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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 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
On the first nontrivial eigenvalue of the ∞Laplacian with Neumann boundary conditions
 CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina. Email address: ldpezzo@dm.uba.ar, jrossi@dm.uba.ar
"... Abstract. We study the limit as p→ ∞ of the first nonzero eigenvalue λp of the pLaplacian with Neumann boundary conditions in a smooth bounded domain U ⊂ Rn. We prove that λ ∞: = limp→+ ∞ λ1/pp = 2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can ..."
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Cited by 7 (4 self)
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Abstract. We study the limit as p→ ∞ of the first nonzero eigenvalue λp of the pLaplacian with Neumann boundary conditions in a smooth bounded domain U ⊂ Rn. We prove that λ ∞: = limp→+ ∞ λ1/pp = 2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think of λ ∞ as the first eigenvalue of the ∞Laplacian with Neumann boundary conditions. We also study the regularity of λ ∞ as a function of the domain U proving that under a smooth perturbation Ut of U by diffeomorphisms close to the identity there holds that λ∞(Ut) = λ∞(U)+O(t). Although λ∞(Ut) is in general not differentiable at t = 0, we prove that in some cases it is so with an explicit formula for the derivative. 1. introduction Denote by λp the first nonzero eigenvalue of the pLaplacian with Neumann boundary conditions in a smooth bounded domain U ⊂ Rn. The aim of this paper is twofold. We first study the asymptotic behaviour of λp as p→∞, obtaining that λ ∞: = lim p→+∞λ
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,
Optimal regularity at the free boundary for the infinity obstacle problem. Preprint
 Department of Mathematics, University of Pittsburgh. Pittsburgh, PA 15260. USA
"... Abstract. This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp C1, 1 3 –regularity at the free boundary. 1. ..."
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Abstract. This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp C1, 1 3 –regularity at the free boundary. 1.
Limits as p(x) → ∞ of p(x)harmonic functions with nonhomogeneous Neumann boundary conditions
"... Abstract. In this paper we study the limit as p(x)→ ∞ of solutions to −∆p(x)u = 0 in a domain Ω, with nonhomogeneous Neumann boundary conditions, ∇up(x) ∂u ∂η = g(x). Our approach consists on considering sequences of variable exponents converging uniformly to + ∞ and then determining the equati ..."
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Cited by 2 (1 self)
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Abstract. In this paper we study the limit as p(x)→ ∞ of solutions to −∆p(x)u = 0 in a domain Ω, with nonhomogeneous Neumann boundary conditions, ∇up(x) ∂u ∂η = g(x). Our approach consists on considering sequences of variable exponents converging uniformly to + ∞ and then determining the equation satisfied by a limit of the corresponding solutions. To Jean Pierre Gossez, with our best wishes in his 65th birthday 1.
An anisotropic infinity Laplacian obtained as the limit of the anisotropic (p, q)−Laplacian
 Comm Contemporary Mathematics
"... Abstract. In this work we study the behaviour of the solutions to the following Dirichlet problem related to the anisotropic (p, q)−Laplacian operator −divx(∇xup−2∇xu) − divy(∇yuq−2∇yu) = 0, in Ω, u = g, on ∂Ω, as p, q → ∞. Here Ω ⊂ RN × RK and ∇xu = ( ∂u∂x1, ∂u ∂x2 ..."
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Abstract. In this work we study the behaviour of the solutions to the following Dirichlet problem related to the anisotropic (p, q)−Laplacian operator −divx(∇xup−2∇xu) − divy(∇yuq−2∇yu) = 0, in Ω, u = g, on ∂Ω, as p, q → ∞. Here Ω ⊂ RN × RK and ∇xu = ( ∂u∂x1, ∂u ∂x2
Limits as p(x) → ∞ of p(x)harmonic functions
"... Abstract. In this note we study the limit as p(x) → ∞ of solutions to −∆p(x)u = 0 in a domain Ω, with Dirichlet boundary conditions. Our approach consists in considering sequences of variable exponents converging uniformly to + ∞ and analyzing how the corresponding solutions of the problem conver ..."
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Abstract. In this note we study the limit as p(x) → ∞ of solutions to −∆p(x)u = 0 in a domain Ω, with Dirichlet boundary conditions. Our approach consists in considering sequences of variable exponents converging uniformly to + ∞ and analyzing how the corresponding solutions of the problem converge and what equation is satisfied by the limit.
A limiting free boundary problem ruled by Aronsson’s equation
 Trans. Amer. Math. Soc
"... In this paper we study the behavior of the free boundary optimal design problem jZ min ∇u(X) Ω p dX ˛ u ∈ W 1,p (Ω), u = f on ∂Ω, L n ff ({u> 0}) ≤ α for large p. We obtain as the limit as p → ∞ the following “limiting problem”: min ˘ Lip(u) ˛ ˛ u ∈ W ..."
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Cited by 1 (0 self)
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In this paper we study the behavior of the free boundary optimal design problem jZ min ∇u(X) Ω p dX ˛ u ∈ W 1,p (Ω), u = f on ∂Ω, L n ff ({u> 0}) ≤ α for large p. We obtain as the limit as p → ∞ the following “limiting problem”: min ˘ Lip(u) ˛ ˛ u ∈ W