Trace operators on bounded subanalytic manifolds

TL;DR

This paper establishes density and trace properties of smooth functions in Sobolev spaces on bounded subanalytic manifolds, extending classical results to singular and disconnected cases.

Contribution

It proves density of smooth functions in Sobolev spaces and constructs trace operators on bounded subanalytic manifolds, including cases with disconnected or singular boundaries.

Findings

Density of $C^ abla(ar{M})$ in $W^{1,p}(M)$ for large $p$

Continuity of restriction maps and trace operators on subanalytic sets

Extension of results to manifolds with disconnected or singular boundaries

Abstract

We prove that if $M\subset \mathbb{R}^n$ is a bounded subanalytic submanifold of $\mathbb{R}^n$ such that $B(x_0,\epsilon)\cap M$ is connected for every $x_0\in\overline{M}$ and $\epsilon>0$ small, then, for $p\in [1,\infty)$ sufficiently large, the space $C^\infty(\overline{M})$ is dense in the Sobolev space $W^{1,p}(M)$. We also show that for $p$ large, if $A\subset \overline{M}\setminus M$ is subanalytic then the restriction mapping $ C^\infty(\overline{M})\ni u\mapsto u_{|A}\in L^p(A)$ is continuous (if $A$ is endowed with the Hausdorff measure), which makes it possible to define a trace operator, and then prove that compactly supported functions are dense in the kernel of this operator. We finally generalize these results to the case where our assumption of connectedness at singular points of $\overline{M}$ is dropped.