  We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic
operator $L=-\rm{div }A\nabla$ in divergence form in a bounded 1-sided Chord
Arc Domain $\Omega$, which considers solutions to $Lu=h-\rm{div}\vec{F}$ in
$\Omega$ with zero Neumann data on the boundary for $h$ and $\vec F$ in some
tent spaces. We give different characterizations of solvability of the $L^p$
Poisson-Neumann problem and its weaker variants, and in particular, we show
that solvability of the weak $L^p$ Poisson-Neumann probelm is equivalent to a
weak reverse H\"older inequality. We show that the Poisson-Neumman problem is
closely related to the $L^p$ Neumann problem, whose solvability is a
long-standing open problem. We are able to improve the extrapolation of the
$L^p$ Neumann problem from Kenig and Pipher by obtaining an extrapolation
result on the Poisson-Neumann problem.
