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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 characterization of pharmonic functions on the Heisenberg group by mean value properties, preprint
, 2012
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On the horizontal Mean Curvature Flow for Axisymmetric surfaces in the Heisenberg Group, preprint
, 2012
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MAXIMAL OPERATORS FOR THE pLAPLACIAN 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)−1Du2−pdiv(Dup−2Du) is the 1homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equation appe ..."
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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)−1Du2−pdiv(Dup−2Du) is the 1homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equation appears naturally when one considers a tugofwar game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the parameters of the game that regulate the probability of playing a usual TugofWar game (without noise) or to play at random. Moreover, the operator 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 problem. 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.
Local regularity results for value functions of tugofwar with noise and running payoff
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THE LIMIT AS p→ ∞ FOR THE EIGENVALUE PROBLEM OF THE 1HOMOGENEOUS pLAPLACIAN
"... Abstract. In this paper we study asymptotics as p→ ∞ of the Dirichlet eigenvalue problem for the 1homogeneous pLaplacian, that is, { − 1 p Du2−pdiv (Dup−2Du) = λu, in Ω, u = 0, on ∂Ω. Here Ω is a bounded starshaped domain in Rn and p> n. There exists a principal eigenvalue λ1,p(Ω), whic ..."
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Abstract. In this paper we study asymptotics as p→ ∞ of the Dirichlet eigenvalue problem for the 1homogeneous pLaplacian, that is, { − 1 p Du2−pdiv (Dup−2Du) = λu, in Ω, u = 0, on ∂Ω. Here Ω is a bounded starshaped domain in Rn and p> n. There exists a principal eigenvalue λ1,p(Ω), which is positive, and has associated a nonnegative nontrivial eigenfunction. Moreover, we show that limp→ ∞ λ1,p(Ω) = λ1,∞(Ω), where λ1,∞(Ω) is the first eigenvalue corresponding to the 1homogeneous infinity Laplacian, that is, −
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.
THE SUBLINEAR PROBLEM FOR THE 1HOMOGENEOUS pLAPLACIAN
"... Abstract. In this paper we prove existence and uniqueness of a positive viscosity solution of the 1homogeneous pLaplacian with a sublinear righthand side, that is, −Du2−pdiv (Dup−2Du) = λuq in Ω, u = 0 on ∂Ω, where Ω is a bounded starshaped domain, λ> 0, p> 2 and 0 < q < 1. 1. ..."
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Abstract. In this paper we prove existence and uniqueness of a positive viscosity solution of the 1homogeneous pLaplacian with a sublinear righthand side, that is, −Du2−pdiv (Dup−2Du) = λuq in Ω, u = 0 on ∂Ω, where Ω is a bounded starshaped domain, λ> 0, p> 2 and 0 < q < 1. 1.