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Multi angle Regularity of the optimal sets for spectral functionals. Part I: sum of eigenvalues

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

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    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
    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

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 01/12/2016
      Date de captation : 24/11/16
      Collection : Research talks
      Format : MP4
      Durée : 00:42:59
      Domaine : Control Theory & Optimization ; PDE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : http://videos.cirm-math.fr/2016-11-24_Terracini.mp4

    Informations sur la rencontre

    Nom du congrès : Shape optimization and isoperimetric and functional inequalities / Optimisation de formes et inégalités isopérimétriques et fonctionnelles
    Organisteurs Congrès : Bucur, Dorin ; Buttazzo, Giuseppe ; Henrot, Antoine ; Pratelli, Aldo
    Dates : 21/11/16 - 25/11/16
    Année de la rencontre : 2016
    URL Congrès : http://conferences.cirm-math.fr/1489.html

    Citation 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. <arXiv:1609.01231> - https://arxiv.org/abs/1609.01231

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