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BIASED TUGOFWAR, THE BIASED INFINITY LAPLACIAN, AND COMPARISON WITH EXPONENTIAL CONES
, 811
"... Abstract. We prove that if U ⊂ R n is an open domain whose closure U is compact in the path metric, and F is a Lipschitz function on ∂U, then for each β ∈ R there exists a unique viscosity solution to the βbiased infinity Laplacian equation on U that extends F, where ∆∞u = ∇u β∇u  + ∆∞u = 0 −2 ..."
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Abstract. We prove that if U ⊂ R n is an open domain whose closure U is compact in the path metric, and F is a Lipschitz function on ∂U, then for each β ∈ R there exists a unique viscosity solution to the βbiased infinity Laplacian equation on U that extends F, where ∆∞u = ∇u β∇u  + ∆∞u = 0 −2 i,j uxiuxixjuxj. In the proof, we extend the tugofwar ideas of Peres, Schramm, Sheffield and Wilson, and define the βbiased ǫgame as follows. The starting position is x0 ∈ U. At the k th step the two players toss a suitably biased coin (in our key example, player I wins with odds of exp(βǫ) to 1), and the winner chooses xk with d(xk, xk−1) < ǫ. The game ends when xk ∈ ∂U, and player II pays the amount F(xk) to player I. We prove that the value u ǫ (x0) of this game exists, and that ‖u ǫ − u‖ ∞ → 0 as ǫ → 0, where u is the unique extension of F to U that satisfies comparison with βexponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with βexponential cones if and only if it is a viscosity solution to the βbiased infinity Laplacian equation. 1.
Inhomogeneous infinity Laplace equation
 Advances in Mathematics
"... We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation ∂xi u∂xj u∂2 xixj u = f in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a cha ..."
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We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation ∂xi u∂xj u∂2 xixj u = f in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity sub and supersolutions of the equation with constant righthand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing righthand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the righthand side and boundary data are perturbed. In the end, we prove the stability of the wellknown homogeneous infinity Laplace equation ∂xi u∂xj u∂2 xixj u = 0, which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its righthand side and boundary data are perturbed simultaneously. © 2007 Elsevier Inc. All rights reserved.
Overdetermined boundary value problems for the ∞Laplacian
, 2009
"... Abstract: We consider overdetermined boundary value problems for the ∞Laplacian in a domain Ω of Rn and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞Laplacian, the normalized or gametheoretic ∞Laplacian and the limit of the pLaplaci ..."
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Cited by 9 (0 self)
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Abstract: We consider overdetermined boundary value problems for the ∞Laplacian in a domain Ω of Rn and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞Laplacian, the normalized or gametheoretic ∞Laplacian and the limit of the pLaplacian as p → ∞ are considered and provide different answers.
INFINITY LAPLACE EQUATION WITH NONTRIVIAL RIGHTHAND SIDE
, 2010
"... We analyze the set of continuous viscosity solutions of the infinity Laplace equation − ∆ N ∞w(x) = f(x), with generally signchanging righthand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron’s const ..."
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Cited by 6 (0 self)
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We analyze the set of continuous viscosity solutions of the infinity Laplace equation − ∆ N ∞w(x) = f(x), with generally signchanging righthand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron’s construction by means of a strict comparison principle. These extremal solutions are proved to be absolutely extremal solutions.
SOLUTIONS OF NONLINEAR PDES IN THE SENSE OF AVERAGES
"... Abstract. We characterize pharmonic functions including p = 1 and p = ∞ by using mean value properties extending classical results of Privaloff from the linear case p = 2 to all p′s. We describe a class of random tugofwar games whose value functions approach pharmonic functions as the step goes ..."
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Cited by 5 (1 self)
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Abstract. We characterize pharmonic functions including p = 1 and p = ∞ by using mean value properties extending classical results of Privaloff from the linear case p = 2 to all p′s. We describe a class of random tugofwar games whose value functions approach pharmonic functions as the step goes to zero for the full range 1 < p <∞. 1.
Local regularity results for value functions of tugofwar with noise and running payoff
"... ar ..."
THE SUBLINEAR PROBLEM FOR THE 1HOMOGENEOUS pLAPLACIAN
"... Abstract. In this paper we prove existence and uniqueness of a positive viscosity solution of the 1homogeneous pLaplacian with a sublinear righthand side, that is, −Du2−pdiv (Dup−2Du) = λuq in Ω, u = 0 on ∂Ω, where Ω is a bounded starshaped domain, λ> 0, p> 2 and 0 < q < 1. 1. ..."
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Abstract. In this paper we prove existence and uniqueness of a positive viscosity solution of the 1homogeneous pLaplacian with a sublinear righthand side, that is, −Du2−pdiv (Dup−2Du) = λuq in Ω, u = 0 on ∂Ω, where Ω is a bounded starshaped domain, λ> 0, p> 2 and 0 < q < 1. 1.
On the best Lipschitz extension problem for a discrete distance and the discrete ∞Laplacian
 J. MATH. PURES APPL
"... This paper is concerned with the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete ∞Laplacian problem, which arise as the dynamic programming formula for the value function of some εtug ..."
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Cited by 1 (1 self)
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This paper is concerned with the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete ∞Laplacian problem, which arise as the dynamic programming formula for the value function of some εtugofwar games. As in the classical case, we obtain the absolutely minimizing Lipschitz extension of a datum f by taking the limit as p → ∞ in a nonlocal p– Laplacian problem.
Calc. Var. DOI 10.1007/s0052601205673 Calculus of Variations On the Aleksandrov–Bakelman–Pucci estimate for the infinity Laplacian
, 2012
"... Abstract We prove L ∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and pLaplacian, namely −ΔNp u = f for n < p ≤ ∞. We are able to provide a stable family of results depending continuously on the parameter p. We also prove the ..."
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Abstract We prove L ∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and pLaplacian, namely −ΔNp u = f for n < p ≤ ∞. We are able to provide a stable family of results depending continuously on the parameter p. We also prove the failure of the classical Alexandrov–Bakelman–Pucci estimate for the normalized infinity Laplacian and propose alternate estimates. Communicated by O. Savin.