A note on splitting-type variational problems with subquadratic growth

Abstract

We consider variational problems of splitting-type, i.e. we want to minimize \int_{\Omega}\left[f(\tilde{\nabla}w)+g(\partial_{n}w)\right]dx where \tilde{\nabla}=(\partial_{1},...,\partial_{n-1}). Thereby f and g are two C^{2}-functions which satisfy power growth conditions with exponents 1<p\leq q<\infty. In case p\geq2 there is a regularity theory for minimizers u:\mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{N} without further restrictions on p and q if n = 2 or N = 1. In the subquadratic case the results are much weaker: we get C^{1,\alpha}-regularity, if we require q\leq2p+2 for n = 2 or q < p + 2 for N = 1. In this paper we show C^{1,\alpha}-regularity under the bounds q<\frac{2p+4}{2-p} resp. q<\infty.