Harnack's Inequality for p-Harmonic Functions via Stochastic Games

被引：32

作者：

Luiro, Hannes ^{[1
]}

Parviainen, Mikko ^{[1
]}

Saksman, Eero ^{[2
]}

机构：

[1] Univ Jyvaskyla, Dept Math & Stat, Jyvaskyla, Finland

[2] Univ Helsinki, Dept Math & Stat, FI-00014 Helsinki, Finland

来源：

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS | 2013年 / 38卷 / 11期

基金：

芬兰科学院;

关键词：

Dynamic programming principle; Harnack inequality; Lipschitz estimates; p-harmonic; Stochastic games; Two-player zero-sum games; TUG-OF-WAR; ELLIPTIC DIFFERENTIAL EQUATIONS; MINIMIZING LIPSCHITZ EXTENSIONS; VISCOSITY SOLUTIONS; INFINITY LAPLACIAN; WEAK SOLUTIONS; REGULARITY; PROOF; EQUIVALENCE; THEOREM;

D O I：

10.1080/03605302.2013.814068

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We give a proof of asymptotic Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Lipschitz continuity and Harnack's inequality for p-harmonic functions in the case p>2. The proof avoids classical techniques like Moser iteration, but instead relies on suitable choices of strategies for the stochastic tug-of-war game.

引用

页码：1985 / 2003

页数：19

共 28 条

[1] An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions [J].

Armstrong, Scott N. ;

Smart, Charles K. .

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 2010, 37 (3-4) :381-384

[2] A STOCHASTIC DIFFERENTIAL GAME FOR THE INHOMOGENEOUS ∞-LAPLACE EQUATION [J].

Atar, Rami ;

Budhiraja, Amarjit .

ANNALS OF PROBABILITY, 2010, 38 (02) :498-531

[3]

De Giorgi E., 1957, Mem. Acad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., V3, P25

[4] C1+ALPHA LOCAL REGULARITY OF WEAK SOLUTIONS OF DEGENERATE ELLIPTIC-EQUATIONS [J].

DIBENEDETTO, E .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1983, 7 (08) :827-850

[5]

DIBENEDETTO E, 1984, ANN I H POINCARE-AN, V1, P295

[6] A NEW PROOF OF LOCAL C-1,ALPHA-REGULARITY FOR SOLUTIONS OF CERTAIN DEGENERATE ELLIPTIC PDE [J].

EVANS, LC .

JOURNAL OF DIFFERENTIAL EQUATIONS, 1982, 45 (03) :356-373

[7]

Grimmett Geoffrey, 2020, Probability and random processes

[8] A New Proof for the Equivalence of Weak and Viscosity Solutions for the p-Laplace Equation [J].

Julin, Vesa ;

Juutinen, Petri .

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2012, 37 (05) :934-946

[9] On the equivalence of viscosity solutions and weak solutions or a quasi-linear equation [J].

Juutinen, P ;

Lindqvist, P ;

Manfredi, JJ .

SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 2001, 33 (03) :699-717

[10] Regularity theory and traces of A-harmonic functions [J].

Koskela, P ;

Manfredi, JJ ;

Villamor, E .

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1996, 348 (02) :755-766

← 1 2 3 →