
Journal of Convex Analysis 14 (2007), No. 4, 785806 Copyright Heldermann Verlag 2007 A Regularity Result in a Shape Optimization Problem with Perimeter Nicolas Landais Dep. de Mathématiques, Ecole Normale Supérieure de Cachan, Campus de KerLann, 35170 Bruz, France nicolas.landais@bretagne.enscachan.fr [Abstractpdf] 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 Lipschitzcontinuous. 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 $\Omega = 0$. [ Fulltextpdf (196 KB)] for subscribers only. 