New results for \(p\) harmonic functions
type: Item
instance of: scholarly article
title: New results for \(p\) harmonic functions
zbMATH Open document ID: 1241.35110
author: Kaj Nyström
author: John L. Lewis
published in: Pure and Applied Mathematics Quarterly
publication date: 2012-01-05T00:00:00Z
review text: Let \(G=\{(y^{\prime },y_{n})\in \mathbb{R}^{n-1}\times \mathbb{R}:y_{n}>\phi (y^{\prime })\}\), where \(\phi :\mathbb{R}^{n-1}\rightarrow \mathbb{R}\) is Lipschitz. Further, let \(p>1\), let \(w\in \partial G\) and \(r>0\), and suppose that \(u,v\) are positive \(p\)-harmonic functions on \(B(w,r)\cap G\) that vanish continuously on \(B(w,r)\cap \partial G\). It is then shown that there exist \(c\geq 1\) and \(\alpha \in (0,1)\), depending only on \(p\), \(n\) and \(\phi \), such that \(\left| \log (u/v)(y_{1})-\log (u/v)(y_{2})\right| \leq c(\left| y_{1}-y_{2}\right| /r)^{\alpha }\) whenever \(y_{1},y_{2}\in B(w,r/c)\cap G\). This implies a result of the authors [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 5, 765--813 (2007; Zbl 1134.31008)] concerning the boundary Harnack principle, and also implies the Hölder continuity up to the boundary of the ratio of two positive \(p\)-harmonic functions that vanish on a portion of a Lipschitz boundary. In addition the authors indicate applications to the Martin boundary for \(p\)-harmonic functions and to free boundary regularity questions.
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 31C35
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5995324
zbMATH Keywords: \(p\)-harmonic function
zbMATH Keywords: boundary behaviour
zbMATH Keywords: Martin boundary
MaRDI profile type: Publication
DOI: 10.4310/PAMQ.2011.V7.N2.A4
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description: scientific article; zbMATH DE number 5995324