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

Abstract

We prove resolvent estimates in $$L^p$$ 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 $$L^p$$ L p -spaces and to localize eigenvalues for perturbations by potentials.

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

“…As noted above in the isotropic and to some extent in the partially anisotropic case, by [16] the Maxwell system possesses Strichartz estimates as the scalar wave equation. However, in the fully anisotropic case already for constant coefficients ε = diag(ε 1 , ε 2 , ε 3 ), µ = diag(µ 1 , µ 2 , µ 3 ) satisfying ( 5)…”

“…Here we use a homogeneous dyadic decomposition (S λ ) λ∈2 Z of spatial frequencies, and (S λ ) λ∈2 Z denotes a decomposition of space-time frequencies, see (18). This surprising behavior is connected to the shape of the characteristic surface depending on the eigenvalues of ε and µ, which has been discussed in [16,Section 2.3]. In [12] the first author and R. Mandel proved the existence of solutions to the time-harmonic Maxwell equations in the fully anisotropic case under the assumption that ε and µ are commuting.…”

“…For the estimate of the high frequencies, we split the initial data into stationary and dispersive components. Let (Φ 1 , Φ 2 ) be the solutions to the elliptic equations in (16). Since we confine to high frequencies, we can replace the homogeneous with inhomogeneous norms.…”