# Méthodes spectrales et théorie des cristaux liquides

Abstract : This thesis aims at studying two kinds of problems. The main aspect of this work is the analysis of the lowest eigenvalue $\la_1(B,\A)$ of the magnetic Neumann Laplacian with variable magnetic field $(i\nabla+B\A)^2$. More precisely, in 2D, we prove a two terms asymptotics of $\la_1(B,\A)$ when $B$ tends to infinity and we prove localizations results for the attached eigenfunctions. In 3D, we give some uniform spectral estimates for a family of magnetic fields appearing in the liquid crystals theory. We also prove an upper bound (with three terms) for the lowest eigenvalue in the generic case of the variable magnetic field. The other part of this work concerns the analysis of the Landau-de Gennes functional introduced to study the liquid crystals. We prove the existence of a critical tempertaure (when the elastic coefficients become large) which permits to determine if the liquid crystal is nematic or smectic.
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https://tel.archives-ouvertes.fr/tel-00424859
Contributor : Nicolas Raymond <>
Submitted on : Monday, October 19, 2009 - 9:13:10 AM
Last modification on : Thursday, January 11, 2018 - 6:12:18 AM
Long-term archiving on : Tuesday, June 15, 2010 - 10:47:51 PM

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• HAL Id : tel-00424859, version 1

### Citation

Nicolas Raymond. Méthodes spectrales et théorie des cristaux liquides. Mathématiques [math]. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00424859⟩

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