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H 1 Dirichlet-Neumann shape optimization problems

Auteurs : Buttazzo, Giuseppe (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : We consider spectral optimization problems of the form

    $\min\lbrace\lambda_1(\Omega;D):\Omega\subset D,|\Omega|=1\rbrace$

    where $D$ is a given subset of the Euclidean space $\textbf{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation

    $\lambda_1(\Omega;D)=\min\lbrace\int_\Omega|\nabla u|^2dx+k\int_{\partial D}u^2d\mathcal{H}^{d-1}:$

    $u\in H^1(D),u=0$ on $\partial\Omega\cap D,||u||_{L^2(\Omega)}=1\rbrace$

    reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains. The case of Dirichlet condition on a $\textit{fixed}$ part and of Neumann condition on the $\textit{free}$ part of the boundary is also considered

    Codes MSC :
    49J20 - Optimal control problems involving partial differential equations
    49N45 - Inverse problems in calculus of variations
    49Q10 - Optimization of shapes other than minimal surfaces

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 15/12/15
      Date de captation : 12/11/15
      Collection : Research talks ; Control Theory and Optimization ; Partial Differential Equations
      Format : MP4
      Durée : 00:55:50
      Domaine : PDE ; Control Theory & Optimization
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2015-11-12_Buttazzo.mp4

    Informations sur la rencontre

    Nom de la rencontre : Controllability of partial differential equations and applications / Contrôle des EDP et applications
    Organisateurs de la rencontre : Dermenjian, Yves ; Cristofol, Michel ; Gaitan, Patricia ; Le Rousseau, Jérôme ; Yamamoto, Masahiro
    Dates : 09/11/15 - 13/11/15
    Année de la rencontre : 2015
    URL Congrès : http://conferences.cirm-math.fr/1368.html

    Citation Data

    DOI : 10.24350/CIRM.V.18892903
    Cite this video as: Buttazzo, Giuseppe (2015). Dirichlet-Neumann shape optimization problems. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18892903
    URI : http://dx.doi.org/10.24350/CIRM.V.18892903


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