Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part II
Pier Domenico Lamberti, Dirk Pauly, Michele Zaccaron
TL;DR
This work develops a Banach-space–parametrized perturbation theory for de Rham complexes, proving Hellmann–Feynman-type differentiability for simple and multiple eigenvalues of self-adjoint operator families and establishing an abstract framework suitable for Lipschitz domain perturbations. It then specializes to 3D Maxwell and Helmholtz problems with mixed boundary conditions and variable coefficients, deriving Hadamard-type shape derivative formulas for eigenvalues and the elementary symmetric functions of eigenvalues, with volume-integral expressions under minimal regularity and surface-integral forms under stronger regularity via Gaffney inequalities. The results include Rellich-type bifurcation analysis for multiple eigenvalues and yield explicit expressions for derivatives in terms of perturbations of material coefficients and domain deformations, including boundary contributions. Overall, the paper provides a rigorous perturbation theory and practical derivative formulas for spectral problems in the de Rham complex, enabling sensitivity analysis and shape optimization in electromagnetics and acoustic settings.
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.