Optimisation of Robin Laplacian eigenvalue with indefinite weight in spherical shell

TL;DR

This paper solves the shape optimization problem for the principal Robin Laplacian eigenvalue with an indefinite weight in a spherical shell across any dimension, revealing the optimal weight distribution.

Contribution

It provides a complete solution to the minimization of the principal eigenvalue with an indefinite weight in spherical shells, leveraging the known bang-bang distribution.

Findings

Optimal weight distribution is bang-bang in spherical shells.

The minimization problem is fully solved for arbitrary dimensions.

The results advance understanding of eigenvalue optimization with indefinite weights.

Abstract

This paper is concerned with an optimisation problem of Robin Laplacian eigenvalue with respect to an indefinite weight, which is formulated as a shape optimisation problem thanks to the known bang-bang distribution of the optimal weight function. The minimisation of the principal eigenvalue of the problem in a spherical shell of an arbitrary dimension is fully solved.