Shape sensitivity analysis for electromagnetic cavities

Abstract

We study the dependence of the eigenvalues of time‐harmonic Maxwell's equations in a cavity upon variation of its shape. The analysis concerns all eigenvalues both simple and multiple. We provide analyticity results for the dependence of the elementary symmetric functions of the eigenvalues splitting a multiple eigenvalue, as well as a Rellich‐Nagy‐type result describing the corresponding bifurcation phenomenon. We also address an isoperimetric problem and characterize the critical cavities for the symmetric functions of the eigenvalues subject to isovolumetric or isoperimetric domain perturbations and prove that balls are critical. We include known formulas for the eigenpairs in a ball and calculate the first one.

Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations

“…Although the adaptation of some of his results is not immediate, one of the necessary optimality conditions we derive is inspired by [25,Theorem 2]. Finally, let us mention the recent work of Lamberti & Zaccaron [35], in which, also using a semi-differential approach, the authors investigate the optimal shape of an electromagnetic cavity.…”

M-dissipative boundary conditions and boundary tuples for Maxwell operators

“…Structural optimization problems for eigenvalues of selfadjoint partial differential operators have been actively studied during several decades [2,11,27,19,38,22,30]. In particular, the H-convergence approach to the existence of optimizers was developed in [11,1].…”