Minimization of the buckling load of a clamped plate with perimeter constraint

We look for minimizers of the buckling load problem with perimeter constraint in any dimension. In dimension 2, we show that the minimizing plates are convex; in higher dimension, by passing through a weaker formulation of the problem, we show that any optimal set is open and connected. For higher eigenvalues, we prove that minimizers exist among convex sets with prescribed perimeter.

In this paper we mainly focus on the first eigenvalue, i.e.
Λ 1 (Ω) = min The minimum above is achieved only on the solutions of Problem (1.1).
Our aim is to show an existence result for a shape optimization problem involving Λ 1 (Ω), whose formulation is appropriated in view of the physical interpretation of the PDE.Indeed, in a 2-dimensional setting, Ω can be thought as a thin elastic plate that is clamped along its boundary ∂Ω and subject to compressive forces (the so called "buckling forces") across ∂Ω; these forces lie on the same plane as Ω, that may deflect out of its plane when these forces reach a certain magnitude.Λ 1 (Ω) is said the "buckling load" of Ω and can be interpreted as the energy associated to the plate Ω in this phenomenon.There are some works in literature treating this problem by constraining the volume of the admissible sets and there is a few information about the minimizers.The first result that uses variational methods is [3], where the authors prove that the problem in R 2 admits a quasi open minimizer without prescribing any bounded design region; moreover, supposing that a minimizer Ω is a sufficiently smooth domain with sufficiently smooth eigenfunctions, the authors show that Ω has to be a disk of maximal area.Several years later, in [15] it is proved an existence result in dimension 2 and 3 with a different technique based on the eigenfunctions, but introducing a (big) bounded design region to assure extra compactness.Recently, [16] provided an existence result for minimizers in the whole of R d based on a mixed strategy: a concentration-compactness argument inspired by [3] to get a limit function u and a regularity argument for u to build the optimal open set using the superlevel sets of continuous functions; moreover, the author shows that the open minimizers are also connected, but nothing is proved about the regularity of the boundary.On the other hand, it seems that there are no available results in literature for higher eigenvalues, even if the admissible shapes satisfy some topological constraint.
Nevertheless, in view of the physical interpretation of the buckling eigenvalues, it seems reasonable to replace the volume constraint with the perimeter constraint; in this work, given p > 0, we focus on the following problem min Λ 1 (Ω) : Ω ⊂ R d open, |Ω| < ∞, P (Ω) ≤ p . ( An appropriate interpretation in R 2 can be the following: given a deformable support (with prescribed length p) wherein the admissible plates can be clamped, we want to find if there exists a plate that minimizes the buckling load due to the buckling forces acting across the support itself; in other words, we look for the optimal plate that can be clamped in the given support.The main result of the paper is the following.We also prove the following existence result for higher eigenvalues in the framework of convex sets.We finally point out that spectral shape optimization problems governed by higher order PDEs are, in general, harder to handle than the second order counterparts and non existence results often appear.To get acquainted on the non existence of optimal shapes for higher order problems, see several results in [11] (and the reference therein).We just mention, for instance, [11,Theorem 3.24], which gives an example of non existence of optimal shapes for a spectral functional neither with volume nor with perimeter constraint among convex sets.
The paper is structured as follows.In Section 2 we give some preliminary results and we fix the notation.In Section 3 we treat separately the existence case in dimension 2 (it is almost straightforward, but we insert the proof for the reader's convenience).In Section 4 we deal with the existence in any dimension in a weak framework and we show that the optimal shapes for the weak problem are in fact minimizers for the original problem, completing the proof of Theorem 1.1.In Section 5 we prove Theorem 1.2.We finally set some open problems and perspectives in Section 6.

