Twodimensional anisotropic variational problems (Q1566343)

Consider the variational functional \[ J(u)= \int _{\Omega} f( Du(x)) dx. \] where the integrand \(f \in C^{2}(\mathbb{R}^{2N})\) satisfies: \[ \lambda (1+|X|^{2})^{\frac {s-2} {2} } \leq D^{2}f(X) (Y,Y)\leq \Lambda (1+|X|^{2})^{\frac {q-2}{2 }} \] where \(1<s<q<\infty\). Using a lemma due to Frehse and Seregin, the authors show that the local minimizers of \(J\) have Hölder continuous first derivatives provided that \(q<2s\).