An asymptotic mean value characterization for $p$-harmonic functions
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- by Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi PDF
- Proc. Amer. Math. Soc. 138 (2010), 881-889 Request permission
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 $\alpha /\beta = (p-2)/(N+2)$.References
- Gunnar Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561 (1967). MR 217665, DOI 10.1007/BF02591928
- Gunnar Aronsson, Michael G. Crandall, and Petri Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 4, 439–505. MR 2083637, DOI 10.1090/S0273-0979-04-01035-3
- E. N. Barron, L. C. Evans, and R. Jensen, The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Amer. Math. Soc. 360 (2008), no. 1, 77–101. MR 2341994, DOI 10.1090/S0002-9947-07-04338-3
- T. Bhattacharya, E. DiBenedetto, and J. Manfredi, Limits as $p\to \infty$ of $\Delta _pu_p=f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino Special Issue (1989), 15–68 (1991). Some topics in nonlinear PDEs (Turin, 1989). MR 1155453
- Fernando Charro, Jesus García Azorero, and Julio D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games, Calc. Var. Partial Differential Equations 34 (2009), no. 3, 307–320. MR 2471139, DOI 10.1007/s00526-008-0185-2
- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis 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, DOI 10.1090/S0273-0979-1992-00266-5
- L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653, viii+66. MR 1464149, DOI 10.1090/memo/0653
- 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 Anal. 66 (2007), no. 2, 349–366. MR 2279530, DOI 10.1016/j.na.2005.11.030
- Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), no. 1, 51–74. MR 1218686, DOI 10.1007/BF00386368
- E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta _\infty (u)=0$, NoDEA Nonlinear Differential Equations Appl. 14 (2007), no. 1-2, 29–55. MR 2346452, DOI 10.1007/s00030-006-4030-z
- 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. 3, 699–717. MR 1871417, DOI 10.1137/S0036141000372179
- Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. MR 2449057, DOI 10.1090/S0894-0347-08-00606-1
- Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J. 145 (2008), no. 1, 91–120. MR 2451291, DOI 10.1215/00127094-2008-048
- Peiyong Wang, A formula for smooth $\infty$-harmonic functions, PanAmer. Math. J. 16 (2006), no. 1, 57–65. MR 2186538
Additional Information
- Juan J. Manfredi
- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- MR Author ID: 205679
- Email: manfredi@pitt.edu
- Mikko Parviainen
- Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Helsinki, Finland
- MR Author ID: 823079
- 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
- MR Author ID: 601009
- ORCID: 0000-0001-7622-2759
- Email: jrossi@dm.uba.ar
- 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 - Communicated by: Matthew J. Gursky
- © Copyright 2009 American Mathematical Society
- 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
- MathSciNet review: 2566554
Dedicated: To the memory of our friend and colleague Fuensanta Andreu