New regularity and uniqueness results in the multidimensional Calculus of Variations

"In the first part of the Thesis we develop a Regularity Theory for a polyconvex functional in compressible elasticity. In particular, we consider energy minimizers/stationary points of the functional\begin{equation}I(u)=\int\limits_\Om{\frac{1}{2}|\grad u|^2+\rho(\det\grad u)\;dx},\label{eq:SA.1.1}\end{equation}where $\Omega\ss\R^2$ is open and bounded, $u\in W^{1,2}(\Om,\R^2)$ and $\rho:\R\ra\R_0^+$ smooth and convex with $\rho(s)=0$ for all $s\le0$ and $\rho$ becomes affine when $s$ exceeds some value $s_0>0.$ Additionally, we may impose boundary conditions.\\The first general result we will establish is that every stationary point needs to be locally Hölder-continuous.Secondly, we prove that if the growth of $\rho$ is `small' s.t.\! the integrand is still uniformly convex, then all stationary points have to be in $W_{loc}^{2,2}.$ Next, a higher-order regularity result is shown.We show that all stationary points that are additionally of class $W_{loc}^{2,2}$ and whose Jacobian is Hölder-continous are of class $C_{loc}^{\infty}.$ In particular, these results show that all stationary points have to be smooth for $\rho'$ `small' enough. \\The theory described above works for fairly general domains and boundary conditions.We specify those by restricting to the unit ball, and we consider M-covering maps, which take the unit sphere to itself, covering the image $M$ times in the process, on the boundary. Under these circumstance, we construct radial symmetric M-covering stationary points to the functional, as given in \eqref{eq:SA.1.1}, which are at least of class $C^1.$ In certain situations, depending mainly on the behaviour of $\rho$ and the stationary point itself, we are even able to guarantee maximal smoothness. \\In the second part, we will concentrate on uniqueness questions in various situations of finite elasticity. Starting in incompressible elasticity, the central point is to showthat for problems with uniformly convex integrands with ""small"" pressure, a unique global minimizer can be guaranteed. We make use of this statement by considering various examples and applications.One such application is the construction of a counterexample to regularity. Indeed, on the unit ball and for smooth boundary conditions we give a non-autonomous uniformly convex functional $f(x,\xi)$ depending smoothly on $\xi$ however discontinously on $x,$ where the unique global minimizer is Lipschitz but no better. Then we generalise the main result in various ways, for instance, we show that if the pressure is too large to guarantee uniqueness in the full class of admissible maps, one can still guarantee uniqueness up to the first Fourier-modes.Lastly, we discuss analogous statements for polyconvex integrands in compressible elasticity.