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BIASED TUG-OF-WAR, 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 β-biased 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 β-biased 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 tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ǫ-game 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 β-exponential 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 β-exponential cones if and only if it is a viscosity solution to the β-biased 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 right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known 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 right-hand 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 game-theoretic ∞-Laplacian and the limit of the p-Laplaci ..."
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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 game-theoretic ∞-Laplacian and the limit of the p-Laplacian as p → ∞ are considered and provide different answers.
INFINITY LAPLACE EQUATION WITH NON-TRIVIAL RIGHT-HAND SIDE
, 2010
"... We analyze the set of continuous viscosity solutions of the infinity Laplace equation − ∆ N ∞w(x) = f(x), with generally sign-changing right-hand 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 sign-changing right-hand 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 p-harmonic 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 tug-of-war games whose value functions approach p-harmonic functions as the step goes ..."
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Cited by 5 (1 self)
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Abstract. We characterize p-harmonic 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 tug-of-war games whose value functions approach p-harmonic functions as the step goes to zero for the full range 1 < p <∞. 1.
Local regularity results for value functions of tug-of-war with noise and running payoff
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THE SUBLINEAR PROBLEM FOR THE 1-HOMOGENEOUS p-LAPLACIAN
"... Abstract. In this paper we prove existence and uniqueness of a posi-tive viscosity solution of the 1-homogeneous p-Laplacian with a sublinear right-hand side, that is, −|Du|2−pdiv (|Du|p−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 posi-tive viscosity solution of the 1-homogeneous p-Laplacian with a sublinear right-hand side, that is, −|Du|2−pdiv (|Du|p−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 ε-tug ..."
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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 ε-tug-of-war 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/s00526-012-0567-3 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 p-Laplacian, 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 p-Laplacian, 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.
