Regularity results for the time-harmonic Maxwell equations with impedance boundary condition

TL;DR

This paper establishes high-order regularity bounds for solutions to the time-harmonic Maxwell equations with impedance boundary conditions, extending previous results to inhomogeneous cases and analyzing dependence on the wave number.

Contribution

It introduces a novel method to derive $H^2$ and $W^{m,p}$ estimates for inhomogeneous boundary conditions, advancing the regularity theory for Maxwell equations.

Findings

Derived $H^2$-norm bounds for solutions

Extended regularity estimates to inhomogeneous boundary conditions

Analyzed the dependence of estimates on the wave number

Abstract

This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and S. Nicaise, Tech. Rep. 10-09, IRMAR (2010)] for the case with homogeneous boundary condition. Unfortunately, their method can not be applied to the inhomogeneous case. The main novelty of this paper is that we follow the spirit of the $H^1$-estimate in [R. Hiptmair, A. Moiola and I. Perugia, Math. Models Methods Appl. Sci., 21(2011), pp. 2263-2287] and modify the proof by applying two inequalities of Friedrichs' type to make the $H^1$-estimate move into $H^2$-estimate and $W^{m, p}$-estimate.Finally, the dependence of the regularity estimates on the wave number is obtained, which will play an important role in the convergence analysis of the numerical solutions for the time-harmonic Maxwell equations.