On the existence and uniqueness of \(p\)-harmonious functions. (English) Zbl 1324.49029

Luiro, Hannes; Parviainen, Mikko; Saksman, Eero

Differ. Integral Equ. 27, No. 3-4, 201-216 (2014).

In this short and nice manuscript, the authors show an easy and self contained proof of existence and uniqueness of \(p\)-harmonious functions, that is, functions that verify \[ u (x) = \frac {\alpha}{2} \left (\sup_{B_\epsilon (x)} u + \inf_{B_\epsilon (x)} u \right) + \beta \frac {1}{| {B_\epsilon (x)}|}\int_{B_\epsilon (x)} u \, dy. \] The proofs only use elementary analytic tools and are not based on game theoretic arguments.

Reviewer: Julio Daniel Rossi (Buenos Aires)

MSC:

49N70	Differential games and control
35A35	Theoretical approximation in context of PDEs
35J92	Quasilinear elliptic equations with \(p\)-Laplacian
91A05	2-person games
91A15	Stochastic games, stochastic differential games
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35Q91	PDEs in connection with game theory, economics, social and behavioral sciences
49L20	Dynamic programming in optimal control and differential games

Keywords:

dynamic programming principle; mean-value iteration; measurability; \(p\)-harmonious functions; \(p\)-Laplacian; tug-of-war with noise