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

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.