Notation and Preliminary Results
In this section we fix the notation and recall some results used throughout the paper.For x ∈ R d and r > 0, B r (x) will denote the open ball of radius r centered in x; when x is omitted, we consider the ball centered in the origin.For every measurable set E ⊆ R d , we will use the symbols χ E for the characteristic function of E, E c for its complement and tE for the rescaled set {tx : x ∈ E}.As usual, |E| and H s (E) (s > 0) stand respectively for the Lebesgue measure and the Hausdorff s-dimensional measure of E; if E is a piecewise regular hypersurface, H d−1 (E) coincides with its area measure.We will denote by H − dim(E) the Hausdorff dimension of the set E; for sufficiently regular sets, it coincides with the topological dimension of the set E, e.g. if E is an open set of R d , H − dim(E) = d (for further details see Chapter 2, Section 8 in [2]).For every open set Ω ⊂ R d , we will denote by L p (Ω) the usual Lebesgue space of (classes of) p-summable functions, by W k,p (Ω) the Sobolev space of functions whose (weak) derivatives are psummable up to order k and by H k (Ω) the (Hilbert) space W k,2 (Ω); whenever f ∈ L p (K) or f ∈ W k,p (K) for any compact set K ⊂ Ω, we say that f ∈ L p loc (Ω) or f ∈ W k,p loc (Ω) respectively.For the convenience of the reader, whenever A, B are open sets, A ⊆ B and u ∈ W k,p 0 (A), we will denote still by u its zero extension to the whole B.Moreover, for any open set Ω and any test function u ∈ H 2 0 (Ω), we denote the Rayleigh quotient in (1.2) by R Ω (u).

Definition 2.1 (sets of finite perimeter)
. Let E ⊆ R d be measurable and let Ω ⊆ R d be open.We define the perimeter of E in Ω as and we say that E is of finite perimeter in Ω if P (E, Ω) < +∞.If Ω = R d we simply say that E is of finite perimeter and denote its perimeter by P (E).
Let us recall that, if E is sufficiently regular (e.g. if E is either a bounded Lipschitz domain or a convex set), it holds In order to minimize Problem (1.3) and its weak version, Problem (4.3), using the direct methods of the Calculus of Variation (or some concentration-compactness argument), we need lower semicontinuity of the buckling eigenvalues with respect to the some suitable topology on the class of sets of R d where the problem is set.As we will see, two good choices for our purposes are the Hausdorff topology of open sets (in the 2-dimensional setting, where the perimeter constraint and the monotonicity of the functional imply the convexity of the optimizers) and the L 1 -topology (in higher dimension, where a weak formulation in the class of measurable sets is needed).Definition 2.2 (convergence in measure).We say that a sequence of measurable sets We say that ( This kind of convergence turns out to be suitable to our purposes to have compactness of minimizing sequences of sets of finite perimeter.Proposition 2.3 (compactness for the convergence in measure and lower semicontinuity of the perimeter, [2],Theorem 3.39).Let A ⊂ R d be an open bounded set and let (E n ) n be a sequence of subsets of A with finite perimeter such that Then, there exists E ⊆ A with finite perimeter in A such that, up to subsequences, In general, one of the main disadvantages of the convergence in measure is that no topological properties of converging sequences can be deduced for the limit set in this framework.To this aim, we introduce the Hausdorff convergences.

The topology induced by this distance is called Hausdorff-complementary topology (or simply H c -topology) on open sets.
This topology guarantees the compactness of sequences of open convex sets under suitable hypotheses.The following proposition contains some results proved in [5], Section 2.4, and shows us that Hausdorff convergences preserve convexity and assure continuity for Lebesgue measure and perimeters of convex sets.
Proposition 2.6.The following results hold for convex sets:  We recall an important result due to F. John (see [12]), involving convex sets.
Theorem 2.7 (John's ellipsoid Theorem).Let K ⊂ R d a compact convex set with nonempty interior.Then, there exists an ellipsoid (where the ellipsoid x 0 + d(E − x 0 ) is obtained by a dilation of E by a factor d and with center x 0 ).
The following properties of the buckling eigenvalues will be useful throughout the paper.
Proposition 2.8 (properties of the buckling eigenvalues).1 (ii) For every set of finite perimeter Ω ⊂ R d and t > 0 Proof.The proof of item (i) is straightforward since H 2 0 (Ω 1 ) ⊆ H 2 0 (Ω 2 ).Item (ii) can be proved via the natural change of variables tΩ x → y = x t ∈ Ω in the Rayleigh quotient and the one-to-one correspondence between u ∈ H 2 0 (tΩ) and u(t Remark 2.9 (equivalent formulations).In view of the scaling properties of the perimeter and of the buckling eigenvalues, Problem (1.3) is equivalent to the scale invariant problem min P (Ω) Indeed, for any t > 0 one has 3) is also equivalent to the penalized problem and show that it attains its minimum in t = 1.By the scaling properties of the perimeter and of the eigenvalues we have To conclude, we choose β > 0 in such a way that the derivative We close this section recalling two useful results involving some properties of the Sobolev spaces.
Proposition 2.10 (see [10] an open set.Then the following statements are equivalent:

