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 $\hat{I}\text{c◯}$ with an analytic boundary $\hat{I}“$, on which we impose impedance boundary conditions. We suppose that the (possibly complex-valued) permeability and permittivity tensor fields $\hat{\mathbf{I}}{¼}^{-1}$ and $\mathbf{\epsilon }$ are piecewise analytic in $\hat{I}\text{c◯}$ and discontinuous only across certain mutually disjoint analytic surfaces inside of $\hat{I}\text{c◯}$. We show that under these circumstances, any weak solution of Maxwell's equations is piecewise analytic in $\hat{I}\text{c◯}$ and that the growth of its derivatives can be controlled explicitly in the wavenumber $k$.