A note on splitting-type variational problems with subquadratic growth

摘要

We consider variational problems of splitting-type, i. e., we want to minimize, where ▽? = (?_1,...,?_(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: ? ? Ω → ?~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 < 2p+4/2-p resp. q < ∞.