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18
Harnack’s inequality for pharmonic functions via stochastic games
 Comm. Partial Differential Equations
, 1985
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TUGOFWAR GAMES AND THE INFINITY LAPLACIAN WITH SPATIAL DEPENDENCE
"... In this paper we look for PDEs that arise as limits of values of TugofWar 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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In this paper we look for PDEs that arise as limits of values of TugofWar 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.
AN OBSTACLE PROBLEM FOR TUGOFWAR GAMES
"... We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinityharmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above t ..."
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We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinityharmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tugofwar.
On the horizontal Mean Curvature Flow for Axisymmetric surfaces in the Heisenberg Group, preprint
, 2012
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Nonlinear elliptic partial differential equations and pharmonic functions on graphs
, 2013
"... In this article we study the wellposedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit ..."
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In this article we study the wellposedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for partial differential equations. The equations include the graph Laplacian, the pLaplacian, the Infinity Laplacian, and the Eikonal operator on the graph.
Local regularity results for value functions of tugofwar with noise and running payoff
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TUGOFWAR GAMES. GAMES THAT PDE PEOPLE LIKE TO PLAY.
"... Abstract. In these notes we review some recent results concerning TugofWar games and their relation to some well known PDEs. In particular, we will show that solutions to certain PDEs can be obtained as limits of values of TugofWar games when the parameter that controls the length of the possibl ..."
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Abstract. In these notes we review some recent results concerning TugofWar games and their relation to some well known PDEs. In particular, we will show that solutions to certain PDEs can be obtained as limits of values of TugofWar games when the parameter that controls the length of the possible movements goes to zero. Since the equations under study are nonlinear and not in divergence form we will make extensive use of the concept of viscosity solutions. 1.