Regularity of vector fields with piecewise regular curl and divergence

TL;DR

This paper establishes piecewise Sobolev regularity for vector fields with piecewise regular curl and divergence in a Lipschitz domain, and applies these results to Maxwell equations with explicit estimates.

Contribution

It introduces a novel approach using parametrices and transmission problem theory to analyze vector fields with discontinuities, extending regularity results to Maxwell equations.

Findings

Piecewise Sobolev regularity of vector fields with discontinuous curl and divergence.

Application of regularity results to time-harmonic Maxwell equations with explicit estimates.

Development of wavenumber-explicit regularity bounds for electromagnetic problems.

Abstract

We consider a bounded Lipschitz domain $\Omega\subseteq\mathbb{R}^3$ with sufficiently smooth boundary and prove piecewise Sobolev regularity of vector fields that have piecewise regular curl and divergence, but may be discontinuous across mutually disjoint and sufficiently smooth surfaces inside of $\Omega$. The main idea behind our approach is to employ recently developed parametrices for the curl-operator and the regularity theory of Poisson transmission problems. We conclude our work by applying our findings to the heterogeneous time-harmonic Maxwell equations with either a) impedance, b) natural or c) essential boundary conditions and providing wavenumber-explicit piecewise regularity estimates for these equations.