A note on splitting-type variational problems with subquadratic growth

Abstract

We consider variational problems of splitting-type, i.e., we want to minimize $$ \int\limits_{\Omega}[f(\widetilde{\nabla} w)+g(\partial_{n}w)]\,dx, $$ where $${\widetilde{\nabla}=(\partial_{1},\ldots,\partial_{n-1})}$$ . Here f and g are two C 2-functions which satisfy power growth conditions with exponents 1 < p ≤ q < ∞. In the case p ≥ 2 there is a regularity theory for locally bounded 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,α -regularity if we require q ≤ 2p + 2 for n = 2 or q < p + 2 for N = 1. In this paper, we show C 1,α -regularity under the bounds $${q<\frac{2p+4}{2-p}}$$ resp. q < ∞.