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50
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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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.
Convexity criteria and uniqueness of absolutely minimizing functions
, 2010
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The 1-Laplacian, the ∞-Laplacian and Differential Games
"... Abstract. This paper considers the p-Laplacian PDE for p = 1, ∞ and some interesting new game theoretic interpretations, due to Kohn–Serfaty [K-S] and to Peres–Schramm–Sheffield–Wilson [P-S-S-W]. ..."
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Cited by 6 (0 self)
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Abstract. This paper considers the p-Laplacian PDE for p = 1, ∞ and some interesting new game theoretic interpretations, due to Kohn–Serfaty [K-S] and to Peres–Schramm–Sheffield–Wilson [P-S-S-W].
INFINITY LAPLACE EQUATION WITH NON-TRIVIAL RIGHT-HAND SIDE
, 2010
"... We analyze the set of continuous viscosity solutions of the infinity Laplace equation − ∆ N ∞w(x) = f(x), with generally sign-changing right-hand 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 sign-changing right-hand 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.
Tug of war with noise: a game theoretic view of the p-Laplacian
- Duke Math. J
"... Fix a bounded domain Ω ⊂ R d, a continuous function F: ∂Ω → R, and constants ǫ> 0 and 1 < p,q < ∞ with p −1 + q −1 = 1. For each x ∈ Ω, let u ǫ (x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and ..."
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Cited by 5 (0 self)
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Fix a bounded domain Ω ⊂ R d, a continuous function F: ∂Ω → R, and constants ǫ> 0 and 1 < p,q < ∞ with p −1 + q −1 = 1. For each x ∈ Ω, let u ǫ (x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and the player who wins the toss chooses a vector v ∈ B(0,ǫ) to add to the game position, after which a random “noise vector ” with mean zero and variance q p |v|2 in each orthogonal direction is also added. The game ends when the game position reaches some y ∈ ∂Ω, and player I’s payoff is F(y). We show that (for sufficiently regular Ω) as ǫ tends to zero the functions u ǫ converge uniformly to the unique p-harmonic extension of F. Using a modified game (in which ǫ gets smaller as the game position approaches ∂Ω), we prove similar statements for general bounded domains Ω and resolutive functions F. These games and their variants interpolate between the tug of war games studied by Peres, Schramm, Sheffield, and Wilson (p = ∞) and the motion-by-curvature games introduced by Spencer and studied by Kohn and Serfaty (p = 1). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about p-capacity and p-harmonic measure. 1
Variational Problems for Maps and the Aronsson PDE System
- October 2012, Pages 2123 - 2139. K2. N. Katzourakis, ∞-Minimal Submanifolds, Proceedings of the AMS
"... Abstract. By employing Aronsson’s Absolute Minimizers of L ∞ functionals, we prove that Absolutely Minimizing Maps u: Rn − → RN solve a “tangential” Aronsson PDE system. By following Sheffield-Smart [SS], we derive ∆ ∞ with respect to the dual operator norm and show that such maps miss information a ..."
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Abstract. By employing Aronsson’s Absolute Minimizers of L ∞ functionals, we prove that Absolutely Minimizing Maps u: Rn − → RN solve a “tangential” Aronsson PDE system. By following Sheffield-Smart [SS], we derive ∆ ∞ with respect to the dual operator norm and show that such maps miss information along a hyperplane when compared to Tight Maps. We recover the lost term which causes non-uniqueness and derive the complete Aronsson system which has discontinuous coefficients. In particular, the Euclidean ∞-Laplacian is ∆∞u = Du ⊗Du: D2u + |Du|2[Du]⊥∆u where [Du] ⊥ is the projection on the null space of Du>. We exibit C ∞ solutions having interfaces along which the rank of their gradient is discontinuous and propose a modification with C0 coefficients which admits varifold solutions. Away from the interfaces, Aronsson Maps satisfy a structural property of local splitting to 2 phases, an horizontal and a vertical; horizontally they possess gradient flows similar to the scalar case and vertically solve a linear system coupled by a scalar Hamilton Jacobi PDE. We also construct singular∞-Harmonic local C1 diffeomorphisms and singular Aronsson Maps.
Non-Local Tug-of-War and the Infinity Fractional Laplacian
- Comm. Pure Applied Math
"... Abstract. Motivated by the “tug-of-war ” game studied in [12], we consider a “non-local” version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving of a fixed amo ..."
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Abstract. Motivated by the “tug-of-war ” game studied in [12], we consider a “non-local” version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving of a fixed amount > 0 (as is done in the classical case), it is a s-stable Levy process which chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically we derive a deterministic non-local integro-differential equation that we call “infinity fractional Laplacian”. We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double obstacle problem, both problems having a natural interpretation as “tug-of-war ” games. 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.
On the value of stochastic differential games
, 2010
"... Abstract: We consider a two player, zero sum stochastic differential game based on a formulation given by Fleming and Souganidis. The saddle point property is introduced, and it is proved that the unique uniformly continuous bounded viscosity solution of the upper Isaacs PDE with boundary condition ..."
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Abstract: We consider a two player, zero sum stochastic differential game based on a formulation given by Fleming and Souganidis. The saddle point property is introduced, and it is proved that the unique uniformly continuous bounded viscosity solution of the upper Isaacs PDE with boundary condition satisfies such a property. Also, it is shown that approximately optimal Markov strategies can be constructed for both players. 1
