LIPSCHITZ CONTINUITY OF THE EIGENFUNCTIONS ON OPTIMAL SETS FOR FUNCTIONALS WITH VARIABLE COEFFICIENTS

. This paper is dedicated to the spectral optimization problem


Introduction and main results
Let D be a bounded open subset of R d and Λ be a positive constant. We consider the spectral optimization problem min λ 1 (Ω) + · · · + λ k (Ω) + Λ|Ω| : Ω ⊂ D quasi-open (1.1) where 0 < λ 1 (Ω) ≤ · · · ≤ λ k (Ω) denote the first k eigenvalues, counted with the due multiplicity, of the operator in divergence form −b(x) −1 div (A x ∇·). This means that for every λ i (Ω) there is an eigenfunction u i ∈ H 1 0 (Ω) such that On the other hand, we show that if Ω * is an optimal set for (1.1), then the vector U = (u 1 , . . . , u k ) of the first k eigenfunctions on Ω * satisfies the almost-minimality condition of Theorem 1.2, and hence that the eigenfunctions u 1 , . . . , u k are Lipschitz continuous. We first state our Lipschitz regularity result for eigenfunctions on optimal sets for (1.1).
Theorem 1.1. Let D ⊂ R d be a bounded open set and let Λ > 0. Let A : D → Sym + d be a matrix valued function satisfying (1.5) and (1.6) and let b ∈ L ∞ (D) be a function satisfying (1.7) (see below). Then the spectral optimization problem (1.1) admits a solution Ω * . Moreover, the first k eigenfunctions on any optimal set Ω * are locally Lipschitz continuous in D. As a consequence, every optimal set for (1.1) is an open set.
In [3], Briancon, Hayouni and Pierre proved the Lipschitz continuity of the first eigenfunction on an optimal set which minimizes the first eigenvalue of the Dirichlet Laplacian among all sets of prescribed volume included in a box. Their proof, which is inspired by the pioneering work of Alt and Caffarelli in [1] on the regularity for a free boundary problem, relies on the fact that the first eigenfunction is the minimum of a variational problem. For spectral optimization problems involving higher eigenvalues, the study of the regularity of the optimal sets and the corresponding eigenfunctions is more involved due to the variational characterization of the eigenvalue λ k through a min-max procedure. In [4] the authors considered the spectral functionals F (λ 1 (Ω), . . . , λ k (Ω)) which are bi-Lipschitz with respect to each eigenvalue λ i (Ω) of the Dirichlet Laplacian, a typical example being the sum of the first k eigenvalues. In particular, they proved the Lipschitz continuity of the eigenfunctions on optimal sets minimizing the sum λ 1 (Ω) + · · · + λ k (Ω) among all shapes Ω ⊂ R d of prescribed measure (see [4], Thm. 6.1). The present paper extends this result to the case of an operator with variable coefficients, but with a completely different proof.
Concerning spectral optimization problems involving an operator with variable coefficients, a regularity result has been obtained in [15], where the authors consider the problem of minimizing the first eigenvalue of the operator with drift −∆ + ∇Φ · ∇, Φ ∈ W 1,∞ (D, R d ), under inclusion and volume constraints. We stress out that our result also applies to this operator with drift since it corresponds to the special case where A = e −Φ Id and b = e −Φ . We would like also to mention a recent work of Lamboley and Sicbaldi in [11] where they prove an existence and regularity result for Faber-Krahn minimizers in a Riemanninan setting.
Let us highlight that the Lipschitz regularity of the eigenfunctions in Theorem 1.1 turned out to be a quite difficult question due to both the min-max nature of the eigenvalues and the presence of the variable coefficients, but it is an important first step for the analysis of the regularity of the free boundary of the optimal shapes for (1.1) which we study in [17].
As already pointed out, the proof of Theorem 1.1 goes through the study of the Lipschitz regularity of vector-valued almost-minimisers for a two-phase functional with variable coefficients. Our approach is to reduce from the non-constant coefficients case to the constant coefficients-one by a change of variables and is inspired by [16], where the authors prove free boundary regularity of almost-minimizers of the one-phase and twophase functionals in dimension 2 using an epiperimetric inequality. The second contribution which was a strong inspiration for our work is of David and Toro in [6]. They in particular prove the Lipschitz regularity of almost-minimizers of the one-phase and the two-phase functionals with constant coefficients (see also [8] for free boundary regularity results).
We have the following result for almost-minimizers of the two-phase functional. -U satisfies the following quasi-minimality condition: for every C 1 > 0, there exist constants ε ∈ (0, 1) and C > 0 such that D A∇U · ∇U + Λ|{|U | > 0}| ≤ 1 + C U −Ũ L 1 D A∇Ũ · ∇Ũ + Λ|{|Ũ | > 0}|, (1.4) for everyŨ ∈ H 1 0 (D, R k ) such that U −Ũ L 1 ≤ ε and Ũ L ∞ ≤ C 1 . Then the vector-valued function U is locally Lipschitz continuous in D. . The quasi-minimality in Theorem 1.2 is not local but naturally arises from the shape optimization problem (1.1) (see Prop. 3.4). We stress out that our conclusion also holds, with exactly the same proof, if the quasi-minimality property (1.4) is replaced by its "local" version, namely: for every C 1 > 0, there exist constants r 0 ∈ (0, 1) and C > 0 such that for every x ∈ D and every r ≤ r 0 such that B r (x) ⊂ D we have We point out that we will only use the assumption (1.3) to prove that U is bounded and to get an almost-monotonicity formula (see Prop. 2.15 and Cor. 2.16).
In ([6], Thm. 6.1), David and Toro proved an almost-monotonicity formula for quasi-minimizers in the case of the Laplacian. It is natural to expect that the same holds for an operator with variable coefficients, but we will not address this question in the present paper since we are mainly interested in the Lipschitz continuity of the eigenfunctions on optimal shapes for the problem (1.1) for which the equation (1.3) is already known.
However, soon before the present paper was published online, a new preprint of the same authors, in collaboration with Engelstein and Smit Vega Garcia (see [7]), appeared on Arxiv. They prove a regularity result for functions satisfying a suitable quasi-minimality condition for operators with variable coefficients. We stress that the present paper and the work in [7] were done in a completely independent way. We notice that our main result neither directly implies nor is directly implied by the main result from [7].
Notations. Let us start by setting the assumptions on the coefficients of the operator that we will use throughout this paper. The matrix-valued function A = (a ij ) ij : D → Sym + d has Hölder continuous coefficients and is uniformly elliptic, where Sym + d denotes the family of all real positive symmetric d × d matrices. Precisely, there exist positive constants δ A , c A > 0 and λ A ≥ 1 such that (1.5) The function b ∈ L ∞ (D) is positive and bounded away from zero: there exists c b > 0 such that (1.7) We now fix some notations and conventions. For x ∈ R d and r > 0 we use the notation B r (x) to denote the ball centred at x of radius r and we simply write B r if x = 0. We denote by |Ω| the Lebesgue measure of a generic set Ω ⊂ R d and by ω d the Lebesgue measure of the unit ball B 1 ⊂ R d . The (d − 1)-dimensional Hausdorff measure is denoted by H d−1 . Moreover, we define the positive and the negative parts of a function u : R → R by u + = max(u, 0) and For a quasi-open set Ω ∈ R d we denote by H 1 0 (Ω) the Sobolev space defined as the set of functions u ∈ H 1 (R d ) which, up to a set of capacity zero, vanishe outside Ω; that is (see e.g. [10] for a definition of the capacity). Notice that if Ω is an open set, then H 1 0 (Ω) is the usual Sobolev space defined as the closure of the smooth real-valued functions with support compact C ∞ c (Ω) with respect to the norm u H 1 = u L 2 + ∇u L 2 . We denote by H 1 0 (Ω, R k ) the space of vector-valued functions U = (u 1 , . . . , u k ) : Ω → R k such that u i ∈ H 1 0 (Ω) for every i = 1, . . . , k, and endowed with the norm We also define the following norms (whenever it makes sense) Moreover, for U = (u 1 , . . . , u k ) : Ω → R k we set |U | = u 2 1 + · · · + u 2 k , |∇U | 2 = |∇u 1 | 2 + · · · + |∇u k | 2 and A∇U · ∇U = A∇u 1 · ∇u 1 + · · · + A∇u k · ∇u k . For f = (f 1 , . . . , f k ) ∈ L 2 (Ω, R k ) we say that U = (u 1 , . . . , u k ) ∈ H 1 0 (Ω, R k ) is solution to the equation if, for every i = 1, . . . , k, the component u i is solution to the equation where the PDE is intended is the weak sense, that is Moreover, we always extend functions of the spaces H 1 0 (Ω) and H 1 0 (Ω, R k ) by zero outside Ω so that we have the inclusions

