Shape Derivatives of Eigenvalues of the de Rham Complex

We study eigenvalue problems for the de Rham complex on varying three dimensional domains. Our analysis includes both the Helmholtz equation and 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. 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 for simple eigenvalues assuming differentiability of the corresponding eigenvectors. In a forthcoming publication we aim for a more general result involving also multiple eigenvalues of suitable families of self-adjoint operators in Hilbert spaces depending on possibly infinite dimensional parameters.