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 ${\mathcal{C}}_0^{\infty }(M)$ 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 $B(x_0,\epsilon )\cap M$ is connected for every $x_0\in \overline{M}$ and $\epsilon >0$ small, we show that ${\mathcal{C}}^{\infty }(\overline{M})$ is dense in $W^{1,p}(M)$ for all such $p$. This yields some duality results between $W^{1,p}(\Omega ,\partial \Omega )$ and $W^{-1,p^{\prime }}(\Omega )$ 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 Poincaré-Friedrichs' inequality. We finish with a generalization of Morrey's Embedding Theorem.

Final version