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Optimisation du spectre du Laplacien avec conditions de Dirichlet et Neumann dans R² et R³

Abstract : The optimization of Laplacian eigenvalues is a classical problem. In fact, at the end of the nineteenth century, Lord Rayleigh conjectured that the first eigenvalue with Dirichlet boundary condition is minimized by a disk. This problem received a lot of attention since this first study and research possibilities are numerous: various conditions, geometrical constraints added, existence, description of optimal shapes... In this document we restrict us to Dirichlet and Neumann boundary conditions in R^2 and R^3. We begin with a state of the art. Then we focus our study on disks and balls. Indeed, these are some of the only shapes for which it is possible to explicitly and relatively easily compute the eigenvalues. But we show in one of the main result of this document that they are not minimizers for most eigenvalues. Finally we take an interest in the possible numerical experiments. Since we can do very few theoretical computations, it is interesting to get numerical candidates. Then we can deduce some theoretical working assumptions. With this in mind we give some keys to understand our numerical method and we also give some results obtained.
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Submitted on : Tuesday, February 2, 2016 - 6:02:06 PM
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Amandine Berger. Optimisation du spectre du Laplacien avec conditions de Dirichlet et Neumann dans R² et R³. Equations aux dérivées partielles [math.AP]. Université de Neuchâtel (Suisse), 2015. Français. ⟨NNT : 2015GREAM036⟩. ⟨tel-01266486⟩

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