On the local character of the extension of traces for Sobolev mappings

Abstract

We prove that a mapping $u:{\mathcal{M}}^{\prime }\to \mathcal{N}$, where ${\mathcal{M}}^{\prime }$ and $\mathcal{N}$ are compact Riemannian manifolds, is the trace of a Sobolev mapping $U:{\mathcal{M}}^{\prime }\times [0,1)\to \mathcal{N}$ if and only if it is on some open covering of ${\mathcal{M}}^{\prime }$. In the global case where $\mathcal{M}$ is a compact Riemannian manifold with boundary, this implies that the analytical obstructions to the extension of a mapping $u:\partial \mathcal{M}\to \mathcal{N}$ to some Sobolev mapping $U:\mathcal{M}\to \mathcal{N}$ are purely local.

16 pages