THE $L^p$ POISSON-NEUMANN PROBLEM AND ITS RELATION TO THE NEUMANN PROBLEM
Résumé
We introduce the Lp Poisson-Neumann problem for an uniformly elliptic operator L = -divA∇ in divergence form in a bounded 1-sided Chord Arc Domain Ω, which considers solutions to Lu = h -div F in Ω with zero Neumann data on the boundary for h and F in some tent spaces. We give different characterizations of solvability of the Lp Poisson-Neumann problem and its weaker variants, and in particular, we show that solvability of the weak Lp Poisson-Neumann probelm is equivalent to a weak reverse Hölder 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 Lp Neumann problem from Kenig and Pipher [KP93] by obtaining an extrapolation result on the Poisson-Neumann problem.
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