Lipschitz continuity of quasi-minimizers
This section is dedicated to the proof of Theorem 1.2. Our approach is to locally freeze the coefficients to reduce to the case where A = Id. More precisely, for every point x ∈ D, an almost-minimizer of the functional with variable coefficients becomes, in a new set of coordinates near x, an almost minimizer for a functional with constant coefficients. We stress out the dealing with the dependence of this change of variables with respect to the point x is not a trivial task. We then adapt the strategy developed by David and Toro in [6] for almost-minimizers of a functional involving the Dirichlet energy.
In this section, u will stand for a coordinate function of the vector U from Theorem 1.2. In Section 2.1 we explicit the change of variables for which u becomes a quasi-minimizer of the Dirichlet energy (in small balls of fixed center). We then prove that u is continuous and we give an estimate of the modulus of continuity from which we deduce that u is locally Hölder continuous in D.
Section 2.2 is addressed to the Lipschitz continuity of u in some region where the function u has a given sign. We show, using in particular the Hölder continuity of u, that most of the estimates proved in Section 2.1 can be improved provided that u keeps the same sign. In this case, we prove that u is Lipschitz continuous and we provide a bound on the Lipschitz constant of u. We also show that u is C 1,β -regular for some β ∈ (0, 1). Next, we show that under some assumption (see the first inequality in (2.45) from Proposition 2.11), if the Dirichlet energy of u in a small ball is big enough, then u keeps the same sign in a smaller ball, which in view of the preceding analysis implies that u is Lipschitz continuous.
In Section 2.3 we complete the proof of the Lipschitz continuity of u. The main missing step is to deal with the case where the Dirichlet energy is big and the first assumption of (2.45) in Proposition 2.11 fails. Using an almost-monotonicity formula for operators with variable coefficients proved by Matevosyan and Petrosyan in ( [13], Thm. III), we show that in this case the value of the Dirichlet energy has to decrease at some smaller scale.
Throughout this section we fix u := u i , for some i = 1, . . . , k, a coordinate function of the vector U = (u 1 , . . . , u k ) from Theorem 1.2. We start by proving that u is a bounded function in D.