Existence of optimal shapes for the first buckling eigenvalue: the planar case
We are able to prove the existence result in dimension two, where the perimeter constraint assures compactness.Proof.We first notice some simplifications that can be done.
we can suppose that they lie at zero distance each other; for the same reason, we can suppose that all the admissible shapes Ω are contained in the same bounded design region.Indeed, for any open set Ω ⊂ R 2 the condition P (Ω) ≤ p implies diam(Ω) < p/2 and then all the admissible domains can be translated in a bounded design region D ⊂⊂ R 2 .• For any admissible Ω, its convex hull Ω is still an admissible set, since it is open and In view of the decreasing monotonicity of the map Ω → Λ 1 (Ω) with respect to the set inclusion, since Ω ⊂ Ω, we have Then, we can reduce ourselves to the class of open convex sets with boundary length less than or equal to p. • In view of the scaling property of Λ 1 and the monotonicity with respect to inclusions, we have that the admissible sets can be assumed to have exactly boundary length equal to p (the perimeter constraint is saturated).
In other words, in R 2 , without loss of generality we can study notice that the convergence is also in measure and that Ω = ∅.Otherwise, in view of the convergence in measure, we would have |Ω n k | → 0 and from the Payne inequality (see, for instance, inequality (3.26) in [11]), we would obtain where λ 2 is the second eigenvalue of the Dirichlet-Laplacian.This would contradict the minimality of the sequence (Ω n ) n .Then, Ω is an open convex set of positive measure and, in addition, it holds so Ω is an admissible set for Problem (3.1).Now, the corresponding subsequence of eigenfunctions (u Then, there exist a further subsequence (still denoted with the same index) and a function In view of the lower semicontinuity of the H 2 0 -norm with respect to the weak convergence in H 2 0 (D) it finally holds proving the thesis.
4. Existence of optimal shapes for the first buckling eigenvalue: the case of higher dimension In general, the perimeter constraint does not imply the boundedness and the convexity of optimal shapes in higher dimension (this is a peculiarity of the 2-dimensional case).To prove the existence of optimal shapes in higher dimension, we follow a different strategy.By following the approach of [3] (later used in [16]), we look for minimizers for Problem (1.3) with neither topological constraint nor bounded design region via a concentrationcompactness argument.The main difference is that in the previous works the authors dealt with a measure constraint, whereas we have to preserve a perimeter constraint.Proposition 2.3 suggests a good tool to this aim, since the result guarantees the lower semicontinuity of the perimeter and the compactness with respect to the convergence in measure for a sequence of measurable sets; in view of this, the original framework of open sets does not seem to be the most appropriate to prove an existence result for Problem (1.3).
A good strategy in this sense is provided in [9], where the authors use a suitable weak formulation of the Dirichlet-Laplacian eigenvalues in the framework of sets of finite perimeter.More precisely, in order to set the problem in the class of measurable sets instead of working only with open sets, for every set of finite perimeter Ω ⊂ R d they introduce the Sobolev-like spaces and they prove the existence of optimal shapes for a weaker version of the functional with perimeter constraint.After proving the existence, they are able to come back to the original problem, showing that weak minimizers are, in fact, open sets.

Definition 4.1 (weak eigenvalues).
Let Ω ⊂ R d be a set of finite perimeter.We define the Sobolev-like space H2 We define the h-th weak buckling eigenvalue by In particular, the first weak buckling eigenvalue of Ω is given by Once given the weaker version of the functional, we look for a right class of admissible sets; our choice is the following: The new problem to consider is thus min Λ1 (Ω) : Ω ∈ Ãp . (4. 3) The choice of this weaker framework has been made in order to ensure the completeness of the class of admissible sets with respect to the local convergence in measure: in other words, a converging sequence of admissible sets converges (locally in measure) to an admissible set.Notice that, in the original statement (1.3), this request fails: the limit set of a sequence of open sets converging in measure is not open, in general.For that reason, it has been necessary to choose also the functional in a weaker sense, keeping into account the new functional space for the test functions.Only the inclusion H2 0 (Ω) ⊇ H 2 0 (Ω) is valid in general, even for open sets.Nevertheless, if the set Ω is open and sufficiently regular, the equality H2 0 (Ω) = H 2 0 (Ω) holds; the equality fails whenever Ω has inner cracks, e.g. if Ω ⊂ R 3 is a ball with an equatorial cut that removes a maximal half-disk, namely Notice that also in Problem (4.3) we avoid the apriori prescription of a bounded design region where the admissible sets are contained.A similar assumption would lead straightforwardly to the compactness of a minimizing sequence of admissible sets, see [2,Theorem 3.39].
We now state some useful properties of the weak eigenvalues.

