A 2D-variant of a theorem of Uraltseva and Urdaletova for higher order variational problems

Abstract

If \Omega is a domain in \mathbb{R}^{2} and if u:\Omega\rightarrow\mathbb{R} locally minimizes the energy \int_{\Omega}\left[h_{1}(\left|(\nabla^{2}u)_{I}\right|)+h_{2}(\left|(\nabla^{2}u)_{II}\right|)\right]dx, where (\nabla^{2}u)_{I}, (\nabla^{2}u)_{II} denotes a decomposition of the Hessian matrix \nabla^{2}u, then we prove the higher integrability and even the continuity of \nabla^{2}u under rather general assumptions imposed on the N-functions h_{1}, h_{2}.