Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions
Abstract
We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, where the bangbang 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 FisherKPP 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 $\partial\Omega$.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.00917
 arXiv:
 arXiv:2205.00917
 Bibcode:
 2022arXiv220500917F
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Optimization and Control
 EPrint:
 27 pages