Extensions of traces for Sobolev mappings into manifolds at the endpoint $p=1$

TL;DR

This paper develops new methods for extending Sobolev mappings from manifolds with boundary into complete manifolds at the critical exponent p=1, using tiling and dyadic cubes, with applications to energy characterization.

Contribution

It provides direct proofs and constructions for trace and extension theorems for Sobolev mappings at p=1, a critical case previously less understood.

Findings

Constructed explicit extension operators for Sobolev maps at p=1

Characterized the asymptotic behavior of the $L^1$-energy of mappings

Applicable to manifolds with boundary, halfspaces, and strips

Abstract

We give direct proofs and constructions of the trace and extension theorems for Sobolev mappings in $W^{1, 1} (M, N)$, where $M$ is Riemannian manifold with compact boundary $\partial M$ and $N$ is a complete Riemannian manifold. The analysis is also applicable to halfspaces and strips. The extension is based on a tiling the domain of the considered applications by suitably chosen dyadic cubes to construct the desired extension. Along the way, we obtain asymptotic characterizations of the $L^1$-energy of mappings.