Lemma 2.1 (Boundedness).
Let Ω ⊂ D be a (non-empty) quasi-open set, f ∈ L p (D) for some p ∈ (d/2, +∞] and let u ∈ H 1 0 (Ω) be the solution of Then, there is a dimensional constant C d such that Proof. Up to arguing with the positive and the negative parts of f , we can assume that f is a non-negative function. By the maximum principle (see [9], Thm. 8.1) we have u ≥ 0 on Ω. Moreover, u is a minimum of the following functional We consider, for every 0 < t < u L ∞ and ε > 0, the test function u t,ε = u ∧ t + (u − t − ε) + ∈ H 1 0 (Ω). Then, by ellipticity of the matrices A x and the inequality J(u) ≤ J(u t,ε ) we get that The end of the proof now follows precisely as in ( [15], Lem. 5.3).

Continuity and Hölder continuity
We change the coordinates and reduce to the case A = Id using in particular the Hölder continuity of the coefficients of A, and we then prove that u is locally Hölder continuous in D. Let us first introduce few notations that we will use throughout this section. For x ∈ D we define the function F x : Moreover, we set u x = u • F x for every x ∈ D.
Remark 2.2. For M ∈ Sym + d we denote by M 1 /2 the square root matrix of M . We recall that if M ∈ Sym + d , then there is an orthogonal matrix P such that P M P t = diag(λ 1 , . . . , λ d ), where P t is the transpose of P and diag(λ 1 , . . . , λ d ) is the diagonal matrix with eigenvalues λ 1 , . . . , λ d . The matrix M 1 /2 is then defined by Remark 2.3 (Notation of the harmonic extension). One of the main ingredient in the proof of Theorem 1.2 is based on small variations of the function u x . Precisely, we will often compare u x in some ball B r with the harmonic extension of the trace of u x to ∂B r . This function will often be denoted by h x,r , or more simply h r if there is no confusion, and is defined by h r = h x,r ∈ H 1 (B r ) and We notice that h r is a minimizer of the Dirichlet energy in the ball B r , that is We now prove that the function u x is in some sense a quasi-minimizer for the Dirichlet energy in small balls centred at the origin.
Proposition 2.4. There exist constants r 0 ∈ (0, 1) and C > 0 such that, if x ∈ D and r ≤ r 0 satisfy B λ A r (x) ⊂ D, then we have where v stands at the i-th position. Set ρ = λ A r and note that F x (B r ) ⊂ B ρ (x) ⊂ D. Then, usingŨ as a test function and observing that u − v ∈ H 1 0 (F x (B r )), we get where C is the constant from Theorem 1.2. Together with A∇v · ∇v this yields

