Maximal operators for the $p$-Laplacian family

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

Vol. 287 (2017), No. 2, 257–295

DOI: 10.2140/pjm.2017.287.257

Abstract

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

$$max\left\{-{\Delta }_{p_1}u\left(x\right),-{\Delta }_{p_2}u\left(x\right)\right\}=f\left(x\right)$$

in a bounded smooth domain $\Omega \subset {ℝ}^N$ with $u=g$ on $\partial \Omega$. Here $-{\Delta }_{p}u={\left(N+p\right)}^{-1}|Du{|}^{2-p}div\left(|Du{|}^{p-2}Du\right)$ is the 1-homogeneous $p$-Laplacian and we assume that $2\le p_1,p_2\le \infty$. 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 $max\left\{-{\Delta }_{p_1}u\left(x\right),-{\Delta }_{p_2}u\left(x\right)\right\}$ 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.

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