Espaces de Sobolev à valeurs variétés

Details

Supervisors

Mironescu, Petru ; Ponce, Augusto

Faculty

Faculté des sciences

Degree label

Master [120] en sciences mathématiques, à finalité approfondie

Abstract

enIn this master thesis we review the current theory about Sobolev spaces into manifolds. These spaces are modelled upon classical Sobolev spaces, by constraining the functions to take their values into an embedded manifold. Their study is motivated by considerations from geometry and physics. The most common example is about a field of liquid crystals, which is described by the orientation of the crystals at each point, and for which it is therefore relevant to use maps with values to the sphere or the projective plane. More precisely, this work is devoted to results and tools concerning three problems raised by the study of those spaces, namely the problems of approximation, extension and lifting. We first motivate and explain each of these problems, before dealing with the most simple cases where one can bring back to classical theory, and showing a typical obstruction for each problem. We then consider more involved methods, including the method of good and bad cubes for approximation, the method of singular projection, the construction of analytical obstructions, and we finish by a review of some methods specific to the lifting problem.