Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle

Abstract

We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains $\Omega$ having prescribed volume and contained in a fixed box $D$; equivalently, we look for the best way to remove a compact set (obstacle) $K\subset\overline{D}$ of Lebesgue measure $|K|=\varepsilon$, $0<\varepsilon<|D|$, in order to minimize the first Dirichlet eigenvalue of the set $\Omega = D \setminus K$. In the small volume regime $\varepsilon\to0$, we prove that the optimal obstacles accumulate, in a suitable sense, to points of $\partial D$ where $|\nabla \phi_0|$ is minimal, where $\phi_0$ denotes the first eigenfunction of the Dirichlet Laplacian on $D$. Moreover, we provide a fairly detailed description of the convergence of the optimal eigenvalues, eigenfunctions and free boundaries. Our results are based on sharp estimates of the optimal eigenvalues, in terms of a suitable notion of relative capacity.