Resolvent estimates for time-harmonic Maxwell's equations in the partially anisotropic case

TL;DR

This paper establishes sharp resolvent estimates for time-harmonic Maxwell's equations with partially anisotropic, spatially homogeneous media in 2D and 3D, enabling analysis of eigenvalues and spectral properties.

Contribution

It provides the first sharp $L^p$ resolvent estimates for anisotropic Maxwell's equations in both two and three dimensions, including the derivation of a Limiting Absorption Principle.

Findings

Sharp resolvent estimates in 2D and 3D for anisotropic Maxwell's equations.

Application of estimates to eigenvalue localization for perturbed operators.

Establishment of a Limiting Absorption Principle in $L^p$-spaces.

Abstract

We prove resolvent estimates in $L^p$-spaces for time-harmonic Maxwell's equations in two spatial dimensions and in three dimensions in the partially anisotropic case. In the two-dimensional case the estimates are sharp up to endpoints. We consider anisotropic permittivity and permeability, which are both taken to be time-independent and spatially homogeneous. For the proof we diagonalize time-harmonic Maxwell's equations to equations involving Half-Laplacians. We apply these estimates to infer a Limiting Absorption Principle in intersections of $L^p$-spaces and to localize eigenvalues for perturbations by potentials.