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Laplacien discret d'un 2-complexe simplicial

Abstract : This thesis gives a general framework for Laplacians defined in terms of the combinatorial structure of a simplicial complex. More precisely, we introduce the notion of orientated triangle face in a connected, orientated and locally finite graph. This structure of a 2-simplicial complex allows to define our discrete Laplacian which acts on the triplets of functions, 1-forms and 2-forms. In this context, we are interested in studying the essential self-adjointness of our Laplacian. Thus, we introduce the geometrical hypothesis of x-completeness on triangulations to ensure the essential self-adjointness of the Gauß-Bonnet operator. This thesis deals also with questions of specral theory of finite triangulations on our Laplacian. We find an estimate for the upper Laplacian spectral gap in a triangulation of a complete graph for which we generalize the definition of the Cheeger constant which gives us an upper bound. Moreover, we obtain a lower bound of this estimate by the first non-zero eigenvalue of the discrete Laplacian defined on the space of functions on the vertices.
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Submitted on : Saturday, May 26, 2018 - 11:30:13 PM
Last modification on : Tuesday, September 21, 2021 - 4:06:03 PM
Long-term archiving on: : Monday, August 27, 2018 - 2:01:31 PM


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Yassin Chebbi. Laplacien discret d'un 2-complexe simplicial . Mathématiques [math]. Université de Nantes, Faculté des sciences et des techniques.; Université de Carthage (Tunisie), 2018. Français. ⟨tel-01800569⟩



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