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

TL;DR

This paper establishes Strichartz estimates for Maxwell equations in fully anisotropic media with H"older-continuous coefficients, using phase space conjugation and oscillatory integral estimates to improve local well-posedness results.

Contribution

It introduces a novel approach combining FBI transform and phase space analysis to handle fully anisotropic Maxwell equations with non-smooth coefficients.

Findings

Proved Strichartz estimates for anisotropic Maxwell equations.

Reduced complex matrix problem to scalar estimates via symmetrizer.

Enhanced local well-posedness results for certain quasilinear Maxwell equations.

Abstract

We prove Strichartz estimates for Maxwell equations in media in the fully anisotropic case with H\"older-continuous coefficients. To this end, we use the FBI transform to conjugate the problem to phase space. After reducing to a scalar estimate by means of a matrix symmetrizer, we show oscillatory integral estimates for a variable-coefficient Fourier extension operator. The characteristic surface has conical singularities for any non-vanishing time frequency. Combined with energy estimates, we improve the local well-posedness for certain fully anisotropic quasilinear Maxwell equations.