Univalent minimizers of polyconvex functionals in two dimensions (Q1337970)

Let \(\Omega\) be a bounded, simply-connected domain in \(\mathbb{R}^ 2\) of class \(C^{2,\alpha}\) for some \(\alpha> 0\). Let \(\psi\) be a diffeomorphism of class \(C^{2,\alpha}\) from \(\partial\Omega\) onto the boundary of a convex domain \({\mathcal M}\subset\mathbb{R}^ 2\). Let \(\psi\) be so oriented that its degree is one. The authors consider the variational problem of minimizing the functional \[ {\mathcal W}(U)= \int_ \Omega F(DU)dX \] in the class \[ {\mathcal A}= \{U W^{1,2m}(\Omega,\mathbb{R}^ 2): U= \psi\text{ on }\partial\Omega\}. \] They prove, among others, that if \(F\) satisfies some conditions (given in the paper) and \(U\) is a minimizer for \({\mathcal W}(\cdot)\) in \(\mathcal A\), then \(U\) is a homeomorphism from \(\overline\Omega\) onto \(\overline{\mathcal M}\). Moreover, \(\lim_{R\to 0} \int_{B_ RX} \text{det}(DU)>0\) for all \(X\in \Omega\).