Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions

TL;DR

This paper analyzes the asymptotic behavior of an optimal principal eigenvalue in a weighted Dirichlet problem, revealing how the optimal ball concentrates at a point maximizing the distance from the boundary as its volume shrinks.

Contribution

It provides sharp asymptotic expansions for the eigenvalue and eigenfunction, and characterizes the concentration point of the optimal ball in the singular limit.

Findings

Optimal eigenvalue concentrates at the point farthest from the boundary.

As the volume of the ball shrinks, the eigenpair exhibits specific asymptotic behavior.

The optimal ball tends to maximize distance from the boundary in the limit.

Abstract

We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, where the bang-bang weight equals a positive constant $\overline{m}$ on a ball $B\subset\Omega$ and a negative constant $-\underline{m}$ on $\Omega\setminus B$. The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher-KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of $B$ in $\Omega$. We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of $B$ vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from $\partial\Omega$.