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

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.

key words: Maxwell compactness property, weak Lipschitz domain, Maxwell estimate, Helmholtz decomposition, electro-magneto statics, mixed boundary conditions, vector potentials. arXiv admin note: text overlap with arXiv:1511.06697