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

TL;DR

This paper develops a framework for analyzing how eigenvalues of the de Rham complex change under Lipschitz domain deformations and variable coefficients, providing formulas for derivatives and extending the Hellmann-Feynman theorem.

Contribution

It introduces an abstract approach for spectral perturbations in de Rham complexes with Lipschitz regularity and derives Hadamard-type formulas for eigenvalue derivatives in Maxwell and Helmholtz problems.

Findings

Derived Lipschitz regularity-based formulas for eigenvalue derivatives.

Extended Hellmann-Feynman theorem to infinite-dimensional parameter spaces.

Reformulated derivative formulas as surface integrals under higher regularity.

Abstract

In this second part of our series of papers, we develop an abstract framework suitable for de Rham complexes that depend on a parameter belonging to an arbitrary Banach space. Our primary focus is on spectral perturbation problems and the differentiability of eigenvalues with respect to perturbations of the involved parameters. As a byproduct, we provide a proof of the celebrated Hellmann-Feynman theorem for both simple and multiple eigenvalues of suitable families of self-adjoint operators in Hilbert spaces, even when these operators depend on possibly infinite-dimensional parameters. We then apply this abstract machinery to the de Rham complex in three dimensions, considering mixed boundary conditions and non-constant coefficients. In particular, we derive Hadamard-type formulas for Maxwell and Helmholtz eigenvalues. First, we compute the derivatives under minimal regularity assumptions - specifically, Lipschitz regularity - on both the domain and the perturbation, expressing the results in terms of volume integrals. Second, under more regularity assumptions on the domains, we reformulate these formulas in terms of surface integrals.