On the Laplace equation on bounded subanalytic manifolds

We prove a trace formula for integration by parts on subanalytic bounded submanifolds of ${\mathbb{R}}^n$, possibly non closed. We also establish density results for ${\mathbf{W}}_{\mathrm{\nabla }}^{1,p}(M)$, $M$ bounded subanalytic manifold, which is the space of the $L^p$ tangent vector fields $v$ on $M$ for which $\mathrm{\nabla }v$ is $L^p$, where $\mathrm{\nabla }$ is the divergence operator. We derive from these results some theorems of existence and uniqueness of solutions of the Laplace equation with Dirichlet and Neumann boundary type conditions. We then study the $p$-Laplace equation, for $p\in [1,\mathrm{\infty })$ large.