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An asymptotic mean value characterization for $p$-harmonic functions

Authors: Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi
Journal: Proc. Amer. Math. Soc. 138 (2010), 881-889
MSC (2010): Primary 35J92, 35J60, 35J70
Published electronically: October 28, 2009
MathSciNet review: 2566554
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Abstract: 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 $\alpha /\beta = (p-2)/(N+2)$.

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Additional Information

Juan J. Manfredi
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
MR Author ID: 205679

Mikko Parviainen
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Helsinki, Finland
MR Author ID: 823079

Julio D. Rossi
Affiliation: IMDEA Matemáticas, C-IX, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Address at time of publication: FCEyN UBA (1428), Buenos Aires, Argentina
MR Author ID: 601009
ORCID: 0000-0001-7622-2759

Keywords: $p$-Laplacian, infinity Laplacian, mean value property, viscosity solutions.
Received by editor(s): January 9, 2009
Published electronically: October 28, 2009
Additional Notes: The second author was supported by the Emil Aaltonen Foundation, the Fulbright Center, and the Magnus Ehrnrooth Foundation
The third author was partially supported by project MTM2004-02223, MEC, Spain; by UBA X066; and by CONICET, Argentina
Dedicated: To the memory of our friend and colleague Fuensanta Andreu
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2009 American Mathematical Society