Abstract

We consider optimal shapes of the functional Eλ(Ω)=J(Ω)+P(Ω)+λ∣∣Ω∣−m∣\mathcal{E}_\lambda(\Omega) = J(\Omega) + P(\Omega) + \lambda ||\Omega| - m| among all the measurable subsets Ω\Omega of a given open bounded domain D⊂RdD \subset \mathbf{R}^d where J(Ω)J(\Omega) is some Dirichlet energy associated with Ω\Omega, P(Ω)P(\Omega) and ∣Ω∣|\Omega| being respectively the perimeter and the Lebesgue measure of Ω\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≡0J \equiv 0). We also consider the minimization of E0\mathcal{E}_0 with Lebesgue measure constraint ∣Ω∣=m|\Omega| = m

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Suggested citation

N. Landais. “A Regularity Result in a Shape Optimization Problem with Perimeter.” Journal of Convex Analysis 14 (2007), No. 4, 785–806. https://doi.org/10.68381/jca14045

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press.