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

TL;DR

This paper investigates how to optimally place small-volume obstacles within a domain to minimize the first Dirichlet Laplacian eigenvalue, revealing that optimal obstacles cluster near boundary points where the eigenfunction gradient is minimal.

Contribution

It provides a rigorous analysis of the asymptotic behavior of optimal obstacles and eigenvalues as obstacle volume shrinks to zero, including their convergence properties.

Findings

Optimal obstacles accumulate near boundary points with minimal eigenfunction gradient.

As obstacle volume approaches zero, eigenvalues and eigenfunctions converge to specific limits.

The study offers sharp estimates of eigenvalues using relative capacity.

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.