF] is the last term. Using Theorem 1.5, we see that (Λµ -(h))h = o(h), so we can choose the same kind of ζ and ε as in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF] to get (30) (see Theorem 5.7 in [START_REF] Brianc ¸on | Regularity of optimal shapes for the Dirichlet's energy with volume constraint[END_REF] for more details).

Proposition 1 . 1

 11 Assume D is open and bounded. The problem (1) has a solution Ω * , and u Ω * is nonnegative and locally Lipschitz continuous in D. If D is connected, Ω * = {x ∈ D, u Ω * > 0}. Moreover, we have

4 .

 4 in the sense of distribution in D, where H d-1 ⌊∂Ω * is the restriction of the (d -1)-Hausdorff measure to ∂Ω * . 3. ∂ * Ω * is an analytic hypersurface in D. If d = 2, then the whole boundary ∂Ω * ∩ D is analytic.

(0, 1 )

 1 |∇u 0 | 2 + Λ|{u 0 = 0} ∩ B(0, 1)| ≤ B(0,1) |∇v| 2 + Λ|{v = 0} ∩ B(0, 1)|,

max{Λ -|∇u| 2 , 2 + {u≥εζ} ε 2 |∇ζ| 2 -λ a {u<εζ} u 2 -λ a {u≥εζ} (εζ) 2 ,

 2222 0} = 0. (30)We argue as in Theorem 6.6 in[START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF]. Let ζ ∈ C ∞ 0 (B) be nonnegative and let v = max{uεζ, 0}. Using[START_REF] Crouzeix | Variational approach of a magnetic shaping problem[END_REF] with this v and h = |0 < u ≤ εζ| ≤ |{ζ = 0}| we get,µ -(h)|0 < u ≤ εζ| ≤ |∇v| 2 -|∇u| 2 + λ a (u 2v 2 ) = |∇ min{εζ, u}| 2 -2 ∇u.∇ min{εζ, u} +λ a {u<εζ} u 2λ a {u≥εζ} (εζ) 2 + 2λ a {u≥εζ} uεζ.Using -∆u = λ a u in Ω u we get:∇u.∇ min{εζ, u} = λ a u min{εζ, u} = λ a {u<εζ} u 2 + λ a {u≥εζ} uεζ,and so,µ -(h)|0 < u ≤ εζ| ≤{u<εζ} |∇u| and so, we can deduce that, {0<u<εζ} (Λ -|∇u| 2 ) ≤ {u≥εζ} ε 2 |∇ζ| 2 + (Λµ -(h))h.

to get (up to a sub-sequence) that v n weakly converges in H 1 0 (D), strongly in L 2 (D) and almosteverywhere to v ∈ F. We have a < |Ω vn | ≤ a + h n and so lim n→∞ |Ω vn | = a. As in step 3 above, we deduce that v is a solution of (6), and using

we get the strong convergence in H 1 0 (D).

Step 4: lim Λ n = Λ. The proof is the same as in step 4 of the first part of the proof. We write the Euler-Lagrange equation for v in D and use u = v outside B R . We get that lim Λ n = Λ by letting n go to +∞ in the Euler-Lagrange equation for v n in B R (using the strong convergence of v n ).

We can now conclude this second part: if Λ > 0, we have, for n large enough,

and so µ + (h n ) ≤ Λ + 2ε.

If Λ = 0 we have

Let h n be a sequence decreasing to 0, and let

and let u t = u(x + tΦ(x)) for t ≥ 0. We have (using the proof of Lemma 2.1),

Now, using [START_REF] Crouzeix | Variational approach of a magnetic shaping problem[END_REF], we have

and we get µ -(h) ≤ Λ.

Let v n be a solution of the following minimization problem,

If |Ω vn | = a we have,

And we get that -∆v n = λ a v n in B R and so, we have

We now deduce (u ≡ 0 on B R ) that λ a ≥ λ 1 (B R ), which is a contradiction, at least for R small enough.

We now study the sequence v n in a very similar way than above.

Step 1: Euler-Lagrange equation for

, so we have an Euler-Lagrange equation ( 13) for v n in B R for some Λ n .

Step 2:

and writting [START_REF] Weiss | Partial regularity for weak solutions of an elliptic free boundary problem[END_REF] with w = v t n we get the result.

