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H 1 Regularity of the optimal sets for spectral functionals. Part I: sum of eigenvalues

Auteurs : Terracini, Susanna (Auteur de la Conférence)
CIRM (Editeur )

Résumé : In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination
of eigenvalues, under a fixed volume constraints. As a model problem, consider
$\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},$
where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure.
We prove that any minimizer $_{opt}$ has a regular part of the topological boundary which is relatively open and
$C^{\infty}$ and that the singular part has Hausdorff dimension smaller than $d-d^*$, where $d^*\geq 5$ is the minimal
dimension allowing the existence of minimal conic solutions to the blow-up problem.

We mainly use techniques from the theory of free boundary problems, which have to be properly extended to the case of
vector-valued functions: nondegeneracy property, Weiss-like monotonicity formulas with area term; finally through the
properties of non tangentially accessible domains we shall be in a position to exploit the ''viscosity'' approach recently proposed by De Silva.

This is a joint work with Dario Mazzoleni and Bozhidar Velichkov.

Codes MSC :
35R35 - Free boundary problems
47A75 - Eigenvalue problems (linear operators)
49Q10 - Optimization of shapes other than minimal surfaces
49R05 - Variational methods for eigenvalues of operators

Ressources complémentaires :
http://www.cirm-math.fr/ProgWeebly/Renc1489/Terracini.pdf

 Informations sur la Vidéo Langue : Anglais Date de publication : 01/12/2016 Date de captation : 24/11/16 Collection : Research talks ; Control Theory and Optimization ; Partial Differential Equations Format : MP4 Durée : 00:42:59 Domaine : Control Theory & Optimization ; PDE Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2016-11-24_Terracini.mp4 Informations sur la rencontre Nom de la rencontre : Shape optimization and isoperimetric and functional inequalities / Optimisation de formes et inégalités isopérimétriques et fonctionnellesOrganisateurs de la rencontre : Bucur, Dorin ; Buttazzo, Giuseppe ; Henrot, Antoine ; Pratelli, AldoDates : 21/11/16 - 25/11/16 Année de la rencontre : 2016 URL Congrès : http://conferences.cirm-math.fr/1489.htmlCitation Data DOI : 10.24350/CIRM.V.19095603 Cite this video as: Terracini, Susanna (2016). Regularity of the optimal sets for spectral functionals. Part I: sum of eigenvalues. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19095603 URI : http://dx.doi.org/10.24350/CIRM.V.19095603

### Voir aussi

Bibliographie

1. Mazzoleni, D., Terracini, S., Velichkov, B. (2016). Regularity of the optimal sets for some spectral functionals. - https://arxiv.org/abs/1609.01231

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