Maximal operators for the \(p\)-Laplacian family. (English) Zbl 1362.35143
Summary: We prove existence and uniqueness of viscosity solutions for the problem:
\[
\max\{-\Delta_{p_1}u (x), \, -\Delta_{p_2}u (x)\}=f(x)
\]
in a bounded smooth domain \(\Omega \subset \mathbb{R}^N\) with \(u=g\) on \(\partial \Omega\). Here \(-\Delta_{p}u = (N+p)^{-1}| Du|^{2-p} \operatorname{div} \bigl(| Du|^{p-2} Du\bigr)\) is the 1-homogeneous \(p\)-Laplacian and we assume that \(2\leq p_1,p_2\leq \infty\). 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 \(\max\{-\Delta_{p_1}u (x), \, -\Delta_{p_2}u (x)\}\) 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.
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.
MSC:
35J70 | Degenerate elliptic equations |
49N70 | Differential games and control |
91A15 | Stochastic games, stochastic differential games |
91A24 | Positional games (pursuit and evasion, etc.) |
References:
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.