  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.
