A New Level Set Formulation for Improved Dirichlet Eigenvalue Minimizers

Abstract

This paper makes several improvements to existing level set based approaches to computing shape optimizers for the Dirichlet eigenvalues subject to a volume constraint. The most notable changes in formulation include an overhaul of the classical level set construction and root-finding procedures as well the use of a regularized approximation to the standard objective function. Our resulting computational minimizers are either comparable to or improvements on the best known minimizers from the literature. We conclude with a survey of subproblems within the field that may benefit from numerical experiments; these include the existence of cusps on the boundary, the end-behavior of eigenfunction weights in the p-parameterized problem, and the nature of Weyl asymptotics as they relate to the Pólya conjecture.

24 pages, 6 figures, 1 table, code and results available at https://github.com/AtharvThakur0/Laplace-LSF-Dirichlet-Eigenvalue-Numerics