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

We give characterizations of the bounded subanalytic ${\mathcal{C}}^{\mathrm{\infty }}$ 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 ${\mathcal{C}}^{\mathrm{\infty }}(\bar{M})$ 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 ${\mathcal{C}}^{\mathrm{\infty }}(M)$ 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.