Partial regularity of the deformation gradient for some model problems in nonlinear twodimensional elasticity

We consider the model problem of minimizing the functional $\int_{\Omega}\frac{1}{2}|\nabla u|^2+h(\operatorname{det}\nabla u)dx$ where $u:\mathbb R^2\supset\Omega\to\mathbb R^2$ and $h:\mathbb R\to[0,\infty]$ denotes a function which is convex and smooth on $(0,\infty)$, $\operatorname{lim}_{t\downarrow 0}h(t)=+\infty$ and $h\equiv+\infty$ on $(-\infty,0]$. In particular, we show that it is possible to introduce an approximation $\int_{\Omega}\frac{1}{2}|\nabla u|^2+h_{\delta}(\operatorname{det}\nabla u)dx$ for the energy whose minimizers $u_{\delta}$ are of class $C^1$ on some open subset $\Omega_{\delta}$ of $\Omega$ and converge strongly in $H^{1,2}(\Omega,\mathbb R^2)$ to a minimizer и of the original problem. Moreover, we have control on the measure of the exceptional set in the sense that $|\Omega-\Omega_{\delta}|\to 0$ as $\delta\to 0$.