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 $$L^p$$ 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.

Improved resolvent estimates for constant-coefficient elliptic operators in three dimensions

“…We shall be brief. It turns out that one can follow along the lines of [15] very closely, substituting k = 3 2 non-vanishing principal curvatures. We prove the following:…”

“…In the first step, we derive a Fourier restriction-extension theorem for surfaces Σ a by following along the lines of the preceding work [15]. We prove strong bounds…”

“…Recently, Castéras-Földes [3] analyzed fourth-order Schrödinger operators (in dimensions d ≥ 2) with smooth characteristic surface, and estimates depending on the number of non-vanishing principal curvatures were proved. A wider range was covered in [15], where also surfaces with conic singularities were treated. Presently, we consider the effect of vanishing Gaussian curvature under a transversality assumption, which was described by Erdős-Salmhofer [6].…”

“…This morally corresponds to a decay from 3 2 principal curvatures bounded from below in modulus and thus improves the previous result for one non-vanishing principal curvature (cf. [15,Theorem 1.3]). In this article we record its consequence for solutions to elliptic differential operators.…”

“…The estimate of B δ L p →L q depends on the order of the elliptic operator. The method of proof is well-known and detailed in [15]; see also [12,10] and references therein. We shall be brief.…”