Problemes de régularité en optimisation de formes

Abstract : This work deals with regularity in shape optimization. Precisely we study the regularity of an open subset which minimizes the energy of Dirichlet's problem among all open sets with fixed measure and included in a big open set (the whole space for example). The first step is to study the regularity of the optimal state function (the solution of Dirichlet's problem on the minimal open subset): we show that, if the state function does not change its sign then it is locally Lipschitz (in the whole space, not only in the optimal open subset). The second step is to study the regularity of the boundary of the optimal open subset. If the state function is Lipschitz, we show that this open subset has finite perimeter. Finally, if the source term in Dirichlet's equation is non-negative, then the Laplacian of the state function is, on the boundary of the optimal open subset, equal to a constant multiplied by the Hausdorff measure of the boundary. This constant comes from the Euler-Lagrange equation. In a weak form this means that the normal derivative of the state function is constant on the boundary. That is what we expect: if we suppose regularity on the boundary we find this result. We can deduce from this that, away from the support of the source term, the free boundary is, except on a negligible subset, an analytic hypersurface.
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https://tel.archives-ouvertes.fr/tel-00002013
Contributor : Tanguy Briancon <>
Submitted on : Monday, November 25, 2002 - 2:24:58 PM
Last modification on : Friday, November 16, 2018 - 1:23:36 AM
Long-term archiving on : Friday, April 2, 2010 - 6:38:20 PM

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  • HAL Id : tel-00002013, version 1

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Tanguy Briançon. Problemes de régularité en optimisation de formes. Mathématiques [math]. Université Rennes 1, 2002. Français. ⟨tel-00002013⟩

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