An asymtotic sharp Sobolev regularity for planar infinity harmonic functions

2018/06/06 by Herbert Koch, Koch, Herbert, Yi Ru-Ya Zhang +3
Mathematics · #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35J60

paper · pdf · doi:10.48550/arxiv.1806.01982

arxiv created 2018/06/06 · arxiv updated 2018/06/07

Abstract

Given an arbitrary planar ∞-harmonic function u, for each α>0 we establish a quantitative local W^1,2-estimate of |Du|^α, which is sharp as α→0. We also show that the distributional determinant of u is a Radon measure enjoying some quantitative lower and upper bounds. As a by-product, for each p>2 we obtain some quantitative local W^1,p-estimates of u, and consequently, an L^p-Liouville property for ∞-harmonic functions in whole plane.

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