The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

TL;DR

This paper proves the Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions, enabling key results like Maxwell estimates, Helmholtz decompositions, and a Fredholm alternative for Maxwell problems.

Contribution

It establishes the Maxwell compactness property in weak Lipschitz domains with mixed boundary conditions, extending classical results to more general geometries.

Findings

Proves Maxwell compactness property in weak Lipschitz domains.

Derives Maxwell estimates and Helmholtz decompositions.

Establishes a Fredholm alternative for time-harmonic Maxwell problems.

Abstract

For a bounded weak Lipschitz domain we show the so called `Maxwell compactness property', that is, the space of square integrable vector fields having square integrable weak rotation and divergence and satisfying mixed tangential and normal boundary conditions is compactly embedded into the space of square integrable vector fields. We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exterior domains formulated in weighted Sobolev spaces are straight forward.