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Theses

Existence et régularité des formes optimales pour des problèmes d'optimisation spectrale

Abstract : In this thesis, we study the existence and the regularity of optimal shapes for some spectral optimization problems involving an elliptic operator with Dirichlet boundary condition.First of all, we consider the problem of minimizing the principal eigenvalue of an operator with bounded drift under inclusion and volume constraints.Whether the drift is fixed or not, this problem admits solutions among the class of quasi-open sets, and if the drift is furthermore the gradient of a Lipschitz continuous function, then the solutions are open sets and C^{1,alpha}-regular except on a set of exceptional points.Next, we study in dimension two the regularity of the solutions to a multi-phase optimization problem for the first eigenvalue of the Dirichlet Laplacian.Finally, we focus on the optimal sets for the sum of the first k eigenvalues of an operator in divergence form. We prove that the first k eigenfunctions on an optimal set are Lipschitz continuous so that the optimal sets are open sets, and we then study the regularity of the boundary of the optimal sets.
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Baptiste Trey. Existence et régularité des formes optimales pour des problèmes d'optimisation spectrale. Algèbres d'opérateurs [math.OA]. Université Grenoble Alpes [2020-..], 2020. Français. ⟨NNT : 2020GRALM019⟩. ⟨tel-03141381⟩

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