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

2022/05/02 by Lorenzo Ferreri, Gianmaria Verzini, Ferreri, Lorenzo +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Stochastic processes and statistical mechanics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2205.00917

Abstract

We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain Ω⊂ ℝ^N, where the bang-bang weight equals a positive constant m on a ball B⊂Ω and a negative constant -\underlinem on Ω∖ 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 Ω. 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 ∂Ω.

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