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

Guillaume Valette

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.

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

Abstract

We give characterizations of the bounded subanalytic submanifolds of 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 , we prove that the Neumann regular elements of are dense in if and only if is connected at almost every boundary point. In the case large, we show that the Neumann regular Lipschitz elements of are dense in if and only if is connected at every boundary point. The proof involves the construction of Lipschitz Neumann regular partitions of unity, which is of independent interest.
Paper Structure (7 sections, 7 theorems, 45 equations, 2 figures)

This paper contains 7 sections, 7 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Some notations and conventions
  3. Subanalytic sets.
  4. Density of $\mathscr{C}^\infty(\mathbb{R}^n)$
  5. Density of Neumann regular functions
  6. The case $p$ small
  7. The case $p$ large.

Key Result

Corollary 1

For $p\in [1,2]$, the space is dense in $W^{1,p}(M)$ if and only if $M$ is connected at $\mathcal{H}^{m-1}$ almost every boundary point.

Figures (2)

  • Figure 1: The germ of ${\Omega}\subset \mathbb{R}^3$ at the origin.
  • Figure 2: The set $Y_t$ is the interior of this polyhedron.

Theorems & Definitions (18)

  • Corollary 1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • Example 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 8 more