Abstract |
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Sobolev spaces between manifolds are a natural tool to study variational problems for maps between manifolds, arising in nonlinear physical models or in geometry. For k≥1 and p≥1, the Sobolev space Wk,p(M,N) between the manifolds M and N can be defined by the set of all maps u:M→N such that the composition map ιou belongs to the Sobolev space Wk,p(M,ℝν), where ιϵCk(N,ℝν) is an isometric embedding of the target manifold N. However, in the higher-order case k≥2, this definition is not intrinsic: it depends on the choice of the embedding. In the first-order case k=1, the definition by embedding is also equivalent to the definition of Sobolev spaces into metric spaces. However, definitions of Sobolev spaces into metric spaces do not provide any notion of weak derivative and thus do not seem adapted to a further definition of higher-order Sobolev spaces by iteration. In the first part of this thesis, we therefore suggest a robust intrinsic definition of first-order Sobolev maps in which the weak derivative plays a central role. This new definition is equivalent to the definition by embedding and to the one of Sobolev maps into metric spaces. We also endow the Sobolev spaces with various intrinsic distances that induce the same topology and for which the space is complete. In the second part of this thesis, in the same spirit of the first order, we define and study intrinsic higher-order weak derivatives and Sobolev spaces. The previous new definition is not equivalent in general with the definition by embedding. In particular, if the manifolds are compact, the intrinsic space is a larger space than the one by embedding. In this setting, we investigate the problem of density of smooth maps. We prove that a necessary condition for the density of smooth maps in the intrinsic space Wk,p(M,N) is that the [kp]th homotopy group of N is trivial. |