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## AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-HARMONIC FUNCTIONS

Citations: | 29 - 13 self |

### Citations

1386 | User’s guide to viscosity solutions of second order partial differential equations - Crandall, Ishii, et al. - 1992 |

162 | Differential equations methods for the Monge-Kantorovich mass transfer problem
- Evans, Gangbo
- 1999
(Show Context)
Citation Context ...ons in the viscosity sense and has applications to best 4 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Lipschitz extensions, image processing and mass transport problems, see [2], [3], [4], =-=[7]-=-, [8], [9]. We observe that the notions of a viscosity solution and a Sobolev weak solution for the p-Laplace equation agree for 1 < p < ∞, see JuutinenLindqvist-Manfredi [11]. Therefore, Theorem 2 ch... |

159 |
Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient.
- Jensen
- 1993
(Show Context)
Citation Context ... viscosity sense and has applications to best 4 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Lipschitz extensions, image processing and mass transport problems, see [2], [3], [4], [7], [8], =-=[9]-=-. We observe that the notions of a viscosity solution and a Sobolev weak solution for the p-Laplace equation agree for 1 < p < ∞, see JuutinenLindqvist-Manfredi [11]. Therefore, Theorem 2 characterize... |

145 | A tour of the theory of absolutely minimizing functions
- Aronsson, Crandall, et al.
- 2004
(Show Context)
Citation Context ...e: Set p =∞ and consider Aronsson’s function u(x, y) = |x|4/3 − |y|4/3 near the point (x, y) = (1, 0). Aronsson’s function is ∞-harmonic in the viscosity sense but it is not of class C2, see Aronsson =-=[1, 2]-=-. It will follow from Theorem 2 below that u satisfies u(x) = 1 2 { max Bε(x) u+ min Bε(x) u } + o(ε2) as ε→ 0, ASYMPTOTIC MEAN VALUE CHARACTERIZATION 3 in the viscosity sense of Definition 1. However... |

141 |
Extensions of functions satisfying Lipschitz conditions
- Aronsson
- 1967
(Show Context)
Citation Context ...e: Set p =∞ and consider Aronsson’s function u(x, y) = |x|4/3 − |y|4/3 near the point (x, y) = (1, 0). Aronsson’s function is ∞-harmonic in the viscosity sense but it is not of class C2, see Aronsson =-=[1, 2]-=-. It will follow from Theorem 2 below that u satisfies u(x) = 1 2 { max Bε(x) u+ min Bε(x) u } + o(ε2) as ε→ 0, ASYMPTOTIC MEAN VALUE CHARACTERIZATION 3 in the viscosity sense of Definition 1. However... |

114 | Tug-of-war and the infinity Laplacian
- Peres, Schramm, et al.
(Show Context)
Citation Context ...might be true, let us formally expand the p-Laplacian as follows (2.1) ∆pu = (p− 2)|∇u|p−4 〈D2u∇u,∇u〉+ |∇u|p−2∆u. This formal expansion was used by Peres and Sheffield in [13] (see also Peres et. al. =-=[12]-=-) to find p−harmonic functions as limits of values of Tug-of-War games. Suppose that u is a smooth function with ∇u 6= 0. We see from (2.1), that u is a solution to ∆pu = 0 if and only if (2.2) (p− 2)... |

82 | On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation
- Juutinen, Lindqvist, et al.
- 2001
(Show Context)
Citation Context ...blems, see [2], [3], [4], [7], [8], [9]. We observe that the notions of a viscosity solution and a Sobolev weak solution for the p-Laplace equation agree for 1 < p < ∞, see JuutinenLindqvist-Manfredi =-=[11]-=-. Therefore, Theorem 2 characterizes weak solutions when 1 < p <∞. Finally, we note that Wang [14] has also used Taylor series to give sufficient conditions for p-subharmonicity in terms of asymptotic... |

