A Regularity Result in a Shape Optimization Problem with Perimeter

We consider optimal shapes of the functional

among all the measurable subsets

\Omega Ω

of a given open bounded domain

D \subset \mathbf{R}^d D ⊂ R d

where

J(\Omega) J ( Ω )

is some Dirichlet energy associated with

\Omega Ω

,

P(\Omega) P ( Ω )

and

|\Omega| ∣ Ω ∣

being respectively the perimeter and the Lebesgue measure of

\Omega Ω

. 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 J ≡ 0

). We also consider the minimization of

\mathcal{E}_0 E 0

with Lebesgue measure constraint

|\Omega| = m ∣ Ω ∣ = m