Quantitative characterization of traces of Sobolev maps

TL;DR

This paper provides a quantitative description of boundary traces of Sobolev maps between compact Riemannian manifolds, characterizing which boundary maps can be extended into the interior based on an extension energy density.

Contribution

It introduces a new characterization of Sobolev trace maps via an extension energy density, linking boundary data to interior Sobolev extension properties.

Findings

Characterization of Sobolev traces using extension energy density.

Extension energy density controls Sobolev extension from boundary subsets.

Provides a criterion for boundary maps to be 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, $\partial \mathcal{M} \neq \emptyset$: the Borel-measurable maps $u\colon \partial \mathcal M \to \mathcal{N}$ 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 $w \colon \partial \mathcal{M} \to [0,\infty]$ 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 $\mathcal{M}$.