Regularity of the optimal sets for some spectral functionals

Mazzoleni, Dario; Terracini, Susanna; Velichkov, Bozhidar

doi:10.1007/s00039-017-0402-2

Cited by 36 publications

(98 citation statements)

References 37 publications

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“…As mentioned above, a regularity result, for the optimal sets for the first eigenvalue of the Laplacian, was proved Briançon and Lamboley in [6]. The regularity of the optimal sets for more general spectral functionals was studied in [10], [34], [? ] and [?].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Existence and regularity of optimal shapes for elliptic operators with drift

2019

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This paper is dedicated to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = −∆+V (x)·∇ with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue λ1(Ω, V ) for a bounded quasi-open set Ω which enjoys similar properties to the case of open sets. Then, given m > 0 and τ ≥ 0, we show that the minimum of the following non-variational problemThe existence when V is fixed, as well as when V varies among all the vector fields which are the gradient of a Lipschitz function, are also proved.The second interest and main result of this paper is the regularity of the optimal shape Ω * solving the minimization problemwhere Φ is a given Lipschitz function on D. We prove that the optimal set Ω * is open and that its topological boundary ∂Ω * is composed of a regular part, which is locally the graph of a C 1,α function, and a singular part, which is empty if d < d * , discrete if d = d * and of locally finite, 6, 7} is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each x ∈ ∂Ω * ∩ ∂D, ∂Ω * is C 1,1/2 in a neighborhood of x.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…for every d > d * (see also [34] for an argument using only the monotonicity of W ). Thus, as a consequence of Lemma 5.37, Lemma 5.38 and the results from [40] and [34], we get Appendix A. Extremality conditions and Lebesgue density…”

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confidence: 99%

Existence and regularity of optimal shapes for elliptic operators with drift

2019

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“…In the last years, in the study of free boundary problems, it has been shown how to prove the same condition, in a weaker sense, without regularity assumptions. Two possible ways are to consider ∆u as a measure concentrated on the boundary (see [1]), or to use the viscosity solutions approach (see [18,37]). To start with, we prove that u is radially symmetric.…”

Section: Remark 214mentioning

confidence: 99%

Asymptotic spherical shapes in some spectral optimization problems

Mazzoleni

Pellacci

Verzini

2020

Journal de Mathématiques Pures et Appliquées

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We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box Ω ⊂ R N , which arises in the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the weight, one is led to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures). We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Using α-symmetrization techniques on cones, we prove that, for suitable choices of the box Ω, the optimal shapes for this second problem are indeed spherical. Moreover, for general Ω, we show that small volume spectral drops are asymptotically spherical, centered near points of ∂Ω having largest mean curvature. RésuméOn etudie l'optimisation de la valeur propre principale positive d'un problème avec un poids indéfini, associé au Laplacien avec conditions au bord de Neumann dans une boîte Ω ⊂ R N , qui apparaît dans l'étude naturellement comme valeur limite de survie en dynamique des populations. En essayant de minimiser une telle valeur propre par rapport au poids, on est amenéà considérer un problème d'optimisation de forme, qui n'admet pas de formes sphériques comme solutions (chose qui contredit certaines conjecturesénoncées précédemment). Nousétudions si des formes sphériques peuventêtre recouvrées comme solutions de certaines perturbations singulières du problème. Notamment, on démontre que, si la partie négative du poids diverge, le problème d'optimisation de forme se rapprocheà la limiteà un problème de "goutte spectrale", cetteà dire un problème de minimisation de la première valeur propre du Laplacien avec conditions mixtes Dirichletet-Neumann au bord. En utilisant techniques de α-symétrisation sur les cônes, on démontre que, pour des choix opportunes de la boîte Ω, les formes optimales pour ce second problème sont bien sphériques. De plus, pour Ω générique, on démontre que les gouttes spectrales s'approchent asymptotiquement -lorsque leur volume devient infinitésimal-à des gouttes sphériques, centrées près des points de ∂Ω ayant courbure moyenne maximale.

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“…Let r > 0, x 0 ∈ R 2 and u ∈ H 1 loc (R d ). The relation between W op and the excess E is given by the following formula, which holds for any function u and can be obtained by a direct computation (see [19] and [16]).…”

Section: 2mentioning

confidence: 99%

“…, n, is C 1,α regular (Theorem 1.8). This line of study has become increasingly important in recent years, where regularity results for solutions of free-boundary problems, and in particular almost-minimizers, have been applied to study the regularity of shape optimization problems involving eigenvalues of the Dirichlet-Laplacian (see for instance [16,14,15,9]).…”

Section: Introductionmentioning

confidence: 99%

Free boundary regularity for a multiphase shape optimization problem

Spolaor

Trey

Velichkov

2019

Communications in Partial Differential Equations

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In this paper we prove a C 1,α regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As a consequence, we deduce the complete regularity of solutions of a multiphase shape optimization problem for the first eigenvalue of the Dirichlet-Laplacian up to the fixed boundary. One of the main ingredient is a new application of the epiperimetric-inequality of [18] up to the boundary. While the framework that leads to this application is valid in every dimension, the epiperimetric inequality is known only in dimension two, thus the restriction on the dimension. 1 Remark 1.2. The Hölder continuity of the (exterior) normal vector n Ω is the best regularity result that one can expect. Indeed, recently Chang-Lara and Savin [10] showed that even for minimizers the regularity of the constrained free boundaries cannot exceed C 1, 1 /2 . Moreover, we notice that the result analogous to Theorem 1.1 was proved in any dimension in [10], by a viscosity approach, but only for minimizers of the functional J op .Analogously, we say that the nonnegative function u : B 2 → R is a almost-minimizer of the one-phase functional J op in B 2 , if u ∈ H 1 (B 2 ) and there are constants r 1 > 0, C 1 > 0 and δ 1 > 0 such that, for every x 0 ∈ B 1 ∩ ∂Ω u and r ∈ (0, r 1 ), we haveThe regularity of the unconstrained one-phase free boundary ∂Ω u follows directly by Theorem 1.1. For the sake of completeness, we give the precise statement in Corollary 1.3 below.Corollary 1.3 (Regularity of the unconstrained one-phase free boundaries). Let B 2 ⊂ R 2 and u : B 2 → R be a non-negative and Lipschitz continuous function. If u is a almost-minimizer of the functional J op in B 2 , then the free boundary B 1 ∩ ∂Ω u is locally the graph of a C 1,α function.

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Regularity of the optimal sets for some spectral functionals

Cited by 36 publications

References 37 publications

Existence and regularity of optimal shapes for elliptic operators with drift

Existence and regularity of optimal shapes for elliptic operators with drift

Asymptotic spherical shapes in some spectral optimization problems

Free boundary regularity for a multiphase shape optimization problem

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