Regularity results for minimal configurations of variational problems involving both bulkand surface energies and subject to a volume constraint are established. The bulk energiesare convex functions with p-power growth, but are otherwise not subjected toany further structure conditions. For a minimal configuration (u,E), Hölder continuity ofthe function u is proved as well as partial regularity of theboundary of the minimal set E. Moreover, full regularity of the boundary of theminimal set is obtained under suitable closeness assumptions on the eigenvalues of thebulk energies.
