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

TL;DR

This paper establishes that solutions to the time-harmonic Maxwell equations with impedance boundary conditions are piecewise analytic inside a domain, with explicit control over derivative growth relative to the wavenumber, under certain regularity assumptions.

Contribution

It provides the first explicit regularity results for Maxwell equations with impedance boundary conditions that depend on the wavenumber, including derivative growth estimates.

Findings

Solutions are piecewise analytic within the domain.

Derivative growth can be explicitly controlled in terms of the wavenumber.

Results apply to complex-valued, piecewise analytic material tensors.

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 $k$.