Resolvent Estimates for Time-Harmonic Maxwell’s Equations in the Partially Anisotropic Case

We prove resolvent estimates in 𝐿𝑝-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 𝐿𝑝-spaces and to localize eigenvalues for perturbations by potentials.