A nonlocal anisotropic eigenvalue problem

TL;DR

This paper investigates a shape optimization problem for the first eigenvalue of an anisotropic Laplacian, revealing a saturation phenomenon where the eigenvalue increases then stabilizes with respect to the weight.

Contribution

It introduces a novel anisotropic eigenvalue problem with a weight-dependent saturation effect and characterizes the optimal shapes minimizing the eigenvalue.

Findings

Eigenvalue increases with weight up to a critical point

Beyond the critical weight, the eigenvalue remains constant

Optimal shapes are characterized within the anisotropic setting

Abstract

We determine the shape which minimizes, among domains with given measure, the first eigenvalue of the anisotropic laplacian perturbed by an integral of the unknown function. Using also some properties related to the associated \lq\lq twisted\rq\rq problem, we show that, this problem displays a \emph{saturation} phenomenon: the first eigenvalue increases with the weight up to a critical value and then remains constant.