Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.

References [Enhancements On Off] (What's this?)

  • 1. G. Aronsson, Extension of functions satisfying Lipschitz conditions. Ark. Mat., 6 (1967), 551-561. MR 0217665 (36:754)
  • 2. G. Aronsson, M.G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc. (N.S.), 41 (2004), 439-505. MR 2083637 (2005k:35159)
  • 3. E.N. Barron, L.C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations. Trans. Amer. Math. Soc., 360 (2008), 77-101. MR 2341994
  • 4. T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $ p \to \infty$ of $ \Delta_p u_p = f$ and related extremal problems. Rend. Sem. Mat. Univ. Politec. Torino, 1989, Special Issue (1991), 15-68. MR 1155453 (93a:35049)
  • 5. F. Charro, J. Garcia Azorero and J. D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games. Calc. Var. Partial Differential Equations, 34 (2009), 307-320. MR 2471139
  • 6. M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. MR 1118699 (92j:35050)
  • 7. L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 137 (1999), no. 653. MR 1464149 (99g:35132)
  • 8. J. García-Azorero, J.J. Manfredi, I. Peral and J.D. Rossi, The Neumann problem for the $ \infty$-Laplacian and the Monge-Kantorovich mass transfer problem. Nonlinear Analysis TM&A, 66(2) (2007), 349-366. MR 2279530 (2008f:35148)
  • 9. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient. Arch. Rational Mech. Anal., 123 (1993), 51-74. MR 1218686 (94g:35063)
  • 10. E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $ \Delta_{\infty}(u)=0$, NoDEA Nonlinear Differ. Equ. Appl., 14 (2007), 29-55. MR 2346452 (2008k:35159)
  • 11. P. Juutinen, P. Lindqvist and J.J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717. MR 1871417 (2002m:35051)
  • 12. Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc., 22 (2009), 167-210. MR 2449057 (2009h:91004)
  • 13. Y. Peres and S. Sheffield, Tug-of-war with noise: A game theoretic view of the $ p$-Laplacian. Duke Math. J. 145(1) (2008), 91-120. MR 2451291
  • 14. P. Wang, A formula for smooth $ \infty$-harmonic functions, PanAmerican Mathematical Journal, 16 (2006), no. 1, 57-65. MR 2186538

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J92, 35J60, 35J70

Retrieve articles in all journals with MSC (2010): 35J92, 35J60, 35J70

Additional Information

Juan J. Manfredi
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

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

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

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

American Mathematical Society