  Given a bounded subanalytic submanifold of $\mathbb{R}^n$, possibly admitting
singularities within its closure, we study the cohomology of $L^p$ differential
forms having an $L^p$ exterior differential (in the sense of currents) and
satisfying Dirichlet or Neumann condition. We show an $L^p$ Hodge decomposition
theorem, an $L^p$ de Rham theorem, as well as a Lefschetz duality theorem
between $L^p$ and $L^{p'}$ forms (with $\frac{1}{p}+\frac{1}{p'}=1$) in the
case where $p$ is large or close to $1$. This is achieved by proving that de
Rham's pairing between complementary $L^p$ differential forms induces a pairing
between cohomology classes which is nondegenerate (for such $p$). The main
difficulty to carry it out is to show the density (in Sobolev spaces of
differential forms) of forms that vanish near some singularities and are smooth
up to the closure of the underlying manifold in the case $p$ large (Theorem
4.1). This result, which is of independent interest, also makes it possible to
give a trace theorem that leads us to some explicit characterizations of
Dirichlet and Neumann conditions in terms of traces and residues.