Fx(Br)
A∇v · ∇v +Cr d , for some constantC. On the other hand, using the Hölder continuity and the ellipticity of A we estimate Br Similarly, we have the following estimate from below A∇v · ∇v.
We now prove that the function u is continuous in D. In the sequel we will often use the following notation: for x ∈ D and r > 0 we set x ∈ D and r ≤ r 0 satisfy B r (x) ⊂ D, then we have |u(y) − u(z)| ≤ C 1 + ω(u, x, r) + log r |y − z| |y − z| for every y, z ∈ B r/2 (x). (2.5) The next Lemma shows that ω(u x , r) cannot grow too fast as r tends to zero and will be useful throughout the proof of the Lipschitz continuity of u. Lemma 2.6. There exist constants r 0 > 0 and C > 0 such that, if x ∈ D and r ≤ r 0 satisfy B λ A r (x) ⊂ D, then we have If, moreover, x is a Lebesgue point for u, then we have Proof. Let t ≤ r and use h t as a test function in (2.1), where h t = h x,t denotes the harmonic extension in B t of the trace of u x to ∂B t , to get where in the last inequality we have used that h t is a minimizer of the Dirichlet energy on B t . Moreover, since Therefore, the triangle inequality, (2.9) and (2.8) give for every s ≤ t ≤ r 0 We then use the estimate (2.10) with the radii r i = 2 −i r, i ≥ 0, and we get This, with an iteration, implies that for every i ≥ 1 we have where we used that the product ∞ j=0 1 + Cr δ A /2 j is bounded by a constant depending on r 0 . The first estimate of the Lemma now follows from (2.11). Indeed, choose i ≥ 0 such that r i+1 < s ≤ r i and note that we have ω(u x , s) ≤ 2 d/2 ω(u x , r i ). If i = 0, this directly implies (2.6); otherwise, i ≥ 1 and use also (2.11).
We now prove the second estimate. For i ≥ 0 we set m i = − Br i u x . By the Poincaré inequality and (2.11) we have (2.12) Furthermore, 0 is a Lebesgue point for u x since x is a Lebesgue point for u and that for every s ≤ r we have In particular, it follows that m i converges to u x (0) = u(x) as i → +∞. Therefore, this with the Cauchy-Schwarz inequality and (2.12) give where in the last inequality we used that +∞ j=i 2 i−j j ≤ C(i + 1). Then, observe that (2.7) is precisely the above inequality with i = 0 to conclude the proof.
Proof of Proposition 2.5. Let y, z ∈ B r/2 (x) and notice that it is enough to prove (2.5) when y and z are Lebesgue points for u. Set δ = |y − z|. We first assume that 4λ 2 A δ ≤ r. Observe that we hence have the inclu- Using a change of variables, the Poincaré inequality and then the ellipticity of A, we estimate On the other hand, since A y ∇u · ∇u 1/2 (2.14) ≤ Cω(u y , 2λ A δ).
We now apply (2.7) to get where we used (2.14) in the last inequality. Therefore, combining the triangle inequality, (2.15), (2.13) and (2. 16) we get that We notice that we can assume the y i to be Lebesgue points for u. Moreover, observe that we can bound the number of points by n ≤ 16λ 4 A + 2. Therefore, applying the estimate (2.20) to each pair (y i , y i+1 ) we have which concludes the proof.
We are now in position to prove the Hölder continuity of u.
Proposition 2.7. The function u is locally α-Hölder continuous in D for every α ∈ (0, 1), that is, for every compact set K ⊂ D, there exist r K > 0 and C K > 0 such that for every x ∈ K we have Proof. Let x ∈ K and set 4r K = r 1 = min{r 0 , dist(K, D c )} where r 0 is given by Proposition 2.5. Since the function r → r 1−α log(r 1 /r) is non-decreasing on (0, c α ) for some constant c α > 0 depending on α and r 1 , it follows from Proposition 2.5 that, if y, z ∈ B r1/2 (x) are such that |y − z| ≤ c α , we have If now |y − z| > c α , then choose n points y 1 = y, . . . , y n = z in B r1/2 (x) such that |y i − y i+1 | = c α r −1 1 |y − z|, with n bounded by some constant depending on α and r 1 . Then apply (2.22) to each pair (y i , y i+1 ) to prove that u is α-Hölder continuous in the ball B r1/2 (x) with a modulus of continuity depending on ω(u, x, r 1 ). Now, (2.21) follows by a compactness argument with the constant C K depending on max{ω(u,

Bound of the Lipschitz constant in {u > 0}
We prove that u is Lipschitz continuous and even C 1,β -regular in the regions where u keeps the same sign. We also provide in this case an estimate of the Lipschitz constant of u in terms of ω(u, x, r) (see Prop. 2.8). Then, we show that under suitable conditions, u keeps the same sign and is therefore Lipschitz continuous (see Prop. 2.11).
Proposition 2.8. Let K ⊂ D be a compact set. There exist constants r K > 0 and C K > 0 such that, if x ∈ K and r ≤ r K satisfy then u is Lipschitz continuous in B r/2 (x) and we have In the next Lemma we compare the Dirichlet energy of u x and of its harmonic extension in small balls where u x has a given sign. The estimate (2.26) in Lemma 2.9 below is similar to (2.1) but with a smaller error term. Thanks to this improvement, the strategy developed in the proof of Lemma 2.6 will lead to a sharper result than estimate (2.6), namely (2.24). Lemma 2.9. Let K ⊂ D be a compact set and let α ∈ (0, 1). There exist constants r K > 0 and C > 0 such that, if x ∈ K and r ≤ r K are such that (2.23) holds, then the function u

