Hadamard variation of eigenvalues with respect to general domain perturbations

TL;DR

This paper rigorously analyzes how Laplacian eigenvalues change under general domain perturbations, providing explicit formulas for derivatives and smooth rearrangements of multiple eigenvalues.

Contribution

It introduces a rigorous second-order Hadamard variation theory for Laplacian eigenvalues under general domain perturbations, applicable to symmetric bilinear forms.

Findings

Existence of eigenvalue derivatives up to second order.

Explicit characterization of derivatives using finite-dimensional eigenvalue problems.

Smooth rearrangement formulas for multiple eigenvalues.

Abstract

We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite dimensional spaces. Then smooth rearrangement of multiple eigenvalues is explicitly given. This result follows from an abstract theory, applicable to general perturbations of symmetric bilinear forms.