Strichartz estimates for Maxwell equations in media: the structured case in two dimensions

Abstract

We prove Strichartz estimates for the two dimensional Maxwell equations with diagonal Lipschitz permittivity of special structure. The estimates have no loss in regularity that occurs in general for $$C^1$$ C 1 -coefficients. In the charge-free case, we recover Strichartz estimates local-in-time for Euclidean wave equations in two dimensions up to endpoints.

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

“…It appears that in the present work the role of the Half-Laplacian is explicitly identified for the analysis of the Maxwell operator the first time. We note that in [27,28], in joint work with R. Schnaubelt, we apply a similar diagonalization to show Strichartz estimates for time-dependent Maxwell's equations with rough coefficients. In these works, due to variable permittivity and permeability, the diagonalization is carried out with pseudo-differential operators, and the present role of the Half-Laplacian is played by the Half-Wave operator.…”

“…are L p -bounded for 1 < p < ∞ with a constant only depending on p, ε, μ as the symbols are linear combinations of Riesz symbols after changes of variables. We find (see (27) for notations)…”