Numerical optimization of Neumann eigenvalues of domains in the sphere

TL;DR

This paper explores numerical methods to optimize the first three Neumann eigenvalues of domains on the sphere, comparing density and level-set approaches, and investigates the shape optimization problem with implications for geometric conjectures.

Contribution

It introduces and compares two numerical optimization methods for Neumann eigenvalues on spherical domains and provides insights into optimal shapes, including extensions to toroidal geometries.

Findings

Optimal shapes vary significantly with surface area proportion.

Geodesic balls are not always optimal for the first eigenvalue.

Algorithms reveal a rich diversity of eigenvalue-optimized shapes.

Abstract

This paper deals with the numerical optimization of the first three eigenvalues of the Laplace-Beltrami operator of domain in the Euclidean sphere in $\mathbb{R}^3$ with Neumann boundary conditions. We address two approaches : the first one is a generalization of the initial problem leading to a density method and the other one is a shape optimization procedure via the level-set method. The original goal of those method was to investigate the conjecture according to which the geodesic ball were optimal for the first non-trivial eigenvalue under certain conditions. These computations give some strong insight on the optimal shapes of those eigenvalue problems and show a rich variety of shapes regarding the proportion of the surface area of the sphere occupied by the domain. In a last part, the same algorithms are used to carry the same survey on a torus.