Wavenumber-explicit analytic regularity of the heterogeneous Maxwell equations with impedance boundary conditions

Abstract

We consider the time-harmonic Maxwell equations at a nonzero wavenumber $k\in\mathbb{C}$ on a bounded and simply connected Lipschitz domain $\Omega$ with an analytic boundary $\Gamma$, on which we impose impedance boundary conditions. We suppose that the (possibly complex-valued) permeability and permittivity tensor fields $\boldsymbol{\mu}^{-1}$ and $\boldsymbol{\varepsilon}$ are piecewise analytic in $\Omega$ and discontinuous only across certain mutually disjoint analytic surfaces inside of $\Omega$. We show that under these circumstances, any weak solution of Maxwell's equations is piecewise analytic in $\Omega$ and that the growth of its derivatives can be controlled explicitly in the wavenumber $k$.