Trace theory for Sobolev mappings into a manifold
Résumé
We 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.
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