26)
where h r stands for the harmonic extension of the trace of u x to ∂B r .
Proof. Set ρ := λ A r for some r > 0 small enough so that ). SetŨ = (u 1 , . . . , v, . . . , u k ) ∈ H 1 0 (D, R k ) and observe that |{|Ũ | > 0}| = |{|U | > 0}| by (2.23) and because v > 0 in F x (B r ). Then, we useŨ as a test function in (1.3) to get where C is the constant from Theorem 1.2. Now, since u is locally α-Hölder continuous, we have the bound Altogether this gives A∇v · ∇v +Cr d+α , for some constantC which involves D A∇U · ∇U . Finally, using the Hölder continuity and the ellipticity of A as in the proof of Proposition 2.4, we get which gives (2.26).
Next Lemma is analogue to Lemma 2.6 with a better estimate of the error term. Its proof is quite similar but we nonetheless sketch the argument since there are small differences. Lemma 2.10. Let K ⊂ D be a compact set and α ∈ (0, 1). There exist constants r K > 0 and C > 0 such that, for every x ∈ K and every r ≤ r K such that (2.23) holds, we have since h t is a minimizer of the Dirichlet energy on B t . Now, for s ≤ t ≤ r 0 we use (2.9) and (2.29) to estimate as in (2.10) which, applied to s = 2 −i r and t = 2 −(i−1) r, gives where we have set r i = 2 −i r. Iterating the above estimate we get for every i ≥ 1 is bounded by a constant depending on r K . This proves (2.27). Finally, (2.28) is proved in the same way than (2.7) but with (2.12) replaced by the estimate Proof of Proposition 2.8. Let us first prove (2.24). We follow the proof of Proposition 2.5 and we only detail the few differences. Let y, z ∈ B r/2 (x) be Lebesgue points for u and set δ = |y − z|. We first assume that 4λ 2 A δ ≤ r. By (2.28) we have and, using also (2.14), Finally, if 4λ 2 A δ > r, we argue as in the proof of Proposition 2.5 and choose a few number of points which connect y and z to prove (2.33).
The strategy to prove Theorem 1.2 is to show that ω(u x , r) cannot become too big as r gets small. In the next Proposition we prove, under some condition (see the first inequality in (2.45) below), that if ω(u x , r) is big enough then u keeps the same sign near the point x and is hence Lipschitz continuous by Proposition 2.8. The case where ω(u x , r) is big and this condition fails is treated in the next subsection. We set for x ∈ D and r > 0 Proposition 2.11. Let K ⊂ D be a compact set and let γ > 0. There exists constants r K , C K > 0 and κ 1 > 0 such that, if x ∈ K and r ≤ r K satisfy

45)
then there exists a constant c > 0 (independent from x and r) such that u is Lipschitz continuous in B cr/2 (x) and we have |u(y) − u(z)| ≤ C K (1 + ω(u, x, r))|y − z| for every y, z ∈ B cr/2 (x). (2.46) Roughly speaking, the condition (2.45) says that the absolute value of the trace of u x to ∂B r is big. This will in fact ensure that u x has, in some smaller ball, the same sign than (the average of) u x on ∂B r . Lemma 2.12. Let γ and τ be two positive constants. There exist r 0 , η ∈ (0, 1) and
Proof. We first prove the second inequality in (2.49). Let us recall that h r = h x,r denotes the harmonic extension of the trace of u x to ∂B r . We want to estimate both − ∂Bρ |h r | and − ∂Bρ |u x − h r | in terms of |b(u x , r)| for some ρ ∈ ( ηr 2 , ηr) defined soon (by (2.52)). If η ≤ 1/2, then by subharmonicity of |∇h r | in B r we have that for every ξ ∈ B ηr Moreover, b(u x , r) = b(h r , r) = h r (0) since h r is harmonic and hence, choosing η such that η2 d/2 ≤ γ/4, we get In view of the two hypothesis in (2.48) we then get where the last inequality holds if we choose r 0 small enough and κ 1 > 0 large enough (both depending on η) such that Now, using (2.51) and (2.53) we have and, using also that h r keeps the same sign on ∂B ρ by (2.50), we have This proves the second inequality in (2.48). Moreover, (2.53) and (2.50) imply that which shows that b(u x , r) and b(u x , ρ) have the same sign.
For the first estimate in (2.49), by (2.55) and the first hypothesis in (2.48), we have which using (2.6) gives (notice that we assumed that B λ A r (x) ⊂ D) Finally, observe that with η small enough (and also r 0 small enough and κ 1 large enough so that (2.54) still holds) we have This completes the proof.
We continue with a self-improvement lemma whose strategy is similar to the one followed in the previous lemma, the main difference being that we now consider u x with different points x. Lemma 2.13. There exist constants r 0 ∈ (0, 1) and τ 0 ≥ 1 with the following property: if x ∈ D, τ ≥ τ 0 and ρ ≤ r 0 satisfy B λ A ρ (x) ⊂ D,
Proof. Firstly, if ε is small enough so thatε := 2λ 2 A ε ≤ 1/4, then by standard estimates on harmonic functions (see [9,Thm. 3.9] Using that b(u x , ρ) = b(h ρ , ρ) = h ρ (0) by harmonicity and the second hypothesis in (2.56), it follows that for every ξ ∈ Bε ρ we have where the last inequality holds if τ 0 is big enough. This implies that Moreover, by (2.1) applied to h ρ (and sinceερ ≤ ρ for τ 0 large enough) we have We now fix some y ∈ B ερ (x). Let F :

Then the coarea formula gives (and because ∂F
We now choose ρ 1 ∈ ( ερ 2 , ερ) such that so that (2.60), (2.61) and the first hypothesis in (2.56) imply where the last inequality holds for τ 0 is large enough. Moreover, because the functions F and F −1 are Lipschitz continuous with Lipschitz constants bounded by λ 2 A , we have for every set E ⊂ R d (see [12], Prop. 3.5) On the other hand, we have by (2.62) Moreover, by (2.58) and since ∂F (B ρ1 ) ⊂ Bε ρ we have Therefore, using the triangle inequality, (2.65) and (2.66) we get This proves that b(u x , ρ) and b(u y , ρ 1 ) have the same sign and also implies that Finally, (2.64) and (2.67) gives which is the second inequality in (2.57).
We now prove the first inequality in (2.57). By (2.67) and the first hypothesis in (2.56) we have

