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Limites d'ensembles quasiminimaux et existence d'ensembles minimaux sous contraintes topologiques

Abstract : In the nineteenth century, Joseph Plateau described the geometrical disposition of soap films. Their shape is explained by their tendency to minimize their area to a reach an equilibrium. Mathematicians have abstracted the concept of "surface with minimal area spanning a boundary" and have named the corresponding minimization problem, "Plateau problem". It has different formulations corresponding to as many ways of defining the class of "surfaces spanning a given boundary" and the "area" to minimize. In this thesis, we generalize to quasiminimizing sequences, the weak limit of minimizing sequences introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi. We show that a weak limit of quasiminimal sets is quasiminimal. This result is analogous to the limiting theorem of David for the local Hausdorff convergence. Our proof is inspired by David's one while being simpler. We deduce a direct method to prove existence of solutions to various Plateau problem, even with a free boundary. We apply it then to two variants of the Reifenberg problem (fixed or free boundary) for all coefficient groups. Furthermore, we propose a structure to build Federer-Fleming projections as well as a new estimate on the choice of projection centers.
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Camille Labourie. Limites d'ensembles quasiminimaux et existence d'ensembles minimaux sous contraintes topologiques. Géométrie métrique [math.MG]. Université Paris-Saclay, 2020. Français. ⟨NNT : 2020UPASS008⟩. ⟨tel-02896598⟩

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