  We investigate the problem of Poincar\'e duality for $L^p$ differential forms
on bounded subanalytic submanifolds of $\mathbb{R}^n$ (not necessarily
compact). We show that, when $p$ is sufficiently close to $1$ then the $L^p$
cohomology of such a submanifold is isomorphic to its singular homology. In the
case where $p$ is large, we show that $L^p$ cohomology is dual to intersection
homology. As a consequence, we can deduce that the $L^p$ cohomology is
Poincar\'e dual to $L^q$ cohomology, if $p$ and $q$ are H\"older conjugate to
each other and $p$ is sufficiently large.
