Mean value property for $p$-harmonic functions
HTML articles powered by AMS MathViewer
- by Tiziana Giorgi and Robert Smits PDF
- Proc. Amer. Math. Soc. 140 (2012), 2453-2463 Request permission
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
- G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991), no.ย 3, 271โ283. MR 1115933, DOI 10.3233/ASY-1991-4305
- W Blaschke. Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logarithmischen Potentials. Leipz. Ber. 68 (1916), 37.
- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, Userโs guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no.ย 1, 1โ67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
- M. Falcone, S. Finzi Vita, T. Giorgi and R. Smits. A semi-Lagrangian scheme for the Game $p$-Laplacian via $p$-averaging. Submitted.
- Petri Juutinen, Peter Lindqvist, and Juan J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal. 33 (2001), no.ย 3, 699โ717. MR 1871417, DOI 10.1137/S0036141000372179
- Juan J. Manfredi, Mikko Parviainen, and Julio D. Rossi, An asymptotic mean value characterization for $p$-harmonic functions, Proc. Amer. Math. Soc. 138 (2010), no.ย 3, 881โ889. MR 2566554, DOI 10.1090/S0002-9939-09-10183-1
- Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J. 145 (2008), no.ย 1, 91โ120. MR 2451291, DOI 10.1215/00127094-2008-048
- I. Privaloff. Sur les fonctions harmoniques. Rec. Math. Moscou (Mat. Sbornik) 32 (1925), 464-471.
- Peiyong Wang, A formula for smooth $\infty$-harmonic functions, PanAmer. Math. J. 16 (2006), no.ย 1, 57โ65. MR 2186538
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
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2898708