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

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 byproduct of our analysis, we give a proof of the celebrated Hellmann–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.