Remark 4.3 (equivalent formulations).
As we proved for Problem (1.3), also Problem (4.3) has two equivalent formulations; more precisely it is equivalent to the scale invariant problem min P (Ω) and to the penalized problem for some β > 0. The proof is the same as in Remark 2.9.
Remark 4.4 (ε-eigenfunctions).We point out that the infimum in (4.2) is not attained, in general.For that reason, we introduce the term 'ε-eigenfunction' to denote a test function for some ε > 0.
The following result is a version of [7, Lemma 3.3] adapted to our Sobolev-like spaces.We use the previous result to get the contradiction in the vanishing case in Theorem 4.6 as follows: we find a sequence (w n ) n in H 1 (R d ) satisfying (after a possible rescaling) n with |Ω n | ≤ C and such that any possible translation of w n weakly converges; in view of Lemma 4.5 (w n ) n can not be bounded in H 1 (R d ) and in particular gradients must be unbounded in L 2 (R d ).
Now, to prove the existence of minimizers for Problem (4.3), we follow a strategy based on the concentration-compactness principle by P.-L.Lions (see [13]) and inspired by [3] (a similar argument has been reprised in [16]).We only have to be careful with the choice of suitable ε-eigenfunctions and to preserve the perimeter constraint.For another result of spectral shape optimization under perimeter constraint by using a concentrationcompactness argument see [6], where the same technique applies in the minimization of the second Dirichlet-Laplace eigenvalue.
The main result of this section is the following.Proof.First of all, we show that every solution Ω ⊂ R d has maximal perimeter.Otherwise, if P ( Ω) < p, the dilated set Ω := p P ( Ω) Ω satisfies P ( Ω ) = p and, in view of the decreasing monotonicity and the scaling property of Λ1 , it holds Λ1 ( Ω ) < Λ1 ( Ω), leading to a contradiction with the minimality of Ω.Now we prove the existence of a minimizer for Problem (4.3).Let (Ω n ) n ⊂ Ãp be a minimizing sequence for Problem (4.3) and let u n ∈ H2 0 (Ω n ) a corresponding sequence of normalized weak 1/n-eigenfunctions, namely we have We infer where we used first the inclusion L 2 * (Ω n ) ⊂ L 2 (Ω n ) and then the Sobolev-Gagliardo-Nirenberg inequality since d > 2; all the constants are independent of n since they depend only on the dimension d and on |Ω n |, that is uniformly bounded.We remark that we do not use directly the Poincaré inequality to show the uniform bound on u n L 2 (Ωn) since Ω n can possibly be unbounded in any direction and, even if Ω n is bounded, the function u n could be outside H 1 0 (Ω n ).Moreover, an integration by parts yields There exists a subsequence (u n k ) k ⊂ H 2 (R d ) such that one of the three following situations occurs.(iii) Dichotomy.There exists α ∈]0, 1[ such that, for every ε > 0, there exist two bounded sequences (u

