  We show that the biharmonic Hilbert complex with mixed boundary conditions on
bounded strong Lipschitz domains is closed and compact. The crucial results are
compact embeddings which follow by abstract arguments using functional analysis
together with particular regular decompositions. Higher Sobolev order results
are also proved. This paper extends recent results of the authors on the de
Rham and elasticity Hilbert complexes with mixed boundary conditions and
results of Pauly and Zulehner on the biharmonic Hilbert complex with empty or
full boundary conditions.
