  In the first part of this doctoral thesis we develop a regularity theory for
a polyconvex functional in compressible elasticity.
  In the second part, we will concentrate on uniqueness questions in various
situations of finite elasticity. Here it is our main objective to establish
uniqueness criteria, which when present, guarantee the uniqueness of the
corresponding global minimizer. Then various applications and generalisations
are discussed one of which is the construction of a counterexample to
regularity.
