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115
A deterministic-control-based approach to motion by curvature
- Comm. Pure Appl. Math
"... by curvature ..."
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An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions
, 2009
"... We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls. ..."
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Cited by 33 (6 self)
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We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls.
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-HARMONIC FUNCTIONS
"... Abstract. We characterize p-harmonic functions in terms of an as-ymptotic mean value property. A p-harmonic function u is a viscosity solution to ∆pu = div(|∇u|p−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 p-harmonic functions in terms of an as-ymptotic mean value property. A p-harmonic function u is a viscosity solution to ∆pu = div(|∇u|p−2∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and only if the expansion u(x) = α 2 max
On the definition and properties of p-harmonious functions
, 2009
"... We consider functions that satisfy the identity uε(x) = ..."
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Cited by 24 (11 self)
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We consider functions that satisfy the identity uε(x) =
A finite difference approach to the infinity Laplace equation and tug-of-war games
- TRANS. AMER. MATH. SOC
, 2009
"... We present a modified version of the two-player “tug-of-war” game introduced by Peres, Schramm, Sheffield, and Wilson [18]. This new tug-ofwar 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 two-player “tug-of-war” game introduced by Peres, Schramm, Sheffield, and Wilson [18]. This new tug-ofwar 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 tug-of-war 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 sign-changing 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.
Random-turn Hex and other selection games
- Amer. Math. Monthly
"... Overview. The game of Hex, invented independently by Piet Hein in 1942 and John Nash in 1948 [9], has two players who take turns placing stones of their respective colors on the hexagons of a rhombus-shaped hexagonal grid (see Figure 1). A player wins by completing a path connecting the two opposite ..."
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Cited by 19 (6 self)
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Overview. The game of Hex, invented independently by Piet Hein in 1942 and John Nash in 1948 [9], has two players who take turns placing stones of their respective colors on the hexagons of a rhombus-shaped hexagonal grid (see Figure 1). A player wins by completing a path connecting the two opposite sides of his or her color. Although it is easy to show
Everywhere differentiability of infinity harmonic functions
"... We show that an infinity harmonic function, that is, a viscosity solution of the nonlinear PDE −∆∞u = −uxiuxj uxixj = 0, is everywhere differentiable. Our new innovation is proving the uniqueness of appropriately rescaled blow-up limits around an arbitrary point. ..."
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Cited by 18 (0 self)
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We show that an infinity harmonic function, that is, a viscosity solution of the nonlinear PDE −∆∞u = −uxiuxj uxixj = 0, is everywhere differentiable. Our new innovation is proving the uniqueness of appropriately rescaled blow-up limits around an arbitrary point.
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS RELATED TO Tug-of-war Games
"... We characterize solutions to the homogeneous parabolic p-Laplace equation ut = |∇u|2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war 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 p-Laplace equation ut = |∇u|2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war 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 Tug-of-war Games
, 2009
"... In this paper we prove that a function u ∈ C(Ω) is the continuous value of the Tug-of-War 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 Tug-of-War game described in [19] if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions
p(x)-harmonic functions with unbounded exponent in a subdomain, Ann
- Inst. H. Poincaré Anal. Non Linéaire
"... Abstract. We study the Dirichlet problem − div(|∇u | p(x)−2 ∇u) = 0 in Ω, with u = f on ∂Ω and p(x) = ∞ in D, a subdomain of the reference domain Ω. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n → ∞ of the solutions un to the corresponding ..."
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Cited by 15 (7 self)
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Abstract. We study the Dirichlet problem − div(|∇u | p(x)−2 ∇u) = 0 in Ω, with u = f on ∂Ω and p(x) = ∞ in D, a subdomain of the reference domain Ω. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n → ∞ of the solutions un to the corresponding problem when pn(x) = p(x) ∧ n, in particular, with pn = n in D. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, ∞-harmonic within D. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of Ω. 1.
