  We study classes of variational problems with energy densities of linear
growth acting on vector-valued functions. Our energies are strictly convex
variants of the TV-regularization model introduced by Rudin, Osher and Fatemi
[15] as a powerful tool in the field of image recovery. In contrast to our
previous work we here try to figure out conditions under which we can solve
these variational problems in classical spaces, e.g. in the Sobolev class
$W^{1,1}$.
