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TUGOFWAR AND THE INFINITY LAPLACIAN
, 2008
"... 1.1. Overview. We consider a class of zerosum twoplayer stochastic games called tugofwar and use them to prove that every bounded realvalued Lipschitz function F on a subset Y of a length space X admits a unique absolutely minimal (AM) extension to X, i.e., a unique Lipschitz extension u: X → R ..."
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Cited by 113 (7 self)
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1.1. Overview. We consider a class of zerosum twoplayer stochastic games called tugofwar and use them to prove that every bounded realvalued Lipschitz function F on a subset Y of a length space X admits a unique absolutely minimal (AM) extension to X, i.e., a unique Lipschitz extension u: X → R for which
A Seeded Image Segmentation Framework Unifying Graph Cuts And Random Walker Which Yields A New Algorithm
 ICCV
, 2007
"... In this work, we present a common framework for seeded image segmentation algorithms that yields two of the leading methods as special cases The Graph Cuts and the Random Walker algorithms. The formulation of this common framework naturally suggests a new, third, algorithm that we develop here. Spe ..."
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Cited by 96 (10 self)
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In this work, we present a common framework for seeded image segmentation algorithms that yields two of the leading methods as special cases The Graph Cuts and the Random Walker algorithms. The formulation of this common framework naturally suggests a new, third, algorithm that we develop here. Specifically, the former algorithms may be shown to minimize a certain energy with respect to either an ℓ1 or an ℓ2 norm. Here, we explore the segmentation algorithm defined by an ℓ ∞ norm, provide a method for the optimization and show that the resulting algorithm produces an accurate segmentation that demonstrates greater stability with respect to the number of seeds employed than either the Graph Cuts or Random Walker methods.
The infinity Laplacian, Aronsson’s equation and their generalizations
"... Abstract. The infinity Laplace equation ∆∞u = 0 arose originally as a sort of Euler–Lagrange equation governing the absolute minimizer for the L ∞ variational problem of minimizing the functional esssup U Du. The more general functional esssup U F (x, u, Du) leads similarly to the socalled Aron ..."
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Cited by 49 (1 self)
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Abstract. The infinity Laplace equation ∆∞u = 0 arose originally as a sort of Euler–Lagrange equation governing the absolute minimizer for the L ∞ variational problem of minimizing the functional esssup U Du. The more general functional esssup U F (x, u, Du) leads similarly to the socalled Aronsson equation AF [u] = 0. In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to L ∞ variational problems, including twoperson game theory with random order of play, rapid switching of states in control problems, etc. The resulting equations can be parabolic and inhomogeneous, equation types precluded in conventional L ∞ variational problems. 1.
Regularity of minima: An invitation to the dark SIDE OF THE CALCULUS OF VARIATIONS
, 2006
"... I am presenting a survey of regularity results for both minima of variational integrals, and solutions to nonlinear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to ..."
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Cited by 41 (4 self)
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I am presenting a survey of regularity results for both minima of variational integrals, and solutions to nonlinear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to
A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions
 Math. Comp
, 2005
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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 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
THE NEUMANN PROBLEM FOR THE ∞LAPLACIAN AND THE MONGEKANTOROVICH MASS TRANSFER PROBLEM
"... Abstract. We consider the natural Neumann boundary condition for the ∞Laplacian. We study the limit as p → ∞ of solutions of −∆pup = 0 in a domain Ω with Dup  p−2 ∂up/∂ν = g on ∂Ω. We obtain a natural minimization problem that is verified by a limit point of {up} and a limit problem that is sati ..."
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Cited by 26 (17 self)
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Abstract. We consider the natural Neumann boundary condition for the ∞Laplacian. We study the limit as p → ∞ of solutions of −∆pup = 0 in a domain Ω with Dup  p−2 ∂up/∂ν = g on ∂Ω. We obtain a natural minimization problem that is verified by a limit point of {up} and a limit problem that is satisfied in the viscosity sense. It turns out that the limit variational problem is related to the MongeKantorovich mass transfer problems when the measures are supported on ∂Ω. 1. Introduction. In this paper we study the natural Neumann boundary conditions that appear when one considers the ∞Laplacian in a smooth bounded domain as limit of the Neumann problem for the pLaplacian as p → ∞. This problem is related to the MongeKantorovich mass tranfer problem when the involved measures are supported
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.
On Transfinite Barycentric Coordinates
, 2006
"... A general construction of transfinite barycentric coordinates is obtained as a simple and natural generalization of Floater's mean value coordinates [Flo03, JSW05b]. The GordonWixom interpolation scheme [GW74] and transfinite counterparts of discrete harmonic and WachspressWarren coordinate ..."
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Cited by 20 (0 self)
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A general construction of transfinite barycentric coordinates is obtained as a simple and natural generalization of Floater's mean value coordinates [Flo03, JSW05b]. The GordonWixom interpolation scheme [GW74] and transfinite counterparts of discrete harmonic and WachspressWarren coordinates are studied as particular cases of that general construction. Motivated by finite element/volume applications, we study capabilities of transfinite barycentric interpolation schemes to approximate harmonic and quasiharmonic functions. Finally we establish and analyze links between transfinite barycentric coordinates and certain inverse problems of di#erential and convex geometry.