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Posted online: 2020-04-11 10:54:41Z by Antoine Henrot11

Cite as: P-200411.3

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

The corresponding minimization problem with the area constraint replaced by a *perimeter* constraint (whatever notion of perimeter one would consider) is in general not well-posed. Indeed, for every $L>0$, one can construct a sequence of smooth connected and closed sets $K_n\subset\overline\Omega$ of perimeter $L$ approaching a subset of $\partial \Omega$ so that $\lambda_1(\Omega\setminus K_n)\downarrow \lambda_1(\Omega)$ as $n\to\infty$ (notice that by regularity there is no doubt on the notion of perimeter of $K_n$).
Therefore, as in Problem 2, we restrict the class of admissible obstacles to convex sets.
For a fixed $L\in (0,P(\overline{\Omega}))$, consider the {minimization} problem
\begin{equation}\label{prob3}
\min \{ \lambda_1(\Omega\setminus K) : \; \text{$K\subset \overline{\Omega}$, $K$ closed and convex, $P(K)=L$}\}.
\end{equation}
The existence of a minimizer is a consequence of the compactness of the class of convex sets and of the continuity of the perimeter w.r.t Hausdorff convergence of convex sets.
Notice that, for particular domains $\Omega$ and small values $L$, it is still possible to have trivial solutions.
For example, if the boundary $\partial\Omega$ contains a segment and if $L$ is smaller
than twice the length of such a segment, then every segment contained in $\partial \Omega$ of perimeter $L$ is a minimizer.
On the other hand, if $L$ is large enough,
every minimizer has positive Lebesgue measure, since minimizing sequences will not be able to degenerate to a segment. In any case, one expects that every minimizer touches the boundary $\partial \Omega$.

Note that the *maximization* problem
$$
\max\{\lambda_1(\Omega\setminus K):\: \text{$K\subseteq \overline{\Omega}$, $K$ compact, connected and non empty set, $P(K)\leq L$}\},
$$
is studied in detail in the paper [1].

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Created at: 2020-04-11 13:16:16Z

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