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 $p\le \mathbf {p_{_M}}$, where $\mathbf {p_{_M}}$ is the codimension in M of the singular locus of ${\overline{M}}{\setminus } M$ (which is always at least 2). In the case where M is normal, i.e. when $\textbf{B}(x_0,\varepsilon )\cap M$ 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 $1< p\le \mathbf {p_{_{\Omega }}}$ and ${\Omega }$ is a bounded subanalytic open subset of $\mathbb {R}^n$. 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.