68)
We then apply Lemma 2.6 (notice that we have B λ A ρ (x) ⊂ D and 2λ 2 A ρ 1 ≤ ρ) and eventually choose τ 0 bigger (depending only on d and δ A ) to get Therefore, with (2.68) this gives which, choosing τ 0 big enough so that τ 1/d 0 ≥ 4C, completes the proof.
We are now in position to prove Proposition 2.11 using the results from Lemmas 2.12, 2.13 and Proposition 2.8.
Let y ∈ Br(x) be fixed. We first apply Lemma 2.12. Now, we apply once Lemma 2.13 at x (notice that we have y ∈ B ερ (x)) and then iteratively at the point y. It follows that there exists a sequence of raddi ρ i > 0 such that and that b(u y , ρ i ) has the same sign than b(u x , r) for every i ≥ 0. Assume that b(u x , r) > 0, the proof in the case b(u x , r) < 0 is identical. Let us denote by h i = h y,ρi the harmonic extension of the trace of the function u y to ∂B ρi . With the same argument as in (2.58) we get Since b(u y , ρ i ) > 0, this implies that for every ξ ∈ B ερi ∩ {u y ≤ 0} we have By the Chebyshev inequality, the Lebesgue measure of B ερi ∩ {u y ≤ 0} is estimate as On the other hand, by (2.60) in this context we have . This shows that the density of the set {u ≤ 0} at every point y ∈ Br(x) is 0 (see [12], Ex. 5.19),, and hence that u > 0 almost-everywhere in Br(x). Now, we set c = λ −1 A τ −1/d 0 η/2, where η and τ 0 are the constants given by Lemma 2.12 and 2.13. Then (2.46) and (2.47) follow from Proposition 2.8 and the fact that ω(u, x, cr) ≤ c −d/2 ω(u, x, r). This concludes the proof.

Lipschitz continuity
In this subsection we prove Theorem 1.2. At this stage, the main work is to deal with the case where ω(u x , r) is big and the first condition in (2.45) fails. In this case, we show in Proposition 2.14 below that the value of ω(u x , r) decreases at some smaller scale. Notice that the extra hypothesis (2.72) is almost irrelevant in view of Proposition 2.8.
Proposition 2.14. Let K ⊂ D be a compact set. There exist positive constants r K , γ ∈ (0, 1) and κ 2 > 0 with the following property: if x ∈ K and r ≤ r K satisfy u x (ξ 0 ) = 0 for some ξ 0 ∈ B (2λ A ) −2 r , (2.72) |b(u x , r)| ≤ γr(1 + ω(u x , r)) and κ 2 ≤ ω(u x , r), (2.73) We will need the following almost-monotonicity formula for operators in divergence form. We refer to ( [13], Thm. III) for a proof (see also [2] and [5] for the case of the Laplacian). Let us set for u + , u − ∈ H 1 (B 1 ) and r ∈ (0, 1) Proposition 2.15. Let B = (b ij ) ij : B 1 → Sym + d be a uniformly elliptic matrix-valued function with Hölder continuous coefficients, that is, for every x, y ∈ B 1 and ξ ∈ R d Let u + , u − be two non-negative and continuous functions in the unit ball B 1 such that div(B∇u ± ) ≥ −1 in B 1 and Then there exist r 0 > 0 and C > 0, depending only on d, c b , δ b and λ b , such that for every r ≤ r 0 we have We now state this almost-monotonicity formula for the functions u ± x . Corollary 2.16. Let Ω ⊂ D be a quasi-open set and K ⊂ D be a compact set. Let A be a matrix-valued function satisfying (1.5) and (1.6). Let f ∈ L ∞ (D). Assume that u ∈ H 1 0 (Ω) is a continuous function solution of the equation − div(A∇u) = f in Ω. (2.75) Then there exists r K > 0 and C m > 0, depending only on d, c A , δ A , λ A , f L ∞ , |D| and dist(K, D c ), such that for every x ∈ K and every r ≤ r K the function u x satisfies Proof. We first prove that we have, in the sense of distributions, Now, the inequality for u + in (2.76) follows by letting n tend +∞, because p n (u) converges almost-everywhere to 1 {u>0} and ∇q n (u) converges in L 2 to ∇u + . The same proof holds for u − .
Then the functions u ± satisfy div(B∇u ± ) ≥ f −1 Therefore, by Proposition 2.15 we have for every r ≤ r K := r 0 ρ Proof of Proposition 2.14. Let us denote as before h r = h x,r the harmonic extension of the trace of u x to ∂B r . Then we have By the quasi-minimality property of u x we can estimate the second term in the right hand side of (2.77) as we did in (2.8), this gives where in the last inequality we have used the second hypothesis in (2.73). On the other hand, estimates for harmonic functions give We now want to estimate b + (u x , r) in terms of r ω(u x , r). Let us assume that ω(u + x , r) ≤ ω(u − x , r), the same proof holds if the opposite inequality is satisfied. We first prove that for ξ 0 ∈ B r/2 and η < 1/2 we have Notice that up to considering the function ξ → u + x (rξ) we can assume that r = 1. Let us define a one to one function F : We set v = u + x • F . For every ξ ∈ ∂B 1 we have by the fundamental theorem of the calculus Note that F is the identity on ∂B 1 and is simply a translation on ∂B η . Therefore, averaging on ξ ∈ ∂B 1 (and which proves (2.80). Now, let ξ 0 ∈ B (2λ A ) −2 r be such that u x (ξ 0 ) = 0 as in (2.72). By Proposition 2.5 we have for every ξ ∈ B ηr (ξ 0 ) (and because F where the last inequality holds for η small enough and since we have Moreover, recall that we assumed that ω(u + x , r) ≤ ω(u − x , r). Using the monotonicity formula in Corollary 2.16 we get which implies by Cauchy-Schwarz's inequality Therefore, combining (2.81), (2.80), (2.82) and using the first hypothesis in (2.73) we have (and also since where in the last inequality we used the second hypothesis in (2.73). We now return to (2.77 Therefore, choosing first η, γ and r K small enough and then κ 2 big enough (depending on η) we obtain (2.74), which concludes the proof.
We are now in position to prove Theorem 1.2 using an iterative argument and Propositions 2.8, 2.11, 2.14.
Proof of Theorem 1.2. Recall that we denote by u any coordinate function of the vector U and that we have to prove that u is locally Lipschitz continuous in D. Let K ⊂ D be a compact set and let x ∈ K. Let r ≤ r K , where r K is smaller than the constants given by Propositions 2.8, 2.11 and 2.14. Set κ = max{κ 1 , κ 2 } where κ 1 , κ 2 are the constants given by Propositions 2.11 and 2.14. We consider the following four cases: Case 1: Case 2:

