Abstract:We consider a function $u:\Omega \to \Bbb R^N$, $\Omega \subset \Bbb R^n$, minimizing the integral $\int _\Omega (|D_1 u|^2 +...+|D_{n-1}u|^2 +|D_n u|^p) dx$, $2(n+1)/(n+3)\leq p<2$, where $D_i u = \partial u/ \partial x_i$, or some more general functional with the same behaviour; we prove the existence of second weak derivatives $D(D_1 u),..., D(D_{n-1} u) \in L^2$ and $D(D_n u) \in L^p$.
Keywords: regularity, minimizers, integral functionals, anisotropic growth
AMS Subject Classification: 49N60, 35J60