Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part I

TL;DR

This paper derives shape derivatives for eigenvalues related to the de Rham complex on three-dimensional domains, including applications to Helmholtz and Maxwell problems, under weak regularity assumptions.

Contribution

It provides Hadamard-type formulas for shape derivatives of eigenvalues in the de Rham complex with variable coefficients and Lipschitz domain deformations, extending existing theories.

Findings

Hadamard-type formulas for shape derivatives

Proof of the Helmann-Feynman theorem for complex eigenvalues

Analysis applicable to Maxwell and Helmholtz eigenproblems

Abstract

We study eigenvalue problems for the de Rham complex on varying three dimensional domains. Our analysis includes the Helmholtz equation as well as the Maxwell system with mixed boundary conditions and non-constant coefficients. We provide Hadamard-type formulas for the shape derivatives under weak regularity assumptions on the domain and its perturbations. Our proofs are based on abstract results adapted to varying Hilbert complexes. As a bypass product of our analysis we give a proof of the celebrated Helmann-Feynman theorem both for simple and multiple eigenvalues of suitable families of self-adjoint operators in Hilbert space depending on possibly infinite dimensional parameters. This series of papers consists of Parts I and II.