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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
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.
An infinity Laplace equation with gradient term and mixed boundary conditions
, 910
"... Abstract. We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation −Δ∞u − β|Du | = f, subject to Dirichlet or mixed Dirichlet-Neumann 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 1-homogeneous infinity Laplace equation −Δ∞u − β|Du | = f, subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation. 1.
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 Du|Du | , Du|Du | 〉 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 Du|Du | , Du|Du | 〉 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 Lp-approximation 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 tug-of-war games we prove that this problem has a solution.
An existence result for the infinity Laplacian with non-homogeneous Neumann boundary conditions using tug-of-war games
, 2009
"... In this paper we show how to use a Tug-of-War 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 Tug-of-War 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,
Local Dynamics in Bargaining Networks via Random-turn Games
"... We present a new technique for analyzing the rate of convergence of local dynamics in bargaining networks. The technique reduces balancing in a bargaining network to optimal play in a randomturn game. We analyze this game using techniques from martingale and Markov chain theory. We obtain a tight p ..."
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Cited by 3 (0 self)
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We present a new technique for analyzing the rate of convergence of local dynamics in bargaining networks. The technique reduces balancing in a bargaining network to optimal play in a randomturn game. We analyze this game using techniques from martingale and Markov chain theory. We obtain a tight polynomial bound on the rate of convergence for a nontrivial class of unweighted graphs (the previous known bound was exponential). Additionally, we show this technique extends naturally to many other graphs and dynamics.
TUG-OF-WAR GAMES AND PARABOLIC PROBLEMS WITH SPATIAL AND TIME DEPENDENCE
"... Abstract. In this paper we use probabilistic arguments (Tug-of-War 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 (Tug-of-War games) to obtain existence of viscosity solutions to a parabolic problem of the form{ K(x,t)(Du)ut(x, t) =
TUG-OF-WAR GAMES AND THE INFINITY LAPLACIAN WITH SPATIAL DEPENDENCE
"... In this paper we look for PDEs that arise as limits of values of Tug-of-War 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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Cited by 2 (1 self)
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In this paper we look for PDEs that arise as limits of values of Tug-of-War 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.
MAXIMAL OPERATORS FOR THE p-LAPLACIAN FAMILY
, 2015
"... We prove existence and uniqueness of viscosity solutions for the following problem: max {−∆p1u(x), −∆p2u(x)} = f(x) in a bounded smooth domain Ω ⊂ RN with u = g on ∂Ω. Here −∆pu = (N + p)−1|Du|2−pdiv(|Du|p−2Du) is the 1-homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equation appe ..."
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Cited by 1 (1 self)
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We prove existence and uniqueness of viscosity solutions for the following problem: max {−∆p1u(x), −∆p2u(x)} = f(x) in a bounded smooth domain Ω ⊂ RN with u = g on ∂Ω. Here −∆pu = (N + p)−1|Du|2−pdiv(|Du|p−2Du) is the 1-homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the pa-rameters of the game that regulate the probability of playing a usual Tug-of-War game (without noise) or to play at random. Moreover, the opera-tor max {−∆p1u(x), −∆p2u(x)} provides a natural analogous with respect to p−Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this prob-lem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory.
