Trace operators on bounded subanalytic manifolds

We prove that if \(M\subset {\mathbb {R}}^n\) $M\subset {\mathbb{R}}^n$ is a bounded subanalytic submanifold of \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ such that \({\textbf{B}}(x_0,\varepsilon )\cap M\) $\mathbf{B}(x_0,\epsilon )\cap M$ is connected for every \(x_0\in {{\overline{M}}}\) $x_0\in \overline{M}$ and \(\varepsilon >0\) $\epsilon >0$ small, then, for \(p\in [1,\infty )\) $p\in [1,\infty )$ sufficiently large, the space \({\mathscr {C}}^\infty ( {{\overline{M}}})\) ${\mathcal{C}}^{\infty }(\overline{M})$ is dense in the Sobolev space \(W^{1,p}(M)\) $W^{1,p}(M)$ . We also show that for p large, if \(A\subset {{\overline{M}}}\setminus M\) $A\subset \overline{M}\backslash M$ is subanalytic then the restriction mapping \( {\mathscr {C}}^\infty ( {{\overline{M}}})\ni u\mapsto u_{|A}\in L^p(A)\) ${\mathcal{C}}^{\infty }(\overline{M})\ni u\mapsto u_{|A}\in L^p(A)$ is continuous (if A is endowed with the Hausdorff measure), which makes it possible to define a trace operator, and then prove that compactly supported functions are dense in the kernel of this operator. We finally generalize these results to the case where our assumption of connectedness at singular points of \( {{\overline{M}}}\) $\overline{M}$ is dropped.