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

We consider a weighted eigenvalue problem for the Dirichlet laplacian in smooth bounded domain Ω⊂RN, where bang–bang weight equals positive constant m¯ on ball B⊂Ω and negative −m̲ Ω∖B. The corresponding principal provides threshold to detect persistence/extinction of species whose evolution is described by heterogeneous Fisher–KPP equation population dynamics. In particular, we study minimization such with respect position B Ω. provide sharp asymptotic expansions optimal eigenpair singularly perturbed regime which volume vanishes. deduce that, up subsequences, concentrates at point maximizing distance from ∂Ω.