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We investigate the problem of Poincar\'e duality for $L^p$ differential formson bounded subanalytic submanifolds of $\mathbb{R}^n$ (not necessarilycompact). 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 thecase where $p$ is large, we show that $L^p$ cohomology is dual to intersectionhomology. As a consequence, we can deduce that the $L^p$ cohomology isPoincar\'e dual to $L^q$ cohomology, if $p$ and $q$ are H\"older conjugate toeach other and $p$ is sufficiently large.

Poincar\'e duality for $L^p$ cohomology on subanalytic singular spaces