Problem 2: maximizing the first eigenvalue with an obstacle of given~area
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Posted online: 2020-04-11 10:54:41Z by Antoine Henrot36
Cite as: P-200411.2
General Description View the group
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: $$ \lambda_1(\Omega\setminus K):=\min_{u\in H_0^1(\Omega\setminus K)\atop u\neq 0} \frac{\int_{\Omega\setminus K} |\nabla u(x)|^2\,dx}{\int_{\Omega\setminus K} u(x)^2\, dx}. $$
Problem's Description
The corresponding maximization problem (of Problem 1) has no solutions. Indeed, one can construct a sequence of closed sets $K_n\subset\overline\Omega$ of Lebesgue measure $A$ so that $\lambda_1(\Omega\setminus K_n)\uparrow \infty$ as $n\to\infty$ (for instance by taking $K_n$ as the union of a given closed set in $\overline \Omega$ of area $A$ with a curve filling $\overline \Omega$ as $n$ increases, see [9], [10] where the limit distribution in $\overline{\Omega}$ of such curves is studied in detail). To guarantee the existence of a maximizer one needs to prevent maximizing sequences to spread out over $\overline\Omega$. This can be achieved by imposing stronger geometrical constraints on the class of admissible obstacles (notice that connectedness is still not sufficient). Therefore, for a fixed $A\in (0, \mathcal{L}(\Omega))$, we are led to consider the maximization problem \begin{equation}\label{prob2} \max \{ \lambda_1(\Omega\setminus K) : \; \text{$K\subset \overline{\Omega}$, $K$ closed and convex, $\mathcal{L}(K)=A$}\}. \end{equation} Now, the existence of a maximizer in the restricted class of convex sets is straightforward (see [3], [5]). Moreover, as convexity seems necessary for the existence, it is natural to expect every solution of this maximization problem to saturate the convexity constraint, in the sense that the boundary of any solution should contain non-strictly convex parts. In particular, it would be interesting to know whether this maximization problem has only polygonal sets as solutions, see [7], [8] for results in this direction for shape optimization problems with convexity constraints.
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Created at: 2020-04-11 10:54:41Z
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