Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

TL;DR

This paper analyzes the properties of optimizers for the principal eigenvalue in indefinite weight problems with Robin boundary conditions, revealing new qualitative insights and solutions, especially in one dimension and through novel rearrangements.

Contribution

It provides a complete solution in 1D, disproves the optimality of the ball in higher dimensions, and introduces a new rearrangement method for better candidate solutions.

Findings

Complete solution in dimension 1

Ball is rarely optimal in dimensions 2 and higher

New rearrangement improves candidate solutions under Neumann conditions

Abstract

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalueproblem in a fixed box $\Omega$ and we consider the general case of Robin boundary conditions on $\partial\Omega$. It is well known that it suffices to consider {\it bang-bang} weights taking two values of different signs, that can be parametrized by the characteristic function of the subset $E$ of $\Omega$ on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to $E$, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.