On Sobolev spaces of bounded 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.