Extension Operators for Fractional Sobolev Spaces on Lipschitz Submanifolds

TL;DR

This paper extends the theory of extension operators for fractional Sobolev spaces from Lipschitz domains to Lipschitz subsets of compact Lipschitz submanifolds, with explicit estimates crucial for numerical applications.

Contribution

It adapts existing extension operator constructions to Lipschitz submanifolds, providing explicit dependence on domain size for improved numerical analysis applications.

Findings

Constructed extension operators for Lipschitz submanifolds.

Provided explicit estimates on the dependence of extension constants.

Applicable to geometry simplification in numerical analysis.

Abstract

A well-known result is that any Lipschitz domain is an extension domain for $W^{s,p}$. This paper extends this result to Lipschitz subsets of compact Lipschitz submanifolds of $\mathbb{R}^n$. We adapt the construction of an extension operator for Lipschitz domains in arXiv:1104.4345v3 to manifolds via local coordinate charts. Furthermore, the dependence on the size of the extension domain is explicit in all estimates. This result is motivated by applications in numerical analysis, most notably geometry simplification, where the explicit dependence of the continuity constant on the domain size is essential.