Variational problems of splitting-type with mixed linear-superlinear growth conditions

Variational problems of splitting-type with mixed linear-superlinear growth conditions are considered. In the twodimensional case the minimizing problem is given by $$J [w] = \int_{\Omega} \Big[f_1\big(\partial_1 w\big) + f_2\big(\partial_2 w\big)\Big] \,dx \to \min$$ w.r.t. a suitable class of comparison functions. Here $f_1$ is supposed to be a convex energy density with linear growth, $f_2$ is supposed to be of superlinear growth, for instance to be given by a $N$-function or just bounded from below by a $N$-function. One motivation for this kind of problem located between the well known splitting-type problems of superlinear growth and the splitting-type problems with linear growth (recently considered in [1]) is the link to mathematical problems in plasticity (compare [2]). Here we prove results on the appropriate way of relaxation including approximation procedures, duality, existence and uniqueness of solutions as well as some new higher integrability results.