Regularity results for an optimal design problem with quasiconvex bulk energies

Regularity results for equilibrium configurations of variational problems involving both bulk and surface energies are established. The bulk energy densities are uniformly strictly quasiconvex functions with quadratic growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u, E), partial Hölder continuity of the gradient of the deformation u is proved, and partial regularity of the boundary of the minimal set E is obtained.