Higher order variational problems on two-dimensional domains

Abstract

Let u:\mathbb{R}^{2}\supset\Omega\rightarrow\mathbb{R}^{M} denote a local minimizer of J[w]=\int_{\Omega}f(\nabla^{k}w)dx, where k\geq2 and \nabla^{k}w is the tensor of all k^{th} order (weak) partial derivatives. Assuming rather general growth and ellipticity conditions for f, we prove that u actually belongs to the class C^{k,\alpha}(\Omega;\mathbb{R}^{M}) by the way extending the result of [BF2] to the higher order case by using different methods. A major tool is a lemma on the higher integrability of functions established in [BFZ].