Density of Neumann regular smooth functions in Sobolev spaces of subanalytic manifolds

TL;DR

This paper characterizes when Neumann regular smooth functions are dense in Sobolev spaces on subanalytic manifolds, revealing that boundary connectivity determines density depending on the integrability exponent p.

Contribution

It provides new characterizations of density of Neumann regular functions in Sobolev spaces on subanalytic manifolds, including the construction of Lipschitz partitions of unity.

Findings

Density depends on boundary connectivity and p value.

Neumann regular functions are dense if the manifold is connected at boundary points.

Construction of Lipschitz Neumann regular partitions of unity.

Abstract

We give characterizations of the bounded subanalytic $\mathscr{C}^\infty$ submanifolds $M$ of $\mathbb{R}^n$ for which the space of Neumann regular functions is dense in Sobolev spaces. By ``Neumann regular function'', we mean a function which is smooth at almost every boundary point and whose gradient is tangent to the boundary. In the case $p\in [1,2]$, we prove that the Neumann regular elements of $\mathscr{C}^\infty(\overline{M})$ are dense in $W^{1,p}(M)$ if and only if $M$ is connected at almost every boundary point. In the case $p$ large, we show that the Neumann regular Lipschitz elements of $\mathscr{C}^\infty(M)$ are dense in $W^{1,p}(M)$ if and only if $M$ is connected at every boundary point. The proof involves the construction of Lipschitz Neumann regular partitions of unity, which is of independent interest.