Step 3: v n strongly converge to some v. As in step 3 above we just write that

to get (up to a sub-sequence) that v n weakly converge in H 1 0 (D), strongly in L 2 (D) and almosteverywhere to v ∈ F. We have ah n < |Ω vn | ≤ a and so lim n→∞ |Ω vn | = a. As in step 3 above, we deduce that v is a solution of (6), and using

we get the strong convergence in H 1 0 (D).

Step 4: lim Λ n = Λ. The proof is exactly the same as in step 4 above in the study of the limit of µ + (h n ). Now we have, using steps 2 and 4, for n large enough,

Fourth part: Λ > 0. We would like to show that Λ > 0 (which implies µ -(h) > 0 for h small enough). We argue by contradiction and we suppose that Λ = 0. The proof is very close to the proof of Proposition 6.1 in [START_REF] Brianc ¸on | Regularity of optimal shapes for the Dirichlet's energy with volume constraint[END_REF]. We start with the following proposition:

Proposition 2.6 Assume Λ = 0. Then, there exists η a decreasing function with lim r→0 η(r) = 0 such that, if

Proof of Proposition 2.6. Let x 0 , r be as above, and we set B r = B(x 0 , r). Let v be defined by,

and v = u outside B r . We have v > 0 on B r . We get, using [START_REF] Silva | A singular energy minimizing free boundary[END_REF],

we also get (using

Now, with the same computations as in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF], [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF] (with λ a u instead of f ) we get,

Now, using (17), ( 18) and ( 19) we get the result.

End of proof of Theorem 1.5. Now, the rest of the proof is the same as Proposition 6.2 in [START_REF] Brianc ¸on | Regularity of optimal shapes for the Dirichlet's energy with volume constraint[END_REF] with λ a u instead of f χ Ωu . The idea is that, from the estimate (16) of Proposition 2.6, ∇u tends to 0 at the boundary, and consequently the measure ∆u does not charge the boundary ∂Ω u . It follows that -∆u = λ a u in B R , which, by strict maximum principle, contradicts that u is zero on some part of B R .

3 Proof of Theorem 1.2

Let Ω * be a solution of [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF]. Then u = u Ω * is a solution of ( 6), and thus satisfies Proposition 1.1 and Theorem 1.5; moreover, Ω * = Ω u . Like in the previous section, we work in B, a small ball centered in ∂Ω u . Since the approach is local, we will show regularity for the part of ∂Ω u included in B; but B can be centered on every point of ∂Ω u ∩ D, so this is of course enough to lead to the announced results in Theorem 1.2. Coupled with Remark 1.3, we conclude that it is sufficient to prove:

(a) Ω * has finite perimeter in B and

We use the same arguments as in [START_REF] Alt -L | Existence and regularity for a minimum problem with free boundary[END_REF] and [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF], but we have to deal with the term in u 2 instead of f u (in [START_REF] Gustafsson | Existence and geometric properties of solutions of a free boundary problem in potential theory[END_REF]). So we first start with the following technical lemma.

u ≤ C 2 then u ≡ 0 on B(x 0 , r/2).

(

Proof. The first point comes directly from the proof of Proposition 2.6. We take the same v and, using equation (19), we see that there exists

For the second part we argue as in Theorem 3.1 in [START_REF] Alt -L | Variational Problems with Two Phases and Their Free Boundaries[END_REF]. We will denote B r for B(x 0 , r). In this proof, C denotes (different) constants which depend only on a, d, D, u and B, but not on x 0 or r.

Let ε > 0 small and such that {u = ε} is smooth (true for almost every ε), let

Appendix

In this appendix, we discuss the hypothesis "D is connected". We begin with the following example, taken from [START_REF] Brianc ¸on -M | Lipschitz continuity of state functions in some optimal shaping[END_REF].

Example 4.1 (from [START_REF] Brianc ¸on -M | Lipschitz continuity of state functions in some optimal shaping[END_REF]) We take

1 +ε, then the solution u of (5) coincides with the first eigenfunction of D 1 and is identically 0 on D 2 , and thus Ω u = D 1 and |Ω u | < a.

In this case, we can choose an open subset ω of D 2 with |ω| = ε. Then Ω * := D 1 ∪ ω is a solution of (1). Since ω may be chosen as irregular as one wants, this proves that optimal domains are not regular in general.

However, we are able to prove the following proposition. Remark 4.4 To summarize, in all cases, there exists a solution Ω * to (1) which is regular in the sense of Theorem 1.2, but there may be some other non regular optimal shape. And if D is connected, any optimal shape is regular. That is -∆u = λ a u in D and the third case is not possible since by maximum principle u > 0 or u = 0 on each connected component of D.