(i) Compactness. There exists a sequence of points (y
Let us show that only compactness does occur. • Vanishing does not occur.Let us assume that vanishing occurs.Then every possible translation of any partial derivative ∂un k ∂x j weakly converges to 0 in L 2 (R d ).Indeed, in view of (4.9), for any φ ∈ C ∞ c (R d ), with supp(φ) contained in some closed ball B R (y), we have , there exists (up to permutations) l ∈ {1, . . ., d} such that Since u n k belongs to H 2 (R d ), by two integration by parts we get, for any i, h ∈ {1, . . ., d}, We thus obtain, by recalling (4.7) (so, that choice satisfies (4.10)).
In view of the previous choice, by using (4.11), (4.12), (4.13) and the inequality we have (possibly switching Now, let us define Clearly, t k ≥ 1; by using the scaling property of the perimeter we have Moreover, in view of the scaling property of the weak eigenvalues, it holds By using this equality in (4.16) we get inf This implies, in view of (4.10), that where we applied the second limit in (4.11) and the fact that dx is uniformly bounded by a positive constant in view of (4.13).
We conclude that (4.18) is true.By (4.17) we obtain inf and then, in view of the arbitrariness of ε > 0, we infer inf Since neither vanishing nor dichotomy occurs, we conclude that compactness takes place.
Then, there exists a sequence (y k ) k ⊂ R d such that, up to subsequences, and there exists u ∈ H 2 (R d ) The equality ∇u L 2 (R d ) = 1 comes from (4.8), the arbitrariness of ε > 0 and the weak lower semicontinuity of the L 2 -norm of the gradient: we deduce that We now prove that there exists a measurable set Ω ⊂ R d such that in order to use u ∈ H2 0 ( Ω) as a test function for Λ1 ( Ω).For any j ∈ N, let us define the set Ω(j) as the limit in measure of the sequence
Then, since u n k (• + y k ) → u strongly in H 1 (R d ) as proved above, recalling (4.7) we finally have concluding the theorem.
The choice of the weaker framework of measurable sets leads us to choose a different approach to the connectedness of the admissible sets: indeed, given Ω ∈ Ãp , for every 'cracked version' Ω of Ω one has Λ1 (Ω ) = Λ1 (Ω); in particular, the same equality holds if we consider cracks splitting the set in two connected components lying at zero distance, e.g. if Ω ⊂ R 3 is a ball and Ω is obtained by removing from Ω a maximal disk.In other words, it does not make sense to talk about connected components in the canonical sense, even for open sets.We point out that the only connected components that can be treated in a classical way are those lying at positive distance (since Λ1 (Ω) is not invariant under relative translations of connected components, unless they remain at positive distance).In view of that, we need to introduce the following definition.Definition 4.7 (well separated sets).Let A, B ⊆ R d .We say that A and B are well separated if there exist two open sets E A , E B and two negligible sets As we expected from the concentration-compactness argument in Theorem 4.6, it can not happen that the optimal set is split in two or more well separated set of positive measure (dichotomy does not occur).We now show that every optimal set for Problem (4.3) is 'connected' in a generalized sense.The proof follows a standard argument for counting the connected components in shape optimization, with the only difference that in our framework Λ1 is an infimum and not a minimum, in general.Proof.Let us suppose that Ω = Ω 1 ∪ Ω 2 , where Ω 1 and Ω 2 are well separated sets of positive measure.Let ε > 0 and let u ε ∈ H2 0 (Ω) an ε-eigenfunction for Λ1 (Ω).Let us define Since Ω 1 and Ω 2 lie at positive distance, then u ε 1 ∈ H2 0 (Ω 1 ) and u ε 2 ∈ H2 0 (Ω 2 ) and so they can be used as test functions for Λ1 (Ω 1 ) and Λ1 (Ω 2 ) respectively.In view of the choice of u ε , by (4.6) we have where we used inequality (4.15).In view of the arbitrariness of ε > 0, either Λ1 (Ω 1 ) ≤ Λ1 (Ω) or Λ1 (Ω 2 ) ≤ Λ1 (Ω).Let us suppose the first case; then, dilating Ω 1 by a factor t > 1 in such a way that P (tΩ 1 ) = p, we get Λ1 (tΩ 2 ) < Λ1 (Ω) contradicting the minimality of Ω.
The previous result about the generalized connectedness of the optimal measurable sets can be proven identically even if we replace the perimeter constraint with the measure constraint.
Once we have assured the existence of optimal shapes in this weak setting, we would like to show that weak solutions are in fact open solutions.To this aim, we follow an approach proposed in [9], introducing the following definition.Definition 4.9 (perimeter supersolution).Let Ω ⊂ R d a set of finite perimeter.We say that Ω is a perimeter supersolution if |Ω| < +∞ and, for every Ω ⊃ Ω of finite perimeter, we have P ( Ω) ≥ P (Ω).
Perimeter supersolutions enjoy good properties for our purposes; one of those is the following density estimate.

