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A finite difference approach to the infinity Laplace equation and tugofwar games
 TRANS. AMER. MATH. SOC
, 2009
"... We present a modified version of the twoplayer “tugofwar” game introduced by Peres, Schramm, Sheffield, and Wilson [18]. This new tugofwar game is identical to the original except near the boundary of the domain ∂Ω, but its associated value functions are more regular. The dynamic programming pri ..."
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Cited by 22 (6 self)
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We present a modified version of the twoplayer “tugofwar” game introduced by Peres, Schramm, Sheffield, and Wilson [18]. This new tugofwar game is identical to the original except near the boundary of the domain ∂Ω, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results. We show that the finite difference equation has unique maximal and minimal solutions, which are identified as the value functions for the two tugofwar players. We demonstrate uniqueness, and hence the existence of a value for the game, in the case that the running payoff function is nonnegative. We also show that uniqueness holds in certain cases for signchanging running payoff functions which are sufficiently small. In the limit ε → 0, we obtain the convergence of the value functions to a viscosity solution of the normalized infinity Laplace equation. We also obtain several new results for the normalized infinity Laplace equation −∆∞u = f. In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous f, and continuous boundary data, as well as the uniqueness of solutions to this problem in the generic case. We present a new elementary proof of uniqueness in the case that f> 0, f < 0, or f ≡ 0. The stability of the solutions with respect to f is also studied, and an explicit continuous dependence estimate from f ≡ 0 is obtained.
An infinity Laplace equation with gradient term and mixed boundary conditions
, 910
"... Abstract. We obtain existence, uniqueness, and stability results for the modified 1homogeneous infinity Laplace equation −Δ∞u − βDu  = f, subject to Dirichlet or mixed DirichletNeumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions o ..."
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Abstract. We obtain existence, uniqueness, and stability results for the modified 1homogeneous infinity Laplace equation −Δ∞u − βDu  = f, subject to Dirichlet or mixed DirichletNeumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation. 1.
Harnack’s inequality for pharmonic functions via stochastic games
 Comm. Partial Differential Equations
, 1985
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The infinity Laplacian with a transport term
 J. MATH. ANAL. APPL
, 2013
"... We consider the following problem: given a bounded domain Ω ⊂ Rn and a vector field ζ: Ω → Rn, find a solution to −∆∞u − 〈Du, ζ 〉 = 0 in Ω, u = f on ∂Ω, where ∆ ∞ is the 1−homogeneous infinity Laplace operator that is formally given by ∆∞u = 〈D2u DuDu  , DuDu  〉 and f a Lipschitz boundary datu ..."
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Cited by 4 (2 self)
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We consider the following problem: given a bounded domain Ω ⊂ Rn and a vector field ζ: Ω → Rn, find a solution to −∆∞u − 〈Du, ζ 〉 = 0 in Ω, u = f on ∂Ω, where ∆ ∞ is the 1−homogeneous infinity Laplace operator that is formally given by ∆∞u = 〈D2u DuDu  , DuDu  〉 and f a Lipschitz boundary datum. If we assume that ζ is a continuous gradient vector field then we obtain existence and uniqueness of a viscosity solution by an Lpapproximation procedure. Also we prove the stability of the unique solution with respect to ζ. In addition when ζ is more regular (Lipschitz continuous) but not necessarily a gradient, using tugofwar games we prove that this problem has a solution.
A GameTree approach to discrete infinity Laplacian with running costs
, 2013
"... We give a selfcontained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tugofwar games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduc ..."
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We give a selfcontained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tugofwar games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce gametrees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the uniqueness. Our arguments are entirely deterministic, and also do not rely on (semi)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for wellposedness.
TUGOFWAR GAMES AND PARABOLIC PROBLEMS WITH SPATIAL AND TIME DEPENDENCE
"... Abstract. In this paper we use probabilistic arguments (TugofWar games) to obtain existence of viscosity solutions to a parabolic problem of the form{ K(x,t)(Du)ut(x, t) = ..."
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Abstract. In this paper we use probabilistic arguments (TugofWar games) to obtain existence of viscosity solutions to a parabolic problem of the form{ K(x,t)(Du)ut(x, t) =
TUGOFWAR GAMES AND THE INFINITY LAPLACIAN WITH SPATIAL DEPENDENCE
"... In this paper we look for PDEs that arise as limits of values of TugofWar games when the possible movements of the game are taken in a family of sets that are not necessarily euclidean balls. In this way we find existence of viscosity solutions to the Dirichlet problem for an equation of the form ..."
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In this paper we look for PDEs that arise as limits of values of TugofWar games when the possible movements of the game are taken in a family of sets that are not necessarily euclidean balls. In this way we find existence of viscosity solutions to the Dirichlet problem for an equation of the form −〈D2v · Jx(Dv); Jx(Dv)〉(x) = 0, that is, an infinity Laplacian with spatial dependence. Here Jx(Dv(x)) is a vector that depends on the the spatial location and the gradient of the solution.
AN OBSTACLE PROBLEM FOR TUGOFWAR GAMES
"... We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinityharmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above t ..."
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We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinityharmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tugofwar.
THE LIMIT AS p→ ∞ FOR THE EIGENVALUE PROBLEM OF THE 1HOMOGENEOUS pLAPLACIAN
"... Abstract. In this paper we study asymptotics as p→ ∞ of the Dirichlet eigenvalue problem for the 1homogeneous pLaplacian, that is, { − 1 p Du2−pdiv (Dup−2Du) = λu, in Ω, u = 0, on ∂Ω. Here Ω is a bounded starshaped domain in Rn and p> n. There exists a principal eigenvalue λ1,p(Ω), whic ..."
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Abstract. In this paper we study asymptotics as p→ ∞ of the Dirichlet eigenvalue problem for the 1homogeneous pLaplacian, that is, { − 1 p Du2−pdiv (Dup−2Du) = λu, in Ω, u = 0, on ∂Ω. Here Ω is a bounded starshaped domain in Rn and p> n. There exists a principal eigenvalue λ1,p(Ω), which is positive, and has associated a nonnegative nontrivial eigenfunction. Moreover, we show that limp→ ∞ λ1,p(Ω) = λ1,∞(Ω), where λ1,∞(Ω) is the first eigenvalue corresponding to the 1homogeneous infinity Laplacian, that is, −