Trace theory for Sobolev mappings into a manifold

Abstract

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 ${\pi }_2(\mathcal{N})\simeq \cdots \simeq {\pi }_{\lfloor p-1\rfloor }(\mathcal{N})\simeq \{0\}$. We present several open problems connected to the extension problem.

15 pages

Cited by in corpus (5)

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• On the existence of extensions for manifold-valued Sobolev maps on perforated domains
• Limiting behavior of minimizing $p$-harmonic maps in 3d as $p$ goes to $2$ with finite fundamental group
• Singular extension of critical Sobolev mappings under an exponential weak-type estimate
• Quantitative characterization of traces of Sobolev maps