We consider several obqtacle 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:
$$
\lambda_1(\Omega\setminus K):=\min_{u\in H_0^1(\Omega\setminus K)\atop u\neq 0} \frac{\int_{\Omega\setminus K} |\nabla u(x)|^2\,dx}{\int_{\Omega\setminus K} u(x)^2\, dx}.
$$
Article
Optimizing the first Dirichlet eigenvalue of the Laplacian with an obstacle
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