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