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

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. 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 localize eigenvalues for perturbations by potentials and to derive a limiting absorption principle in intersections of L p -spaces.

Strichartz estimates for Maxwell equations in media: The partially anisotropic case

“…The diagonalization argument follows the two-dimensional case, but is more involved. The eigenpairs had been computed in case of constant coefficients in [10].…”

“…P (x, ∂) is conjugated to half-wave equations whose dispersive properties depend on the number of different eigenvalues of ε. This was previously analyzed in the constant-coefficient case by Liess [6] and Lucente-Ziliotti [7]; see also [10,8]. It was proved that for ε(x) ≡ ε satisfying (3) solutions to (6) with ε having less than three different eigenvalues decay like solutions to the three-dimensional wave equation.…”