Definition 4.11 (exterior density estimate).
Let Ω ⊂ R d a set of finite perimeter.We say that Ω satisfies an exterior density estimate if there exists a positive dimensional constant c = c(d) such that, for every x ∈ R d , one of the following situations occurs: (i) there exists r > 0 such that B r (x) ⊂ Ω a.e.; (ii) for every r > 0, it holds |B r (x) The next results link the previous density estimate with the perimeter supersolutions, ensuring that there exist open optimal shapes for Problem (4.3) and that such an open solution is, in fact, a solution for Problem (1.3).The proof of the following proposition is omitted, as it can be found in [9,Lemma 4.4].Proposition 4.12.Let Ω ⊂ R d be a perimeter supersolution.Then, Ω satisfies an exterior density estimate.In particular, if Ω ⊂ R d is a solution of (4.3), then Ω satisfies an exterior density estimate.
The following crucial result states that the measure theoretic interior Ω 1 of a perimeter supersolution Ω is an open set and that the test space H2 0 (Ω) is, in fact, the classical Sobolev space H 2 0 (Ω 1 ).The proof is based on [9,Proposition 4.7], where the authors show the equality between the spaces H1 0 (Ω) and H 1 0 (Ω 1 ).
Proof.The fact that Ω 1 is open follows from the exterior density estimate.
To show the equality H2 0 (Ω) = H 2 0 (Ω 1 ) it is sufficient to prove that H2 0 (Ω) ⊆ H 2 0 (Ω 1 ).Moreover, since |Ω∆Ω 1 | = 0, the equality H2 0 (Ω) = H2 0 (Ω 1 ) holds and so we prove the inclusion H2 0 (Ω 1 ) ⊆ H 2 0 (Ω 1 ).Let u ∈ H2 0 (Ω 1 ), so, in particular u ∈ H 1 (R d ), u = 0 a.e. in Ω c 1 and thus u ∈ H1 0 (Ω 1 ).By [9, Proposition 4.7], since Ω is a perimeter supersolution, we have that H 1 0 (Ω 1 ) = H1 0 (Ω 1 ) and so u ∈ H 1 0 (Ω 1 ).Then, by Proposition 2.11, we have u = 0 everywhere in Ω c 1 and so, in view of Proposition 2.10, we get ∇u = 0 a.e. in Ω c 1 .Moreover, D j u ∈ H 1 (R d ) for any j = 1, . . ., d, but this implies that D j u ∈ H1 0 (Ω 1 ) = H 1 0 (Ω 1 ) and so that ∇u = 0 everywhere in Ω c 1 .By Proposition 2.11 we get that u ∈ H 2 0 (Ω 1 ).Now we are ready to show that weak minimizers are equivalent to minimizing open sets for Problem (1.3). 5. Existence of optimal shapes for the higher buckling eigenvalues: the case of convex sets The existence of optimal shapes for higher eigenvalues needs a more careful investigation in the framework of sets of finite perimeter.It seems necessary an analysis of the boundedness of minimizers in order to apply an inductive concentration compactness argument as applied for instance in [9].At the moment we are not able to get the required boundedness of the optimal sets since for these fourth order problems the common tools of surgery introduced in the H 1 -setting fail (to get an overview for the Dirichlet-Laplace eigenvalues see, for instance, [9] for the problem with perimeter constraint or [14] for the problem with volume constraint).
Nevertheless, Theorem 1.2 ensures the existence of minimizers for higher eigenvalues in the framework of convex sets.The variational argument used to prove this result is based on a standard application of the direct methods which is inspired by previous works in which higher eigenvalues for the Laplace operator are minimized among convex sets (see, for instance, [8] for the case of Robin eigenvalues or [5] for the Dirichlet case).
In order to apply the direct methods of the Calculus of variations we show that Λ h is lower semicontinuous with respect to the Hausdorff convergence.