64 |
Limits as p →∞ of ∆pup = f and related extremal problems,
- Bhattacharya, DiBenedetto, et al.
- 1989
(Show Context)
Citation Context ...unctions in the viscosity sense and has applications to best 4 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Lipschitz extensions, image processing and mass transport problems, see [2], [3], =-=[4]-=-, [7], [8], [9]. We observe that the notions of a viscosity solution and a Sobolev weak solution for the p-Laplace equation agree for 1 < p < ∞, see JuutinenLindqvist-Manfredi [11]. Therefore, Theorem... |

52 |
Tug-of-war with noise: a game theoretic view of the pLaplacian,
- Peres, Sheffield
- 2008
(Show Context)
Citation Context ...symptotic mean value formula might be true, let us formally expand the p-Laplacian as follows (2.1) ∆pu = (p− 2)|∇u|p−4 〈D2u∇u,∇u〉+ |∇u|p−2∆u. This formal expansion was used by Peres and Sheffield in =-=[13]-=- (see also Peres et. al. [12]) to find p−harmonic functions as limits of values of Tug-of-War games. Suppose that u is a smooth function with ∇u 6= 0. We see from (2.1), that u is a solution to ∆pu = ... |

50 | The infinity laplacian, Aronssons equation and their generalizations.
- Barron, Evans, et al.
- 2008
(Show Context)
Citation Context ...nic functions in the viscosity sense and has applications to best 4 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Lipschitz extensions, image processing and mass transport problems, see [2], =-=[3]-=-, [4], [7], [8], [9]. We observe that the notions of a viscosity solution and a Sobolev weak solution for the p-Laplace equation agree for 1 < p < ∞, see JuutinenLindqvist-Manfredi [11]. Therefore, Th... |

26 | The Neumann problem for the ∞-Laplacian and the Monge-Kantorovich mass transfer problem
- Garćıa-Azorero, Manfredi, et al.
(Show Context)
Citation Context ...n the viscosity sense and has applications to best 4 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Lipschitz extensions, image processing and mass transport problems, see [2], [3], [4], [7], =-=[8]-=-, [9]. We observe that the notions of a viscosity solution and a Sobolev weak solution for the p-Laplace equation agree for 1 < p < ∞, see JuutinenLindqvist-Manfredi [11]. Therefore, Theorem 2 charact... |

16 | A mixed problem for the infinity Laplacian via tug-of-war games
- Charro, Garćıa-Azorero, et al.
(Show Context)
Citation Context ...d of x. Let xε1 and x ε 2 be the point at which φ attains its minimum and maximum in Bε(x) respectively; that is, φ(xε1) = min y∈Bε(x) φ(y) and φ(xε2) = max y∈Bε(x) φ(y). Next, we use some ideas from =-=[5]-=-. Consider the Taylor expansion of the second order of φ φ(y) = φ(x) +∇φ(x) · (y − x) + 1 2 〈D2φ(x)(y − x), (y − x)〉+ o(|y − x|2) as |y − x| → 0. Evaluating this Taylor expansion of φ at the point x a... |

13 | On absolutely minimizing Lipschitz extensions and PDE ∆∞(u) = 0
- Gruyer
(Show Context)
Citation Context ...φ+ min Bε(x) φ } + β ∫ Bε(x) ψ(y) dy + o(ε2). This condition is actually weaker than asking for the asymptotic expansion to hold in the classical sense as the following example suggested in Le Gruyer =-=[10]-=- shows. Example: Set p =∞ and consider Aronsson’s function u(x, y) = |x|4/3 − |y|4/3 near the point (x, y) = (1, 0). Aronsson’s function is ∞-harmonic in the viscosity sense but it is not of class C2,... |

2 |
A formula for smooth ∞-Harmonic Functions, PanAmerican Mathematical Journal, Number 1
- Wang
(Show Context)
Citation Context ... Sobolev weak solution for the p-Laplace equation agree for 1 < p < ∞, see JuutinenLindqvist-Manfredi [11]. Therefore, Theorem 2 characterizes weak solutions when 1 < p <∞. Finally, we note that Wang =-=[14]-=- has also used Taylor series to give sufficient conditions for p-subharmonicity in terms of asymptotic mean values of (u(x)− u(0))p. 2. Proof of Theorem 2 To gain some intuition on why such asymptotic... |