  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.
