Weck's Selection Theorem: The Maxwell Compactness Property for Bounded Weak Lipschitz Domains with Mixed Boundary Conditions in Arbitrary Dimensions

TL;DR

This paper proves a compact embedding theorem for differential forms with mixed boundary conditions on weak Lipschitz domains, enabling advanced analysis in electromagnetism and PDEs in arbitrary dimensions.

Contribution

It establishes Weck's selection theorem for weak Lipschitz domains with mixed boundary conditions, including Maxwell estimates and Helmholtz decompositions in arbitrary dimensions.

Findings

Proves compact embedding of differential forms with weak exterior and co-derivative.

Establishes Maxwell estimates and Helmholtz decompositions.

Shows existence of regular potentials and decompositions.

Abstract

It is proved that the space of differential forms with weak exterior and co-derivative, is compactly embedded into the space of square integrable differential forms. Mixed boundary conditions on weak Lipschitz domains are considered. Furthermore, canonical applications such as Maxwell estimates, Helmholtz decompositions and a static solution theory are proved. As a side product and crucial tool for our proofs we show the existence of regular potentials and regular decompositions as well.