Faber-Krahn and Lieb-type inequalities for the composite membrane problem, Analysis,

. The classical Faber-Krahn inequality states that, among all domains with given measure, the ball has the smallest ﬁrst Dirichlet eigenvalue of the Laplacian. Another inequality related to the ﬁrst eigenvalue of the Laplacian has been proved by Lieb in 1983 and it relates the ﬁrst Dirichlet eigenvalues of the Laplacian of two diﬀerent domains with the ﬁrst Dirichlet eigenvalue of the intersection of translations of them. In this paper we prove the analogue of Faber-Krahn and Lieb inequalities for the composite membrane problem.


Introduction
The composite membrane problem is an eigenvalue optimization problem that received a considerable attention starting from the works of Chanillo and al. [3][4][5][6]16]. In physical terms the problem can be stated as follows: build a membrane of prescribed shape and mass using materials of varying densities, in such a way that the basic frequency is the smallest possible. As shown in [3] and [4] the composite membrane problem can be considered as a special instance of a more general eigenvalue optimization problem, which we are going to introduce in R n , n ≥ 2, keeping in mind that the physically relevant case is n = 2.
Let Ω ⊂ R n be a non-empty open, bounded and connected set with Lipschitz boundary ∂Ω. Any minimizer D in (1.2) is called an optimal configuration for the data (Ω, α, A). If moreover u satisfies (1.1) then (u, D) is called an optimal pair. Due to the variational characterization of (1.2), changing D by a set of measure zero does not affect λ Ω (α, D) nor u. Therefore we consider sets D that differs by a null-set as equal.
In this context, uniqueness of optimal pairs is a delicate issue and cannot concern the function u. Indeed, if (u, D) is an optimal pair, then for every constant c = 0, also (cu, D) is an optimal pair of the same problem. Therefore, it is interesting to look only for uniqueness of optimal configurations D. Nevertheless, this cannot be expected in general, as shown in [3,Theorem 7], but on balls there is a unique optimal configuration, see [3,Corollary 5]. Moreover, if u solves (1.1) then, denoted by u ⋆ (u ⋆ ) its Schwarz decreasing (increasing) symmetrization, by Ω ⋆ the ball centered at the origin and volume |Ω ⋆ | = |Ω| and by D ⋆ the set defined through its characteristic function, is an optimal pair of (1.2) on Ω ⋆ , see the proof of [3,Theorem 4]. We stress that quite simple computations show that the set D ⋆ is an annuli containing the boundary, see [3,Corollary 5] and Remark 2.7.
In this note we address a similar problem for the problem (1.2) with suitable constraints on the parameter α > 0. To state our result, we need to define the constant α Ω (A). Given Ω and A ∈ [0, |Ω|), there exists a unique positive number, denoted by α Ω (A), such that see Section 2. Notice that in the interval (0, α Ω (A)), where α takes its values, problem (1.2) and the composite membrane problem are in one-to-one correspondence. We refer to Section 2 for more details.
To prove Theorem 1.1 we adapt to our setting the proof of the classical Faber-Krahn inequality due to Kesavan [12], which relies on a well known result by Talenti, see Theorem 2.3 in Section 2. We stress that, in our situation, we need to pay attention to the presence of the infimum among all the sets D ⊂ Ω of given measure |D| = A. The crucial ingredient of the proof is Proposition 3.3, which provides an explicit expression of the Schwarz symmetrization of a function on which we apply the above mentioned theorem of Talenti. Recently, in [7,8], some of the results proved for the composite problem in [3] have been extended to the fourth-order case of the composite plate problem. It would be interesting to address Faber-Krahn-type problems in that context as well.
Our second result is a Lieb-type inequality for the intersection of two composite membranes, see Theorem 1.2. To state it precisely, we have to introduce the following notation: for every non-empty set Ω ⊂ R n and for every x ∈ R n , we denote the translated of Ω by x as Ω x := Ω + x = {y ∈ R n : y = z + x, z ∈ Ω}.
Our result in this context reads as follows: Let Ω 1 , Ω 2 ⊂ R n be two open connected and bounded sets in R n with Lipschitz boundary. Fixed A i ∈ [0, |Ω i |] and α i > 0, i ∈ {1, 2}, let D 1 and D 2 be optimal configurations for (Ω 1 , α 1 , A 1 ) and (Ω 2 , α 2 , A 2 ), respectively. Then there exists a set Σ ⊆ R n , |Σ| > 0, such that for a.e. x ∈ Σ it is |Ω 1 ∩ Ω 2,x | > 0 and, for every α ∈ (0, α 1 + α 2 ] and for every A ∈ [0, |D 1 ∩ D 2,x |], we have We point out that both in Theorem 1.1 and in Theorem 1.2 we required the boundaries of Ω, Ω 1 and Ω 2 to be Lipschitz continuous. Even if this assumption does not explicitly play a role along the proofs of our results, we must ask for it because it guarantees the existence of optimal pairs, see [3,Theorem 1]. Notice that this regularity on the sets is not necessary neither for the Faber-Krahn inequality nor the Lieb inequality for the Laplacian −∆.
We finally remark that sort of reversed Faber-Krahn and Lieb inequality have been recently proved for the first eigenvalue of a degenerate operator called truncated Laplacian, see [2].
The paper is organized as follows: in Section 2 we recall basic facts on Steiner/Schwarz rearrangements and the composite membrane problem studied in [3]. In Section 3 we prove a few technical results that are needed in Section 4, where we prove Theorem 1.1. Finally, in Section 5 we prove Theorem 1.2.