Proof. Without loss of generality, we can assume sup
Notice that u 1 , . . ., u h are linearly independent in H 2 0 (Ω), since Ω n converges to Ω also in measure; hence, the linear space V := span {u 1 , . . ., u h } is a competitor for the computation of Λ h (Ω).Let w = α i u i realizing the maximum of the Rayleigh quotient R Ω (•) on V and let w n := α i u n i ∈ V n .Let us observe that the Dirichlet integral at the denominator converges and the numerator is lower semicontinuous and so the Rayleigh quotient is lower semicontinuous as well.Since w n ∈ V n , we conclude that obtaining the required lower semicontinuity of the buckling eigenvalues.Now we are able to prove Theorem 1.2.In order to apply the direct methods of the Calculus of Variations, we need a compactness property for a minimizing sequence (Ω n ) n .To this aim, we just need a careful analysis about the non degeneracy and the uniform boundedness of the sequence (Ω n ) n .(5.1) We prove (5.1) arguing by contradiction.Let us suppose that the limit in (5.1) is zero; in view of John's Ellipsoid Theorem 2.7, there exists an ellipsoid E n such that, up to rotations and translations Then, since also the width of E n vanishes, we have → +∞, against the minimality of Ω n .Then (5.1) holds.
To prove that the diameters of the Ω n sets are uniformly bounded, we argue again by contradiction.Let us suppose that the sequence of the diameters is unbounded.Since the sets Ω n are convex and uniformly bounded in measure, the product has to be uniformly bounded.In view of our assumptions, as the diameter of Ω n tends to infinity, necessarily the first term of the product diverges.We deduce that at least the smallest term among the remaining d − 1 terms has to vanish.In other words, we have Then (Ω n ) n is an equibounded sequence of convex sets.In view of Proposition 2.6(v), (Ω n ) n converges (up to subsequences) in the sense of Hausdorff to a bounded convex set Ω; moreover, by Proposition 2.6(ii), the convergence is also in measure.In addition, thanks to (5.1), the limit set Ω is not degenerate (i.e. it has positive measure) and In view of Proposition 5.1 Ω is the required minimizer.Moreover, if the perimeter constraint in Problem (1.4) is replaced by a volume constraint, the existence of optimal shapes for problem min Λ h (Ω) : Ω ⊂ R d open and convex, |Ω| ≤ m , can be proved by using the same arguments in this section.

Further remarks and open problems
Once proved the existence of minimizers, some interesting questions arise about the regularity or the precise shape of the minimizers.For the first eigenvalue the two questions are related, as highlighted in [3] for the planar case with measure constraint: provided that an optimal shape is regular enough, it must coincide with the disk of given measure.In our framework we start from a better situation, since optimal planar sets are convex.In this case (and, more generally, in the case of Problem (1.4)) it seems necessary at least to remove the possible corners to get more regularity of the boundary.Unfortunately, at the moment the cutting techniques that are known in the H 1 -setting (see, for instance, [4] for the Dirichlet-Laplace eigenvalues, or [8] for the Robin-Laplace eigenvalues) do not seem to apply since they are based on surgery arguments that are not available in the H 2 -setting.
Another aspect which is worth to analyze is the regularity of the optimal sets in higher dimension.Due to the choice of the perimeter constraint, it would be interesting to understand if it was possible to see minimizers for Problem (1.3) as quasi-minimizers of the perimeter in the sense of De Giorgi, as done in [9], in order to obtain that optimal open sets have C 1,α boundary up to a singular set whose dimension is less then or equal to d−7.
To this aim, it seems necessary to prove that optimal sets are bounded.Unfortunately, as highlighted at the beginning of Section 5, up to our knowledge, there are no available techniques to reach this goal at the moment.
We conclude giving the following list of some open problems.
Open Problem 6.1.Are optimal shapes for Problem (1.3) smooth?Is it possible to remove the corners from the boundary of the convex minimizers in the planar case?
Open Problem 6.2.Provided that an optimal shape for Problem (1.3) is smooth enough, can we prove that it is a ball, at least in the planar case?
Open Problem 6.3.Are optimal shapes for Problem (1.3) bounded also in higher dimension?
Open Problem 6.4.Do minimizers for Λ h exist among open sets with prescribed perimeter in higher dimension?Are they bounded?
Declarations.S.C. has been partially supported by the ACROSS project Cod.ARS01-00702.A.L. has been partially supported by the Italian M.U.R. PRIN: grant number 2017KC8WMB.The authors have no competing interests to declare that are relevant to the content of this article.

Figure 1 .
Figure 1.A clamped plate subject to the buckling forces in dimension d = 2; the deflection can be both upward and downward.

Theorem 1 . 1 .
Problem (1.3) admits a solution.Every minimizer is an open connected set.In R 2 any optimal set is open, bounded and convex.

Definition 2 . 4 (
Hausdorff topology on closed sets).Let A, B ⊆ R d be closed.We define the Hausdorff distance between A and B by d H (A, B) := max sup x∈A dist(x, B), sup x∈B dist(x, A) .The topology induced by this distance is called Hausdorff topology (or simply H-topology) on closed sets.The counterpart of the Hausdorff topology for open sets is defined below.

