## Trace theory for Sobolev mappings into a manifold

### Abstract

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