  We study the regularity of the interface for optimal energy configurations of
functionals involving bulk energies with an additional perimeter penalization
of the interface. It is allowed the dependence on $(x,u)$ for the bulk energy.
For a minimal configuration $(E,u)$, the H\"{o}lder continuity of $u$ is well
known. We give an estimate for the singular set of the boundary $\partial E$.
Namely we show that the Hausdorff dimension of the singular set is strictly
smaller than $n-1$.