Preliminaries
The first part of this section is devoted to a brief summary of the definitions and the basic properties of Steiner and Schwarz rearrangements needed for the proof of Theorem 1.1. We refer to the monographs [11,13] and the references therein for a more comprehensive introduction to the subject. The second part of this section contains part of the results proved in [3] on the composite membrane problem.
Let Ω ⊂ R n be a measurable set with finite n-dimensional Lebesgue measure |Ω| < +∞. We denote by Ω ⋆ the open ball centered at the origin and measure |Ω ⋆ | = |Ω|. We also denote by ω n the measure of the unit ball.
Let u : Ω → R be a measurable function. The distribution function µ u : To simplify the notation, in the following we will write {u > τ } in place of {x ∈ Ω : u(x) > τ }, and similarly for (sub-)level sets.
The decreasing Steiner rearrangement u ♯ : [0, |Ω|] → R of u is defined as (2. 2) The increasing Steiner rearrangement u ♯ : [0, |Ω|] → R of u is defined as The decreasing Schwarz symmetrization u ⋆ : Ω ⋆ → R of u is defined as and the increasing Schwarz symmetrization u ⋆ : Ω ⋆ → R of u is defined as It follows from the previous definitions that the increasing and decreasing Steiner rearrangements are related as follows: Due to its importance in the proof of Theorem 1.1, we recall here the following result due to Talenti, see [17]: Let Ω ⊂ R n be an open set and let Ω ⋆ be the ball centered at the origin and measure |Ω ⋆ | = |Ω|. Let f ∈ L 2 (Ω) and let f ⋆ be its Schwarz symmetrization.
on ∂Ω, We end this section with a brief recap on the optimization eigenvalue problem (1.2). We refer to [3] for proofs and more results. The next theorem condensates parts of the content of [3, Theorem 1, Theorem 2, Proposition 10].

Technical Lemmas
In this section we prove several technical results needed to prove Theorem 1.1. Let Ω ⊂ R n be an open and bounded connected set with Lipschitz boundary. Let α ∈ (0, α Ω (A)), and let (u, D) be an optimal pair which realizes the double infimum in (1.2). To simplify the notation, in this section we will denote Λ Ω (α, A) by Λ.
We recall that by Theorem 2.5 (ii) there exists t > 0 such that This implies µ u (t) = |Ω \ D|.
Case III: τ < (Λ − α)t. We have By (3.6), (3.8) and (3.1) (3.10) Let us now consider the first set at the right hand side of (3.9). By (3.7) On the other hand, from the characterization of D, see (3.1), it follows that Thus, we get Combining (3.9), (3.10) and (3.11), we get the desired result.  (3.14) where in the last equality we used the change of variable σ = τ Λ .
Proof. We have that On the other hand, where U ♯ has been explicitly determined in Proposition 3.2. Therefore, and this closes the proof.

