Solvability of the divergence equation implies John via Poincaré inequality

2013/07/04 by Renjin Jiang, Jiang, Renjin, Aapo Kauranen +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Nonlinear Partial Differential Equations #math.AP #math.CA

paper · pdf · doi:10.48550/arxiv.1307.1340

arxiv created 2013/07/04 · arxiv updated 2013/07/05

Abstract

Let Ω⊂ \rr² be a bounded simply connected domain. We show that, for a fixed (every) p∈ (1,\fz), the divergence equation div v=f is solvable in W^1,p₀(Ω)² for every f∈ L^p₀(Ω), if and only if Ω is a John domain, if and only if the weighted Poincaré inequality ∫_Ω|u(x)-u_Ω|^q dx≤ C∫_Ω|∇ u(x)|^q\dist(x,∂ Ω)^q dx holds for some (every) q∈ [1,\fz). In higher dimensions similar results are proved under some additional assumptions on the domain in question.

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