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)

 
 

 

Mean value property for $ p$-harmonic functions


Authors: Tiziana Giorgi and Robert Smits
Journal: Proc. Amer. Math. Soc. 140 (2012), 2453-2463
MSC (2010): Primary 35J92, 35D40, 35J60, 35J70
DOI: https://doi.org/10.1090/S0002-9939-2011-11181-X
Published electronically: November 21, 2011
MathSciNet review: 2898708
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We derive a mean value property for $ p$-harmonic functions in two dimensions, $ 1<p<\infty $, which holds asymptotically in the viscosity sense. The formula coincides with the classical mean value property for harmonic functions, when $ p=2$, and is a consequence of a representation for the Game $ p$-Laplacian obtained via $ p$-averaging.


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

  • 1. G. Barles and P.E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991), 271-283. MR 1115933 (92d:35137)
  • 2. W Blaschke. Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logarithmischen Potentials. Leipz. Ber. 68 (1916), 37.
  • 3. 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), no. 1, 1-67. MR 1118699 (92j:35050)
  • 4. M. Falcone, S. Finzi Vita, T. Giorgi and R. Smits. A semi-Lagrangian scheme for the Game $ p$-Laplacian via $ p$-averaging. Submitted.
  • 5. P. Juutinen, P. Lindqvist and J. Manfredi. On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal. 33 (2001), no. 3, 699-717. MR 1871417 (2002m:35051)
  • 6. J. Manfredi, M. Parviainen and J. D. Rossi. An asymptotic mean value characterization for $ p$-harmonic functions. Proc. Amer. Math. Soc. 138 (2010), no. 3, 881-889. MR 2566554 (2010k:35200)
  • 7. Y. Peres and S. Sheffield. Tug-of-war with noise: a game-theoretic view of the $ p$-Laplacian. Duke Math. J. 145 (2008), 91-120. MR 2451291 (2010i:35100)
  • 8. I. Privaloff. Sur les fonctions harmoniques. Rec. Math. Moscou (Mat. Sbornik) 32 (1925), 464-471.
  • 9. 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, 35D40, 35J60, 35J70

Retrieve articles in all journals with MSC (2010): 35J92, 35D40, 35J60, 35J70


Additional Information

Tiziana Giorgi
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001
Email: tgiorgi@nmsu.edu

Robert Smits
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001
Email: rsmits@nmsu.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11181-X
Received by editor(s): November 1, 2010
Received by editor(s) in revised form: February 26, 2011
Published electronically: November 21, 2011
Additional Notes: Funding for the first author was provided by National Science Foundation Grant #DMS-0604843
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society