Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part I
Pier Domenico Lamberti, Dirk Pauly, Michele Zaccaron
TL;DR
The paper develops a unified framework to study how the eigenvalues of the de Rham complex, including Maxwell and Helmholtz problems with mixed boundary conditions and variable coefficients, depend on Lipschitz domain perturbations. It uses domain transplantation and Hilbert-complex techniques to establish unitary equivalence of spectra under pullbacks and derives Hadamard-type shape derivative formulas, initially in a formal setting and later planned to be made rigorous in Part II via a Helmann–Feynman approach. Key contributions include a general transformation theorem for differential operators, a detailed spectral analysis of the de Rham complex with mixed BC, and a Rayleigh-quotient representation of eigenvalues that accommodates non-constant parameters ε, μ, ν. The results lay the groundwork for robust shape optimization in electromagnetics and related PDE systems on non-smooth domains, with implications for design under geometric perturbations and material heterogeneity.
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.