85)
Case 3: |b(u x , r)| ≤ γr(1 + ω(u x , r)) and κ ≤ ω(u x , r), (2.86) For k ≥ 0 we set r k = 3 −k r. We denote by k 0 , if it exists, the smallest integer k ≥ 0 such that the pair (x, r k ) satisfies either (2.84) or (2.85), and we set k 0 = +∞ otherwise. If k 0 > 0, then for every k < k 0 we have that: if (x, r k ) satisfies (2.86) then by Proposition 2.14 we have (notice that (2.72) holds since u is continuous and that (2.84) is not satisfied) while if (x, r k ) satisfies (2.87), then we have Therefore, with an induction we get that for every 0 ≤ k ≤ k 0 Assume that k 0 = +∞. If x is a Lebesgue point for ∇u, then 0 is a Lebesgue point of u x and it follows from (2.88) that Assume now that k 0 < +∞. Then, by definition of k 0 , the pair (x, r k0 ) satisfies either (2.84) or (2.85). If (2.84) holds, then Proposition 2.8 infers that u is C 1,β near x and that we have (using also (2.88)) Moreover, by Proposition 2.11 the same estimate holds if the pair (x, r k0 ) satisfies (2.85). Therefore, in all cases it follows that for almost every point x ∈ K and every r ≤ r K we have |∇u(x)| ≤ C K (1 + ω(u x , r)). (2.89) Let now x 0 ∈ K. Then, for almost every x ∈ B r K /2 (x 0 ), it follows by (2.89) that With a compactness argument this proves that u is locally Lipschitz continuous in D and completes the proof.

Lipschitz continuity of the eigenfunctions
This section is dedicated to the proof of Theorem 1.1. Precisely, we prove that the vector U = (u 1 , . . . , u k ) of the first k eigenfunctions on an optimal set for (1.1) is locally Lipschitz continuous in D. Using an idea taken from [14], we show that U is a quasi-minimizer in the sense of (1.4), and we then apply Theorem 1.2 to get the Lipschitz continuity of U .

