Spectral optimization for weighted anisotropic problems with Robin conditions

TL;DR

This paper investigates spectral optimization for anisotropic weighted eigenvalue problems with Robin boundary conditions, revealing optimal bang-bang weights and solving the problem explicitly in one dimension, with applications to resource distribution in habitats.

Contribution

It establishes the existence of eigenvalues under Robin conditions, characterizes optimal weights as bang-bang functions, and solves the optimization problem explicitly in one dimension, highlighting new anisotropic effects.

Findings

Optimal eigenvalues are equal for positive and negative cases.

Optimal weights are piece-wise constant, taking only two values.

Complete solution of the optimization problem in one dimension.

Abstract

We study a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains $\Omega\subset \mathbb{R}^{N} $, $N\ge1$, under Robin boundary conditions, proving the existence of two positive eigenvalues $\lambda^{\pm}$ respectively associated with a positive and a negative eigenfunction. Next, we analyze the minimization of $\lambda^{\pm}$ with respect to the sign-changing weight, showing that the optimal eigenvalues $\Lambda^{\pm}$ are equal and the optimal weights are of bang-bang type, namely piece-wise constant functions, each one taking only two values. As a consequence, the problem is equivalent to the minimization with respect to the subsets of $\Omega$ satisfying a volume constraint. Then, we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann conditions, showing new phenomena induced by the presence of the anisotropic diffusion. The optmization problem for $\lambda^{+}$ naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in a heterogeneous habitat.