Proof of Theorem 1.1
In this section we prove Theorem 1.1. To simplify the notation and since α and A are fixed, we will write Λ Ω in place of Λ Ω (α, A).
Proof of Theorem 1.1. The proof consists of two steps. The first one is to show that for every Ω ⊂ R n such that |Ω| = |Ω ⋆ |. To this aim, let (u, D) be an optimal pair which realizes Λ Ω . This means that Let D ⋆ be defined through its characteristic function in (1.3). Since the decreasing Schwarz symmetrization u ⋆ ∈ H 1 0 (Ω ⋆ ), and, by definition, D ⋆ ⊂ Ω ⋆ with |D ⋆ | = |D| = A, we have that (u ⋆ , D ⋆ ) is an admissible pair for (1.2) on Ω ⋆ . Moreover, since u > 0 then u ⋆ > 0 and hence we get that Therefore, we have where in the second inequality we used (4.2), Proposition 2.2 and Pólya-Szegö. This shows that (4.1) holds true. The second step is to prove that the ball Ω ⋆ is the unique minimizer. In other words, we have to prove that if Λ Ω = Λ Ω ⋆ , then Ω = Ω ⋆ .
Since we assume Λ Ω = Λ Ω ⋆ , we can simply write Λ omitting the dependance on the set. Let us consider an optimal pair (u, D) of the double minimization problem (1.2) on Ω. Clearly, they satisfy the second order Euler-Lagrange equation associated to (1.2), i.e.
Let us now consider the function v ∈ H 1 0 (Ω ⋆ ) which solves the following auxiliary boundary value problem, A direct application of Theorem 2.3 implies that Now, by Proposition 3.3, we have that v actually solves Therefore, by (2.5) and (4.3), we have for almost every x ∈ Ω ⋆ . Therefore, multiplying by v the former inequality and integrating by parts,ˆΩ for v ∈ H 1 0 (Ω ⋆ ) and D ⋆ ⊂ Ω ⋆ with |D ⋆ | = A. Since (v, D ⋆ ) is an admissible pair, then in (4.5) the equality holds and (v, D ⋆ ) must be an optimal pair. Therefore v solves By (4.4) and (4.6), subtracting term by term, and keeping in mind (2.6) we get The assumption α < α Ω implies Λ − αχ D⋆ > 0, see (2.5). Therefore the equality above, together with (4.3), implies v = u * a.e. in Ω ⋆ .
By Remark 2.4, this is enough to conclude that Ω = Ω ⋆ . This closes the proof.
Remark 4.1. We point out that the validity of (4.1) is essentially already contained in [17,Theorem 3] and in the proof of [3,Theorem 4]. We also want to stress that the assumption α < α Ω ⋆ (A) is needed only in the second step of the proof, because we exploit that Λ−α > 0.

Proof of Theorem 1.2
In this section we will prove Theorem 1.2, adapting to our setting the Lieb's proof in [15] of the similar inequality for the lowest eigenvalue of −∆.
Proof of Theorem 1.2. We know from [3] that for every i = 1, 2, Λ Ω i (α i , A) is actually achieved by (at least) one optimal pair (v i , D i ), with v i ∈ H 1 0 (Ω i ) and |D i | = A i . The functions v i are uniquely determined, up to a scalar multiple, by D i , and may be chosen to be positive in Ω i , see [3]. Without loss of generality, we can assume that v i L 2 (Ω i ) = 1.
As in [15], from now on and with a slight abuse of notation, we assume the functions v i 's to be defined on the whole of R n , with For a.e. x ∈ R n we define h x : R n → R, Notice that h x = 0 a.e. in R n \ (Ω 1 ∩ Ω 2,x ), see (1.5), and h x ∈ L 1 . Since v 1 , v 2 , ∇ y v 1 and ∇ y v 2 are L 2 functions, then w x (y) : . Moreover, ∇ y h x = w x in the sense of distributions. Even more, it holds h x ∈ H 1 (R n ) and ∇ y h x = w x ∈ L 2 , see [15, p. 445].