# Obstacle problem for the first eigenvalue of the Dirichlet-Laplacian

Posted online: 2020-04-11 10:54:41Z by Antoine Henrot34

Cite as: G-200411.1

• Spectral Theory
• Analysis of PDEs
• Optimization and Control

### Problem's Description

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: $$\lambda_1(\Omega\setminus K):=\min_{u\in H_0^1(\Omega\setminus K)\atop u\neq 0} \frac{\int_{\Omega\setminus K} |\nabla u(x)|^2\,dx}{\int_{\Omega\setminus K} u(x)^2\, dx}.$$

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

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

No solutions added yet

No remarks yet