Definition 2 . 5 (
Hausdorff topology on open sets).Let A, B ⊆ R d be open.We define the Hausdorff-complementary distance between A and B by d H c (A, B) := d H (A c , B c ).

. 1 )
From now on, we denoteA p := {Ω ⊂ D, Ω open and convex, H 1 (∂Ω) = p}.Let now (Ω n ) n be a minimizing sequence for (3.1) and, for any n ∈ N, let u n ∈ H 2 0 (Ω n ) an eigenfunction for Λ 1 (Ω n ) with ∇u n 2 = 1.In view of the properties of the Hausdorff convergence (Proposition 2.6), there exist a subsequence (Ω n k ) k and an open convex set Ω ⊂ D such that Ω n k → Ω in the sense of Hausdorff; Then, in general, for any open set Ω it holds Λh (Ω) ≤ Λ h (Ω).

Lemma 4 . 5 .
Let (w n ) n be a bounded sequence in H 1 (R d ) such that w n L 2 (R d ) = 1 and w n = 0 a.e. in Ω c n with |Ω n | ≤ C.There exists a sequence of vectors (y n ) n ⊂ R d such that the sequence (w n (• + y n )) n does not possess a weakly convergent subsequence in H 1 (R d ).

Proposition 4 . 8 (
generalized connectedness of the minimizers).Every solution Ω of Problem (4.3) is 'connected' in the sense of Definition 4.7, i.e. if Ω is union of well separated sets, only one has positive Lebesgue measure.

Proposition 5 . 1 (
Lower semicontinuity of Λ h among convex sets).Let (Ω n ) n be a sequence of open convex sets converging to an open, non empty, convex set Ω in the Hausdorff topology and let Ω n , Ω be contained in a compact set D ⊂ R d .Then, for every

Proof of Theorem 1 . 2 .
Let (Ω n ) n be a minimizing sequence of admissible open convex sets for Problem (1.4) such that H d−1 (∂Ω n ) = p.Let us show that, up to subsequences, Ω n converges in the sense of Hausdorff (and then in measure) to a nonempty admissible open convex set Ω with H d−1 (∂Ω) = p.Without loss of generality, up to translations and rotations, we can assume that diam(Ω n ) = H 1 (Ω n ∩ {x 2 = . . .= x d = 0}) and that min i=2,...,d max Ωn x i − min Ωn x i = max Ωn x d − min Ωn x d i.e. the length of the one dimensional projection of Ω n on the axis x 1 is equal to the diameter of Ω n and the projection on the axis x d has minimal length.We claim that sup n diam(Ω n ) < +∞ and that, up to subsequences, lim n max Ωn x d − min Ωn x d > 0.
We point out that in dimension d = 2 Theorem 1.2 proves the existence of open minimizers for Λ h in the whole of R d .Indeed, the problem min Λ h (Ω) : Ω ⊂ R 2 open, |Ω| < ∞, P (Ω) ≤ p reduces to the minimization problem among convex sets with prescribed boundary length p.
are closed (respectively open) and convex and A n → A with respect to the H-topology (respectively H c -topology), then χ An → χ A in L 1 ; moreover, if for every n ∈ N it holds where ρ is the radius of the biggest ball contained in A.(iv) If a sequence (A n ) n of closed convex sets H-converges to a closed set A, then A is a closed convex set; if a sequence (B n ) n ofopen convex sets H c -converges to an open set B, then B is an open convex set.(v) Let D ⊂ R d a fixed compact set.Then, the class of the closed convex sets contained in D is compact in the H-topology and the class of the open convex sets contained in D is compact in the H c -topology is also bounded from below away from zero; this implies that the infimum of Problem (4.3) is strictly positive.Analogously, also u n L 2 (R d ) is larger than a positive constant and this avoids the degeneracy of the eigenfunctions u n in L 2 (R d ).As already highlighted, due to the lack of a bounded design region D ⊂⊂ R d , we apply the concentration-compactness Lemma [13, Lemma I.1] to the sequence (|∇u n | 2 ) n : (R d) and in H 1 (R d ) as well (every translation of u n k is bounded in H 2 (R d ), so every translation [13]n the statement of the dichotomy case (iii) and let ε > 0. The sequences(u 1 k ) k , (u 2 k ) k ∈ H 2 (R d) can be chosen as follows, see[13]I.1 and [3 lim sup