$L^p$ Hodge theory for bounded subanalytic manifolds

Abstract

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.