A Regularity Result in a Shape Optimization Problem with Perimeter

We consider optimal shapes of the functional $$\mathcal{E}_\lambda(\Omega) = J(\Omega) + P(\Omega) + \lambda ||\Omega| - m|$$ among all the measurable subsets $\Omega$ of a given open bounded domain $D \subset \mathbf{R}^d$ where $J(\Omega)$ is some Dirichlet energy associated with $\Omega$, $P(\Omega)$ and $|\Omega|$ being respectively the perimeter and the Lebesgue measure of $\Omega$. We prove here that for some optimal shape, the state function associated with the Dirichlet energy is Lipschitz-continuous. Then we deduce the same regularity properties for the boundary of the optimal shape as in the pure isoperimetric problem (case $J \equiv 0$). We also consider the minimization of $\mathcal{E}_0$ with Lebesgue measure constraint $|\Omega| = 0$