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Trace theory for Sobolev mappings into a manifold

By Petru Mironescu and Jean Van Schaftingen


To appear at Annales de la Faculté des Sciences de ToulouseWe review the current state of the art concerning the characterization of traces of the spaces $W^{1,p}({\mathbb B}^{m-1}\times (0,1), {\mathcal N})$ of Sobolev mappings with values into a compact manifold ${\mathcal N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p$ < $m$ is an integer and the homotopy group $\pi_p({\mathcal N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $\pi_1({\mathcal N})$ is finite and $\pi_2({\mathcal N})=\cdots=\pi_{\lfloor p \rfloor}({\mathcal N})\simeq\{ 0\}$. We present several open problems connected to the extension problem

Topics: maps with valued into manifolds, trace theory, obstructions, Sobolev spaces, MSC 2010 classification: 46T10 (46E35, 58D15), [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA]
Publisher: HAL CCSD
Year: 2020
OAI identifier: oai:HAL:hal-02431628v2
Provided by: HAL-UJM
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