Wavenumber-explicit analytic regularity of the heterogeneous Maxwell equations with impedance boundary conditions
This provides foundational mathematical insights for computational electromagnetics, particularly in high-frequency scattering and wave propagation problems, though it is incremental as it extends prior regularity results to more general settings.
The paper tackles the problem of establishing analytic regularity for weak solutions of the heterogeneous Maxwell equations with impedance boundary conditions on domains with analytic boundaries and piecewise analytic material parameters. The result shows that any weak solution is piecewise analytic, with explicit wavenumber-dependent control over derivative growth.
We consider the time-harmonic Maxwell equations at a nonzero wavenumber $k\in\mathbb{C}$ on a bounded and simply connected Lipschitz domain $Ω$ with an analytic boundary $Î$, on which we impose impedance boundary conditions. We suppose that the (possibly complex-valued) permeability and permittivity tensor fields $\boldsymbolμ^{-1}$ and $\boldsymbol{\varepsilon}$ are piecewise analytic in $Ω$ and discontinuous only across certain mutually disjoint analytic surfaces inside of $Ω$. We show that under these circumstances, any weak solution of Maxwell's equations is piecewise analytic in $Ω$ and that the growth of its derivatives can be controlled explicitly in the wavenumber $k$.