Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions. (November 2022)

• Record Type: Journal Article
• Authors: Ferreri, Lorenzo; Verzini, Gianmaria
• Is Part Of: Nonlinear analysis. Volume 224(2022)
• Publication Date: 2022-11
• Subjects: 49R05 -- 92D25 -- 47A75 -- 35B40; Spectral optimization -- Blow-up analysis -- Concentration phenomena -- Indefinite weight -- Survival threshold; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113103
• Languages: English
• ISSNs: 0362-546X

Abstract

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 − m ̲ 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 .