Quantitative characterization of traces of Sobolev maps

Abstract

We give a quantitative characterization of traces on the boundary of Sobolev maps in ${\dot{W}}^{1,p}(\mathcal{M},\mathcal{N})$, where $\mathcal{M}$ and $\mathcal{N}$ are compact Riemannian manifolds, $\partial \mathcal{M}\ne \varnothing$: the Borel-measurable maps $u:\partial \mathcal{M}\to \mathcal{N}$ that are the trace of a map $U\in {\dot{W}}^{1,p}(\mathcal{M},\mathcal{N})$ are characterized as the maps for which there exists an extension energy density $w:\partial \mathcal{M}\to [0,\infty ]$ that controls the Sobolev energy of extensions from $\lfloor p-1\rfloor$-dimensional subsets of $\partial \mathcal{M}$ to $\lfloor p\rfloor$-dimensional subsets of $\mathcal{M}$.

arXiv comment26 pages, minor corrections

References in corpus (5)

• Trace and extension theorems for Sobolev-type functions in metric spaces
• Strong density for higher order Sobolev spaces into compact manifolds
• Trace theory for Sobolev mappings into a manifold
• Lifting in compact covering spaces for fractional Sobolev mappings
• Uniform boundedness principles for Sobolev maps into manifolds