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Existence et régularité pour des problèmes d'optimisation de formes

Abstract : The first part of this work deals with the existence and the Lipschitz regularity of the state function in a shape optimization problem. This problem consists in fin ding an open subset of R[exponent]N with a prescribed measure, which minimizes the energy associated to the Dirichlet problem on such sets. We start by using a variational approach to get the existence result. Then, we introduce an approximated variational problem and prove that its solutions are regular since the EulerLagrange equation is a semi-linear partial differential equation. Provided that those solutions do not change their signs, we show that they are uniformly Lipschitz regular and therefore, converges to a Lipschitz solution to the initial variational problem. Moreover, the set where this state function does not vanish is a solution to the considered shape optimisation problem. The second part is devoted to the study, in 2 and 3 dimensions, of the continuity, with respect to the variations of a bounded domain (in Hausdorff sense), of the solutions of the biharmonic problem with homogenous Dirichlet boundary conditions. We first give a necessary and sufficient condition on the domaine under which the continuity holds. Then we bring out some simple and sufficient conditions on the boundary of the domain. Pinally, we give an explicit example of homogenization of the gradient in 2 dimension al case.
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Submitted on : Thursday, March 29, 2018 - 10:58:08 AM
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  • HAL Id : tel-01747370, version 1



Mohammed Hayouni. Existence et régularité pour des problèmes d'optimisation de formes. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 1997. Français. ⟨NNT : 1997NAN10089⟩. ⟨tel-01747370⟩



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