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22
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.
On the definition and properties of pharmonious 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) =
Vector Valued Optimal Lipschitz Extensions
 2012, 128  154. 30 NICHOLAS KATZOURAKIS
"... Abstract. Consider a bounded open set U ⊂ Rn and a Lipschitz function g: ∂U → Rm. Does this function always have a canonical optimal Lipschitz extension to all of U? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, ..."
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Cited by 14 (1 self)
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Abstract. Consider a bounded open set U ⊂ Rn and a Lipschitz function g: ∂U → Rm. Does this function always have a canonical optimal Lipschitz extension to all of U? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs.
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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Cited by 9 (1 self)
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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.
INFINITY LAPLACE EQUATION WITH NONTRIVIAL RIGHTHAND SIDE
, 2010
"... We analyze the set of continuous viscosity solutions of the infinity Laplace equation − ∆ N ∞w(x) = f(x), with generally signchanging righthand 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 signchanging righthand 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.
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.
An existence result for the infinity Laplacian with nonhomogeneous Neumann boundary conditions using tugofwar games
, 2009
"... In this paper we show how to use a TugofWar game to obtain existence of a viscosity solution to the infinity laplacian with nonhomogeneous mixed boundary conditions. For a Lipschitz and positive function g there exists a viscosity solution of the mixed boundary value problem, 8 −∆∞u(x) = 0 in Ω ..."
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Cited by 3 (0 self)
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In this paper we show how to use a TugofWar game to obtain existence of a viscosity solution to the infinity laplacian with nonhomogeneous mixed boundary conditions. For a Lipschitz and positive function g there exists a viscosity solution of the mixed boundary value problem, 8 −∆∞u(x) = 0 in Ω,