Vol. 287, No. 2, 2017

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Maximal operators for the $p$-Laplacian family

Pablo Blanc, Juan P. Pinasco and Julio D. Rossi

Vol. 287 (2017), No. 2, 257–295
Abstract

We prove existence and uniqueness of viscosity solutions for the problem:

max{Δp1u(x), Δp2u(x)} = f(x)

in a bounded smooth domain Ω N with u = g on Ω. Here Δpu = (N + p)1|Du|2p div(|Du|p2Du) is the 1-homogeneous p-Laplacian and we assume that 2 p1,p2 . This equation appears naturally when one considers a tug-of-war 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 tug-of-war game (without noise) or playing at random. Moreover, the operator max{Δp1u(x), Δp2u(x)} provides a natural analogue 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.

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Keywords
Dirichlet boundary conditions, dynamic programming principle, p-Laplacian, tug-of-war games
Mathematical Subject Classification 2010
Primary: 35J70, 49N70, 91A15, 91A24
Milestones
Received: 5 June 2015
Revised: 23 June 2016
Accepted: 22 September 2016
Published: 9 March 2017
Authors
Pablo Blanc
Departamento de Matemática FCEyN
Universidad de Buenos Aires
Ciudad Universitaria, Pabellòn 1 (1428)
Buenos Aires
Argentina
Juan P. Pinasco
Departamento de Matemática FCEyN
Universidad de Buenos Aires
Ciudad Universitaria, Pabellòn 1 (1428)
Buenos Aires
Argentina
Julio D. Rossi
Departamento de Matemática FCEyN
Universidad de Buenos Aires
Ciudad Universitaria, Pabellòn 1 (1428)
Buenos Aires
Argentina