Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions. (English) Zbl 1496.49013
Summary: 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 \(\Omega \backslash B\). 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 \(\partial \Omega \).
MSC:
| 49K30 | Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
| 49R05 | Variational methods for eigenvalues of operators |
| 92D25 | Population dynamics (general) |
| 47A75 | Eigenvalue problems for linear operators |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
spectral optimization; blow-up analysis; concentration phenomena; indefinite weight; survival thresholdReferences:
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