New regularity and uniqueness results in the multidimensional Calculus of Variations

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.