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We study the regularity of the interface for optimal energy configurations offunctionals involving bulk energies with an additional perimeter penalizationof 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 wellknown. 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 strictlysmaller than $n-1$.

Regularity Results for an Optimal Design Problem with lower order terms