A note on splitting-type variational problems with subquadratic growth

We consider variational problems of splitting-type, i.e., we want to minimize

integral(Omega)[f((del) over tildew) + g(partial derivative(n)w)] dx,

where (del) over tilde = (partial derivative(1),..., partial derivative(n-1)). Here f and g are two C-2-functions which satisfy power growth conditions with exponents 1 <p = 2 there is a regularity theory for locally bounded minimizers u : R-n superset of Omega -> 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,C-alpha-regularity if we require q