# Problem 2: maximizing the first eigenvalue with an obstacle of given~area

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Posted online: 2020-04-11 10:54:41Z by Antoine Henrot10

Cite as: P-200411.2

• Spectral Theory
• Analysis of PDEs
• Optimization and Control

### 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 maximization problem (of Problem 1) has no solutions. Indeed, one can construct a sequence of closed sets $K_n\subset\overline\Omega$ of Lebesgue measure $A$ so that $\lambda_1(\Omega\setminus K_n)\uparrow \infty$ as $n\to\infty$ (for instance by taking $K_n$ as the union of a given closed set in $\overline \Omega$ of area $A$ with a curve filling $\overline \Omega$ as $n$ increases, see [9], [10] where the limit distribution in $\overline{\Omega}$ of such curves is studied in detail). To guarantee the existence of a maximizer one needs to prevent maximizing sequences to spread out over $\overline\Omega$. This can be achieved by imposing stronger geometrical constraints on the class of admissible obstacles (notice that connectedness is still not sufficient). Therefore, for a fixed $A\in (0, \mathcal{L}(\Omega))$, we are led to consider the maximization problem $$\label{prob2} \max \{ \lambda_1(\Omega\setminus K) : \; \text{K\subset \overline{\Omega}, K closed and convex, \mathcal{L}(K)=A}\}.$$ Now, the existence of a maximizer in the restricted class of convex sets is straightforward (see [3], [5]). Moreover, as convexity seems necessary for the existence, it is natural to expect every solution of this maximization problem to saturate the convexity constraint, in the sense that the boundary of any solution should contain non-strictly convex parts. In particular, it would be interesting to know whether this maximization problem has only polygonal sets as solutions, see [7], [8] for results in this direction for shape optimization problems with convexity constraints.

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

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• Created at: 2020-04-11 10:54:41Z