An asymptotic mean value characterization for p-harmonic functions

We characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to \Delta _p u = \mbox {div} ( |\nabla u|^{p-2} \nabla u)=0 with 1< p \leq \infty in a domain \Omega if and only if the expansion u(x) = \frac {\alpha }{2} \left { \...

Full description

Saved in:
Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2010-03, Vol.138 (3), p.881-889
Main Authors: MANFREDI, JUAN J., PARVIAINEN, MIKKO, ROSSI, JULIO D.
Format: Article
Language:English
Subjects:
Asymptotic value
Calculus of variations and optimal control
Continuous functions
Exact sciences and technology
General mathematics
General, history and biography
Infinity
Laplacians
Mathematical analysis
Mathematical functions
Mathematical minima
Mathematics
Sciences and techniques of general use
Viscosity
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to \Delta _p u = \mbox {div} ( |\nabla u|^{p-2} \nabla u)=0 with 1< p \leq \infty in a domain \Omega if and only if the expansion u(x) = \frac {\alpha }{2} \left { \max _{\overline {B_\varepsilon (x)}} u + \min _{\overline {B_\varepsilon (x)}} u \right } + \frac {\beta }{|B_\varepsilon (x)|} \int _{B_\varepsilon (x)} u d y + o (\varepsilon ^2) holds as \varepsilon \to 0 for x\in \Omega in a weak sense, which we call the viscosity sense. Here the coefficients \alpha , \beta are determined by \alpha + \beta =1 and \alpha /\beta = (p-2)/(N+2).
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-09-10183-1