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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
DOI: https://doi.org/10.1090/S0002-9939-09-10183-1
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 =$   div$ ( \vert\nabla u\vert^{p-2} \nabla u)=0$ with $ 1< p \leq \infty$ in a domain $ \Omega$ if and only if the expansion

$\displaystyle u(x) = \frac{\alpha}{2} \left\{ \max_{\overline{B_\varepsilon (x)... ...ert B_\varepsilon (x)\vert} \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
Email: manfredi@pitt.edu

Mikko Parviainen
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Helsinki, Finland
Email: Mikko.Parviainen@tkk.fi

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
Email: jrossi@dm.uba.ar

DOI: https://doi.org/10.1090/S0002-9939-09-10183-1
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

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