2015/05/20 by Simon Bortz, Steve Hofmann, Bortz, Simon +1
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1505.05419
openalex publication_date 2015/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present an alternative proof of a result of Kenig and Toro, which states that if Ω⊂ ℝn+1 is a two sided NTA domain, with Ahlfors-David regular boundary, and the log of the Poisson kernel associated to Ω as well as the log of the Poisson kernel associated to Ω\rm ext are in VMO, then the outer unit normal ν is in VMO . Our proof exploits the usual jump relation formula for the non-tangential limit of the gradient of the single layer potential. We are also able to relax the assumptions of Kenig and Toro in the case that the pole for the Poisson kernel is finite: in this case, we assume only that ∂Ω is uniformly rectifiable, and that ∂Ω coincides with the measure theoretic boundary of Ω a.e. with respect to Hausdorff Hn measure.