The C^r dependence problem of eigenvalues of the Laplace operator on domains in the plane

TL;DR

This paper investigates how eigenvalues of the Laplace operator depend smoothly on domain perturbations, providing detailed descriptions for specific shapes and introducing a new implicit function theorem for analysis.

Contribution

It offers a comprehensive analysis of the C^r dependence of Laplace eigenvalues on domains, including new results for disks, squares, and higher-dimensional balls, using a novel implicit function theorem.

Findings

Eigenvalues on disks and squares are thoroughly characterized.

Second eigenvalue behavior on higher-dimensional balls is described.

A new degenerate implicit function theorem is developed for Banach spaces.

Abstract

The C^r dependence problem of multiple Dirichlet eigenvalues on domains is discussed for elliptic operators by regarding smooth one-parameter families of C^1 perturbations of domains in R^n. As applications of our main theorem (Theorem 1), we provide a fairly complete description for all eigenvalues of the Laplace operator on disks and squares and also for its second eigenvalue on balls in R^n for any n >= 3. The central tool used in our proof is a degenerate implicit function theorem on Banach spaces of independent interest.