Investigation of Degree Theory of Orlicz-sobolev Mappings between Riemannian Manifolds

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Abedin Nejad, Mohammad Mohsen | 2013

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 46899 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ranjbar Motlagh, Alireza
Abstract:
In this thesis, we are going to study Sobolev classes of weakly differentiable mappings, specially Orlicz-Sobolev class, between compact Riemannian manifolds without boundary. These mappings posses less regularity than the mappings in the borderline case. Two major themes we investigate are smooth approximation of these mappings and the integrability of the Jacobian determinant. We impose no topological restrictions on manifolds in the approximation issue. We characterize classes of weakly differentiable mappings satisfying the approximation property and claim the Orlicz-Sobolev class is in these classes. The importance of our approach is that we are able to find tiny sets on which Sobolev mappings are continuous. These tiny sets are called web like structure of the manifold in the domain associated with the given mapping. In the integrability of the Jacobian issue, we demonstrate if the target manifold admits cohomology groups like n-sphere, it is possible that the Jacobian is not integrable. Then, we are able to study degree theory of these mappings, specially Orlicz-Sobolev class
Keywords:
Orlicz-Sobolev Space ; Cartan Form ; Rational Homology Sphere ; Distributional Jacobian

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