Shape Derivatives for Maxwell's Equations with Nonlinear Boundary Conditions

Abstract

This paper develops a trace-regular variational framework for time-harmonic Maxwell scattering problems involving pointwise nonlinear boundary and interface responses. We investigate three canonical classes of models: nonlinear impedance, nonlinear perfect electric conductor, and nonlinear transmission conditions. Since the standard Maxwell tangential trace belongs to a space of negative order, the nonlinearities are formulated in refined functional spaces where the tangential electric field admits an $L^2(\Gamma)$-trace. Under the assumption of a sufficiently small Lipschitz constant for the nonlinear response, we establish the well-posedness of the direct problems via fixed-point arguments leveraging the mapping properties of the associated linear Maxwell operators. Within this framework, we perform a rigorous sensitivity analysis of the electromagnetic fields with respect to perturbations of the scattering interface. By employing the covariant Piola transform, we prove the continuity and Fr\'echet differentiability of the pulled-back solutions with respect to domain variations. The material derivative is characterized as the unique solution to a corresponding $\mathbb{R}$-linearized Maxwell system, and the shape derivative is shown to satisfy explicit boundary or interface conditions for each of the three nonlinear models. We further demonstrate that the resulting sensitivity expressions possess the Hadamard structure, depending exclusively on the normal component of the boundary deformation. The resulting derivative characterizations provide a mathematical basis for subsequent adjoint-based sensitivity analysis, shape optimization, and gradient-driven inverse reconstruction.