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

TL;DR

This work addresses trace and extension problems for manifold-valued Sobolev mappings at the endpoint $p=1$, introducing a direct, constructive approach to surjectivity of the trace for $\dot{W}^{1,1}(M,N)$ and providing explicit extension operators. A dyadic cube tiling is used to glue boundary data into the interior via a BV-join, which is then smoothed to a ${\dot{W}}^{1,1}$-map with quantitative energy bounds; these bounds connect the interior energy to boundary double-integrals involving the target's intrinsic distance. The results include trace inequalities on balls and manifolds, and extend to half-spaces and general manifold domains, yielding BBM-type energy characterizations for integrable manifold-valued maps. The framework removes some compactness assumptions on $N$ and offers explicit, quantitative tools for geometric analysis and variational problems involving singular Sobolev maps into manifolds.

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.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof : Proof of Proposition \ref{['proposition_b_unique']}
• Proposition 2.2
• proof : Proof of Proposition \ref{['prop:tobezerogeneral']}
• Proposition 2.3
• proof : Proof of Proposition \ref{['prop:ballBBMII']}