We study multiple valued functions with values in a Hilbert space. We introduce a possibledefinition of Sobolev spaces and the rightful notion of p energy. We prove the existence of pminimizers. Then we consider two-valued real functions of two variables which are stationarywith respect to both domain and range transformations. We prove their local Lipschitzcontinuity and use it to establish strong convergence in W1,2 to their unique blow-up at anypoint. We claim that the branch set of any such function consists of finitely many real analyticcurves meeting at nod points with equal angles. We also provide an example showing thatstationarity with respect to domain transformations only does not imply continuity.In a second part, we prove that there does not exist a uniformly continuous retractionfrom the space of continuous vector fields onto the subspace of vector fields whose divergencevanishes in the distributional sense. We then generalise this result using the concept of mcharges on any subset X _ Rn satisfying a mild geometric condition, there is no uniformlycontinuous representation operator for mcharges in X.
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-01015943 |
| Date | 07 January 2014 |
| Creators | Bouafia, Philippe |
| Publisher | Université Paris Sud - Paris XI |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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