Obstacle problem for the first eigenvalue of the Dirichlet-Laplacian

Posted online: 2020-04-11 10:54:41Z by Antoine Henrot30

Cite as: G-200411.1

  • Spectral Theory
  • Analysis of PDEs
  • Optimization and Control
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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: $$ \lambda_1(\Omega\setminus K):=\min_{u\in H_0^1(\Omega\setminus K)\atop u\neq 0} \frac{\int_{\Omega\setminus K} |\nabla u(x)|^2\,dx}{\int_{\Omega\setminus K} u(x)^2\, dx}. $$

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  1. Open Problem 1: minimizing the first eigenvalue with an obstacle of given area

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

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

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