Preliminaries and existence of an optimal set
We start with some properties about the spectrum of the operator in divergence form defined in 1. Moreover, if Ω = R d we will simply write u L 2 (m) = u L 2 (R d ;m) . We notice that, by the hypothesis (1.7) on the function b, the norms · L 2 (Ω;m) and · L 2 (Ω) are equivalent. We stress out that the choice of these norms is natural in view of (1.2) and is motivated by the variational formulation of the sum of the first k eigenfunctions (see (3.1) below). Now, the Lax-Milgram theorem and the Poincaré inequality imply that for every f ∈ L 2 (Ω, m) there exists a unique solution u ∈ H 1 0 (Ω, m) to the problem The resolvent operator R Ω : f ∈ L 2 (Ω; m) → H 1 0 (Ω; m) ⊂ L 2 (Ω; m) defined as R Ω (f ) = u is a continuous, self-adjoint and positive operator. Since the Sobolev space H 1 0 (Ω; m) is compactly embedded into L 2 (Ω; m) (because we have assumed that b ≥ c b > 0, see (1.7)), the resolvent R Ω is in addition a compact operator. We say that a complex number λ is an eigenvalue of the operator (1.2) in Ω if there exists a non-trivial eigenfunction u ∈ H 1 0 (Ω; m) solution of the equation − div(A∇u) = λu b in Ω, u ∈ H 1 0 (Ω; m).
The above properties of the resolvent ensure that the spectrum of the operator (1.2) in Ω is given by an increasing sequence of eigenvalues which are strictly positive real numbers, non-necessarily distinct, and which we denote by The eigenvalues λ k (Ω) are variationnaly characterized by the following min-max formula Moreover, we denote by u k the normalized (with respect to the norm · L 2 (Ω;m) ) eigenfunctions corresponding to the eigenvalues λ k (Ω) and note that the family (u k ) k form an orthonormal system in L 2 (Ω; m), that is As a consequence, we have the following variational formulation for the sum of the first k eigenvalues on a quasi-open set Ω for which the minimum is attained for the vector U = (u 1 , . . . , u k ). We now deduce from this characterization that the minimum in (1.1) is reached. Proof. Let (Ω n ) n∈N be a minimizing sequence of quasi-open sets to the problem (1.1) and denote by U n = (u n 1 , . . . , u n k ) the first k eigenfunctions on Ω n . Since the matrices A x , x ∈ D, are uniformly elliptic, we have the following inequality which infers that the norm U n H 1 is uniformly bounded. Therefore, up to a subsequence, U n converges weakly in H 1 (D, R k ) and strongly in L 2 (D, R k ) to some V ∈ H 1 (D, R k ). Notice that V is an orthonormal vector. Set Ω * := {|V | = 0}. Then using (3.1), the weak convergence in H 1 of U n to V and the semi-continuity of the Lebesgue measure we have which concludes the proof.
In the next Lemma we prove that the eigenfunctions are bounded. This result is a consequence of Lemma 2.1 and we refer to ( [15], Lem. 5.4) for a proof which is based on an interpolation argument.
In particular, if u is an eigenfunction on Ω normalized by u L 2 (m) = 1, then u ∈ L ∞ (Ω) and where λ(Ω) denotes the eigenvalue corresponding to u.
To conclude this subsection, we show that the first eigenfunction on an optimal set Ω * keeps the same sign on every connected component of Ω * . Notice that Ω * may not be connected and has at most k connected components. Then u is non-negative in Ω (up to a change of sign).
Proof. We assume that u + = 0 (if not, take −u instead of u) and we set and Since u is variationally characterized by λ 1 (Ω) = Ω A∇u · ∇u dx = min Ω A∇ũ · ∇ũ dx :ũ ∈ H 1 0 (Ω), we have Ω A∇u · ∇u dx ≤ Ω A∇u + · ∇u + dx and Then, it follows that the two above inequalities are in fact equalities since otherwise we have which is absurd. In view of the minimization characterization (3.2), this ensures that u + is solution of the equation Then, the strong maximum principle (see [9], Thm. 8.19) and the connectedness of Ω imply that u + is strictly positive in Ω, which completes the proof.

Quasi-minimality and Lipschitz continuity of the eigenfunctions
We prove that the vector U = (u 1 , . . . , u k ) of normalized eigenfunctions on an optimal set Ω * for the problem (1.1) is a local quasi-minimizer of the vector-valued functional in the sense of the Proposition below. The Lipschitz continuity of the eigenfunctions is then a consequence of Theorem 1.2. We notice that, in view of the variational formulation (3.1), the vector U is solution to the following problem Proposition 3.4 (Quasi-minimality of U ).
The next Lemma, in which we get rid of the orthogonality constraint in (3.3), is similar to Lemma 2.5 in [14] with only slight modifications, but we decided to recall the whole proof for a sake of completeness.
We are now in position to prove the Lemma by induction. For k = 1, we ask that ε 1 ≤ (4δ) −1 , so that we have D A∇v 1 · ∇v 1 dx ≤ ũ 1 Suppose now that the Lemma holds for 1, . . . , k − 1. Thanks to the first estimate in (3.6) of the preceding claim we have where the last inequality holds if ε k ≤ (4δC k ) −1 . On the other hand, for every i = 1, . . . , k − 1, we have by the inductive assumption Therefore, using the estimate (3.8) we get D A∇w k · ∇w k dx We then ask that ε k ≤ (2C k ) −1 (1 + C k−1 ) −1/2 so that we get D A∇v k · ∇v k dx = w k This, using once again the inductive hypothesis, proves (3.5) and concludes the proof.
Proof of Proposition 3.4. LetŨ be a vector satisfying the hypothesis of Proposition 3.4 and let V ∈ H 1 0 (D, R k ) be the vector given by Lemma 3.5 and obtained by orthonormalizingŨ . Using V as a test function in (3.3) and then Lemma 3.5, we have D A∇U · ∇U dx + Λ|{|U | > 0}| ≤ D A∇V · ∇V dx + Λ|{|V | > 0}| where we have used in the last inequality that {|V | > 0} ⊂ {|Ũ | > 0} (which holds by construction of V ).
We now conclude the proof of Theorem 1.1.
Proof of Theorem 1.1. The proof follows from Proposition 3.4 and Theorem 1.2 (see also Lem. 3.2).