Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error constant estimation

TL;DR

This paper presents a computer-assisted proof that the equilateral triangle minimizes the first Laplacian eigenvalue among triangles with a fixed diameter, with applications to interpolation error constants.

Contribution

It introduces a computer-assisted method for shape optimization of Laplacian eigenvalues over triangles and provides an elementary proof of the Hadamard shape derivative.

Findings

The equilateral triangle minimizes the first eigenvalue among triangles with unit diameter.

The method applies to both Dirichlet and Neumann boundary conditions.

The approach validates shape monotonicity using the Hadamard shape derivative.

Abstract

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The proof utilizes recently developed guaranteed computation methods for both eigenvalues and eigenfunctions of differential operators. The paper also provides an elementary and concise proof of the Hadamard shape derivative, which helps to validate the monotonicity of eigenvalue with respect to shape parameters. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix--Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.