On Sobolev spaces of bounded subanalytic manifolds

We focus on the Sobolev spaces of bounded subanalytic submanifolds of \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . We prove that if M is such a manifold then the space \(\mathscr {C}_0^\infty (M)\) ${\mathcal{C}}_0^{\infty }(M)$ is dense in \(W^{1,p}(M,\partial M)\) $W^{1,p}(M,\partial M)$ (the kernel of the trace operator) for all \(p\le \mathbf {p_{_M}}\) $p\le {\mathbf{p}}_{{}_{\mathbf{M}}}$ , where \(\mathbf {p_{_M}}\) ${\mathbf{p}}_{{}_{\mathbf{M}}}$ is the codimension in M of the singular locus of \( {\overline{M}}{\setminus } M\) $\overline{M}\backslash M$ (which is always at least 2). In the case where M is normal, i.e. when \(\textbf{B}(x_0,\varepsilon )\cap M\) $\mathbf{B}(x_0,\epsilon )\cap M$ is connected for every \(x_0\in {\overline{M}}\) $x_0\in \overline{M}$ and \(\varepsilon >0\) $\epsilon >0$ small, we show that \(\mathscr {C}^\infty ( {\overline{M}})\) ${\mathcal{C}}^{\infty }(\overline{M})$ is dense in \(W^{1,p}(M)\) $W^{1,p}(M)$ for all such p. This yields some duality results between \(W^{1,p}({\Omega },\partial {\Omega })\) $W^{1,p}(\mathrm{\Omega },\partial \mathrm{\Omega })$ and \(W^{-1,p'}({\Omega })\) $W^{-1,p^{\prime }}(\mathrm{\Omega })$ in the case where \(1< p\le \mathbf {p_{_{\Omega }}}\) $1<p\le {\mathbf{p}}_{{}_{\mathrm{\Omega }}}$ and \({\Omega }\) $\mathrm{\Omega }$ is a bounded subanalytic open subset of \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . 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 \(p\in [1,\infty )\) $p\in [1,\infty )$ ), and derive some versions of Poincaré–Friedrichs’ inequality. We finish with a generalization of Morrey’s Embedding Theorem.