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50
A deterministiccontrolbased approach to motion by curvature
 Comm. Pure Appl. Math
"... by curvature ..."
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR pHARMONIC FUNCTIONS
"... Abstract. We characterize pharmonic functions in terms of an asymptotic mean value property. A pharmonic function u is a viscosity solution to ∆pu = div(∇up−2∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and only if the expansion u(x) = α 2 max ..."
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Cited by 29 (13 self)
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Abstract. We characterize pharmonic functions in terms of an asymptotic mean value property. A pharmonic function u is a viscosity solution to ∆pu = div(∇up−2∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and only if the expansion u(x) = α 2 max
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 ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS RELATED TO Tugofwar Games
"... We characterize solutions to the homogeneous parabolic pLaplace equation ut = ∇u2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tugofwar games with noise in which the number of rounds is bounded. The value functions for t ..."
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Cited by 18 (9 self)
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We characterize solutions to the homogeneous parabolic pLaplace equation ut = ∇u2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tugofwar games with noise in which the number of rounds is bounded. The value functions for these game approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.
A MIXED PROBLEM FOR THE INFINITY LAPLACIAN VIA Tugofwar Games
, 2009
"... In this paper we prove that a function u ∈ C(Ω) is the continuous value of the TugofWar game described in [19] if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions ..."
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Cited by 16 (8 self)
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In this paper we prove that a function u ∈ C(Ω) is the continuous value of the TugofWar game described in [19] if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions
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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Cited by 14 (1 self)
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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.
A PDE perspective of the normalized infinity Laplacian, preprint
"... The inhomogeneous normalized infinity Laplace equation was derived from the tugofwar game in [PSSW] with the positive righthandside as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [PSSW] by the game theory. In this paper, the normalized ..."
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Cited by 12 (2 self)
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The inhomogeneous normalized infinity Laplace equation was derived from the tugofwar game in [PSSW] with the positive righthandside as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [PSSW] by the game theory. In this paper, the normalized infinity Laplacian, formally written as △N ∞u =  ▽ u  −2 �n i,j=1 ∂xiu∂xju∂2xixju, is defined in a canonical way with the second derivatives in the local maximum and minimum directions, and understood analytically by a dichotomy. A comparison with polar quadratic polynomials property, the counterpart of the comparison with cones property, is proved to characterize the viscosity solutions of the inhomogeneous normalized infinity Laplace equation. We also prove that there is exactly one viscosity solution of the boundary value problem for the infinity Laplace equation △ N ∞u = f with positivef in a bounded open subset of R n. The stability of the inhomogeneous infinity Laplace equation △ N ∞u = f with strictly positive f and of the homogeneous equation △ N ∞u = 0 by small perturbation of the righthandside and the boundary data is established in the last part of the work. Our PDE method approach is quite different from those in [PSSW]. The stability result in this paper appears to be new.
On the evolution governed by the infinity Laplacian
 Math. Ann
"... Abstract. We investigate the basic properties of the degenerate and singular evolution equation ut = ..."
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Cited by 11 (1 self)
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Abstract. We investigate the basic properties of the degenerate and singular evolution equation ut =
A stochastic differential game for the inhomogeneous
, 2008
"... ∞Laplace equation ..."
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A VISIT WITH THE ∞LAPLACE EQUATION
"... In these notes we present an outline of the theory of the archetypal L ∞ variational problem in the calculus of variations. Namely, given an open U ⊂ IR n and b ∈ C(∂U), find u ∈ C(U) which agrees with the boundary function b on ∂U and minimizes (0.1) F∞(u, U): = �Du�L ∞ (U) ..."
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Cited by 9 (1 self)
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In these notes we present an outline of the theory of the archetypal L ∞ variational problem in the calculus of variations. Namely, given an open U ⊂ IR n and b ∈ C(∂U), find u ∈ C(U) which agrees with the boundary function b on ∂U and minimizes (0.1) F∞(u, U): = �Du�L ∞ (U)