Quasiconvex Bulk and Surface Energies with subquadratic growth

We establish partial Hölder continuity of the gradient for equilibrium configurations of vectorial multidimensional variational problems, involving bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with $p$-growth, $1<p<2$, without any further structure conditions. The anisotropic surface energy is defined by means of an elliptic integrand $\mathrm{\Phi }$ not necessarily regular.