Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces

Abstract

We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations of Fresnel's wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds, but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner-Riesz estimates with negative index for non-elliptic surfaces.

42 pages, 7 figures, MAPLE code attached; revised version was accepted to Ann. Henri Poincaré