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A GameTree approach to discrete infinity Laplacian with running costs
, 2013
"... We give a selfcontained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tugofwar games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduc ..."
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We give a selfcontained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tugofwar games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce gametrees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the uniqueness. Our arguments are entirely deterministic, and also do not rely on (semi)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for wellposedness.
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.