Low Frequency Asymptotics and Electro-Magneto-Statics for Time-Harmonic Maxwell's Equations in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions

TL;DR

This paper demonstrates that solutions to time-harmonic Maxwell's equations in exterior weak Lipschitz domains approach static solutions as frequency approaches zero, using weighted Sobolev spaces and new compactly supported fields.

Contribution

It introduces a novel approach for analyzing low-frequency limits of Maxwell's equations in exterior domains, including convergence in operator norm and construction of new fields.

Findings

Solutions converge to static solutions as frequency tends to zero

Convergence is established in operator norm

New compactly supported fields for Dirichlet-Neumann problems are constructed

Abstract

We prove that the time-harmonic solutions to Maxwell's equations in a 3D exterior domain converge to a certain static solution as the frequency tends to zero. We work in weighted Sobolev spaces and construct new compactly supported replacements for Dirichlet-Neumann fields. Moreover, we even show convergence in operator norm.