Boundary behavior of non-negative solutions to degenerate sub-elliptic equations
type: Item
instance of: scholarly article
title: Boundary behavior of non-negative solutions to degenerate sub-elliptic equations
zbMATH Open document ID: 1273.35141
author: Elin Götmark
author: Kaj Nyström
published in: Journal of Differential Equations
publication date: 2013-04-17T00:00:00Z
review text: The authors study the nonnegative solutions to the equation \[ Lu=\sum^m_{i,j=1}X^{\ast}_i(a_{ij}X_ju)=\sum^m_{i,j=1}X^{\ast}_i(x)(a_{ij}(x)X_j(x)u(x))=0 \] in \(\Omega\) where \(X=\{X_1,\dots ,X_m\}\) is a system of \(C^{\infty}\) vector fields satisfying Hörmander's finite rank condition, \(X^{\ast}_i\) is the formal adjoint of \(X_i\) and \(\Omega\) is a non-tangentially accessible domain with respect to the Carnot--Carathéodory distance \(d\) induced by \(X\). Moreover \(\{a_{ij}(x)\}\) is assumed to be real, symmetric and such that \[ \beta^{-1}\lambda (x)|\xi |^2\leq \sum^m_{i,j=1}a_{ij}(x)\xi_i\xi_j\leq \beta \lambda (x)|\xi |^2 \] where \(\beta \geq 1\) and \(\lambda\) defines an \(A_2\)-weight with respect to the metric introduced by the system of vector fields \(X=\{X_1,\dots ,X_m\}\). More precisely the authors study the doubling property of the associated elliptic measure and the boundary behaviour of nonnegative solutions. More precisely, they prove the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. This result generalizes results proved in the eighties by Fabes, Kenig, Jerison and Serapioni and more recently by Capogna and Garofalo. The aim of the authors is to extend to the most possible general case of very recent results proved by Lewis, Lundström and Nyström concerning the boundary behavior of non-negative solutions to equations of \(p\)-Laplace type.
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6154866
zbMATH Keywords: Hörmander condition
zbMATH Keywords: boundary Harnack inequality
zbMATH Keywords: elliptical measure
zbMATH Keywords: sub-elliptical PDEs
zbMATH Keywords: Muckenhoupt weights
zbMATH Keywords: quasi-linear equations
zbMATH Keywords: \(p\)-Laplace
MaRDI profile type: Publication
full work available at URL: j.jde.2013.01.030
OpenAlex ID: W1998957519
DOI: 10.1016/J.JDE.2013.01.030
description: scientific article; zbMATH DE number 6154866