Problem 3: minimizing the first eigenvalue with an obstacle of given perimeter

Open

Posted online: 2020-04-11 10:54:41Z by Antoine Henrot60

Cite as: P-200411.3

  • Spectral Theory
  • Analysis of PDEs
  • Optimization and Control
View pdf

General Description View the group

We consider several obstacle problems for the first eigenvalue of the Dirichlet-Laplacian: how to place an obstacle $K$ into a domain $\Omega$ to minimize or maximize the first Dirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We introduce the following notations: Let $\Omega\subset \mathbb{R}^2$ a bounded open set and $K\subset \Omega$ a compact subset included in $\Omega$. Here $\Omega$ is fixed and $K$ is considered as the unknown. We are interested in $\lambda_1(\Omega\setminus K)$ the first Dirichlet eigenvalue of the set $\Omega\setminus K$ which can be defined as: $$ \lambda_1(\Omega\setminus K):=\min_{u\in H_0^1(\Omega\setminus K)\atop u\neq 0} \frac{\int_{\Omega\setminus K} |\nabla u(x)|^2\,dx}{\int_{\Omega\setminus K} u(x)^2\, dx}. $$

Problem's Description

The corresponding minimization problem with the area constraint replaced by a perimeter constraint (whatever notion of perimeter one would consider) is in general not well-posed. Indeed, for every $L>0$, one can construct a sequence of smooth connected and closed sets $K_n\subset\overline\Omega$ of perimeter $L$ approaching a subset of $\partial \Omega$ so that $\lambda_1(\Omega\setminus K_n)\downarrow \lambda_1(\Omega)$ as $n\to\infty$ (notice that by regularity there is no doubt on the notion of perimeter of $K_n$). Therefore, as in Problem 2, we restrict the class of admissible obstacles to convex sets. For a fixed $L\in (0,P(\overline{\Omega}))$, consider the {minimization} problem \begin{equation}\label{prob3} \min \{ \lambda_1(\Omega\setminus K) : \; \text{$K\subset \overline{\Omega}$, $K$ closed and convex, $P(K)=L$}\}. \end{equation} The existence of a minimizer is a consequence of the compactness of the class of convex sets and of the continuity of the perimeter w.r.t Hausdorff convergence of convex sets. Notice that, for particular domains $\Omega$ and small values $L$, it is still possible to have trivial solutions. For example, if the boundary $\partial\Omega$ contains a segment and if $L$ is smaller than twice the length of such a segment, then every segment contained in $\partial \Omega$ of perimeter $L$ is a minimizer. On the other hand, if $L$ is large enough, every minimizer has positive Lebesgue measure, since minimizing sequences will not be able to degenerate to a segment. In any case, one expects that every minimizer touches the boundary $\partial \Omega$.

Note that the maximization problem $$ \max\{\lambda_1(\Omega\setminus K):\: \text{$K\subseteq \overline{\Omega}$, $K$ compact, connected and non empty set, $P(K)\leq L$}\}, $$ is studied in detail in the paper [1].

  1. Article Optimizing the first Dirichlet eigenvalue of the Laplacian with an obstacle

    Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 19, 1535-1559, 2019

  2. Article Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints

    Annales de l'institut Henri Poincaré (C) Analyse non linéaire 26, 1149-1163, 2009fulltext

  3. Book Variational methods in shape optimization problems

    pp. viii+216, year of publication: 2005

  4. Article An existence result for a class of shape optimization problems

    Archive for Rational Mechanics and Analysis 122, 183-195, 1993fulltext

  5. Book Extremum problems for eigenvalues of elliptic operators

    pp. x+202, year of publication: 2006

  6. Article Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions

    Archive for Rational Mechanics and Analysis 169, 73-87, 2003fulltext

  7. Article Polygons as optimal shapes with convexity constraint

    SIAM Journal on Control and Optimization 48, 3003-3025, 2009/10fulltext

  8. Article Regularity and singularities of optimal convex shapes in the plane

    Archive for Rational Mechanics and Analysis 205, 311-343, 2012fulltext

  9. Article Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length

    SIAM Journal on Mathematical Analysis 45, 3266-3282, 2013fulltext

  10. Article Where best to place a Dirichlet condition in an anisotropic membrane?

    SIAM Journal on Mathematical Analysis 47, 2699-2721, 2015fulltext


No solutions added yet

No remarks yet

  • Created at: 2020-04-11 13:16:16Z