Solvability of the divergence equation implies John via Poincaré inequality

Abstract

Let Ω⊂R2 be a bounded simply connected domain. We show that, for a fixed (every) p∈(1,∞), the divergence equation divv=f is solvable in W01,p(Ω)2 for every f∈L0p(Ω), if and only if Ω is a John domain, if and only if the weighted Poincaré inequality ∫Ω|u(x)−uΩ|qdx≤C∫Ω|∇u(x)|q dist (x,∂Ω)qdx holds for some (every) q∈[1,∞). This gives a positive answer to a question raised by Russ (2013) in the case of bounded simply connected domains. In higher dimensions similar results are proved under some additional assumptions on the domain in question.