On Sobolev spaces of bounded subanalytic manifolds

TL;DR

This paper investigates Sobolev spaces on bounded subanalytic manifolds, establishing density, duality, and embedding results, and applying these to solve PDEs and extend classical inequalities in this geometric setting.

Contribution

It provides new density theorems, duality results, and embeddings for Sobolev spaces on subanalytic manifolds, generalizing classical analysis tools to singular geometric contexts.

Findings

Density of smooth functions in Sobolev spaces on subanalytic manifolds

Duality and reflexivity of Sobolev spaces in this setting

Sobolev and Poincaré inequalities, and embeddings for subanalytic manifolds

Abstract

We focus on the Sobolev spaces of bounded subanalytic submanifolds of $\mathbb{R}^n$. We prove that if $M$ is such a manifold then the space $\mathscr{C}_0^\infty(M)$ is dense in $W^{1,p}(M,\partial M)$ (the kernel of the trace operator) for all $p\le p_M$, where $p_M$ is the codimension in $M$ of the singular locus of $\overline{M}\setminus M$. In the case where $M$ is normal, i.e. when $B(x_0,\varepsilon)\cap M$ is connected for every $x_0\in\overline{M}$ and $\varepsilon>0$ small, we show that $\mathscr{C}^\infty(\overline{M})$ is dense in $W^{1,p}(M)$ for all such $p$. This yields some duality results between $W^{1,p}(\Omega,\partial \Omega)$ and $W^{-1,p'}(\Omega)$ in the case where $1< p\le p_\Omega$ and $\Omega$ is a bounded subanalytic open subset of $\mathbb{R}^n$, and consequently that $W^{1,p}(\Omega,\partial \Omega)$ is reflexive for such $p$. As a byproduct, we deduce uniqueness of the (weak) solution of the Dirichlet problem associated with the Laplace equation. We then prove a version of Sobolev's Embedding Theorem for subanalytic bounded manifolds, show Gagliardo-Nirenberg's inequality (for all $p\in [1,\infty)$), and derive some versions of Poincar\'e-Friedrichs' inequality. We finish with a generalization of Morrey's Embedding Theorem.