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Posted online: 2020-04-11 10:54:41Z by Antoine Henrot20

Cite as: P-200411.1

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

For a fixed $A\in (0, \mathcal{L}(\Omega))$, with $\mathcal{L}$ the Lebesgue measure, consider the *minimization* problem
$$
\min \{ \lambda_1(\Omega\setminus K) : \; \text{$K\subset \overline{\Omega}$, $K$ closed, $\mathcal{L}(K)=A$}\}.
$$
This problem is related to the minimization of the first eigenvalue among *open* sets constrained to lie in a given {box} (and also with a given area), see [5]. Indeed, passing to the complementary set $O=\Omega\setminus K$ this problem becomes equivalent to the minimization of $\lambda_1(O)$ among open sets $O\subseteq \Omega$ of area $\mathcal{L}({\overline{\Omega}})-A$ (in this framework $\Omega$ represents the box).
Therefore, from what is known on the minimizers contained into a box, we infer the existence of a solutionfor that minimization problem and some of its qualitative properties.
We have to distinguish two cases, depending on the existence of disks of area $\mathcal{L}(\overline{\Omega})-A$ that are contained inside $ \Omega$ (to this aim we introduce the inradius $\rho(\Omega)$ of $\Omega$).

1. Let $A\geq \mathcal{L}(\overline{\Omega})-\pi \rho(\Omega)^2$. Thanks to the Faber-Krahn inequality, a closed set $K_{opt}$ is a minimizer if and only if $K_{opt}=\overline{\Omega}\setminus B$ with the *open* set $B$ that is (up to sets of capacity zero) any disk in $\Omega$ of area $\mathcal{L}(\overline{\Omega})-A$.
This imply that, in general, this problem does not have a unique solution.

2. Let $A< \mathcal{L}(\overline{\Omega})-\pi \rho(\Omega)^2$. By [5], [2] and [6], every minimizer $K_{opt}$ touches the boundary of $\Omega$, its free boundary (i.e., the part of the boundary of $K_{opt}$ which is inside $\Omega$) is analytic in dimension 2 and does not contain any arc of circle.

**Open problems:**

1. Regularity: Study the regularity of the free boundary of the minimizer $K_{opt}$ in higher dimension $N\geq 3$ (existence is provided by the classical Buttazzo-Dal Maso theorem, see [4]).

2. Geometric property: prove that $\Omega\setminus K_{opt}$ is convex when $\Omega$ is convex

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Created at: 2020-04-11 10:54:41Z

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