Everywhere regularity results for a polyconvex functional in finite elasticity

Here we develop a regularity theory for a polyconvex functional in $2\times2-$dimensional compressible finite elasticity. In particular, we consider energy minimizers/stationary points of the functional $I(u)=\int\limits_{\Omega}{\frac{1}{2}|\nabla u|^2+\rho(\det\nabla u)\;dx},$ where $\Omega\subset\mathbb{R}^2$ is open and bounded, $u\in W^{1,2}(\Omega,\mathbb{R}^2)$ and $\rho:\mathbb{R}\rightarrow\mathbb{R}_0^+$ 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 result we show is that every stationary point needs to be locally H\"older-continuous. Secondly, we prove that if $\|\rho'\|_{L^\infty(\mathbb{R})}<1$ 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. Indeed, we show that all stationary points that are additionally of class $W_{loc}^{2,2}$ and whose Jacobian is suitably H\"older-continuous are of class $C_{loc}^{\infty}.$ As a consequence, these results show that in the case when $\|\rho'\|_{L^\infty(\mathbb{R})}<1$ all stationary points have to be smooth.