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

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.


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
Full Text: arXiv