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

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.

💡 Research Summary

The paper investigates the density of Neumann‑regular smooth functions in Sobolev spaces on bounded subanalytic C^∞ manifolds M⊂ℝⁿ. A “Neumann‑regular” function is smooth at (almost) every boundary point and has a gradient tangent to the boundary (i.e., orthogonal to the outward normal). The authors give a complete topological characterization of when such functions are dense in W^{1,p}(M).

First, they recall that subanalytic sets form an o‑minimal structure, admit cell decompositions, and have a stratification into smooth pieces. The boundary ∂M is split into a Lipschitz‑regular part Γ (the smooth part) and a singular part δM\Γ of codimension at least two. A point x∈δM is said to be “connected” if for sufficiently small ε, the intersection B(x,ε)∩M is connected.

The main results are twofold.

1. **Small‑p regime (p∈