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Preprints, Working Papers, ... Year : 2023

Numerical optimization of Neumann eigenvalues of domains in the sphere

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.
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Dates and versions

hal-04038780 , version 1 (21-03-2023)

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Eloi Martinet. Numerical optimization of Neumann eigenvalues of domains in the sphere. 2023. ⟨hal-04038780⟩
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