Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 114, pp. 1-11.

Title: p-harmonious functions with drift on graphs via games

Author: Alexander P. Sviridov (Univ. of Pittsburgh, PA, USA)

Abstract:
 In a connected finite graph $E$ with set of vertices
 $\mathfrak{X}$, choose a nonempty subset, not equal to 
 the whole set,  $Y\subset \mathfrak{X}$,
 and call it the  boundary  $Y=\partial\mathfrak{X}$.
 Given a real-valued function  $F: Y\to \mathbb{R}$, our objective 
 is to find a function $u$, such that   $u=F$ on $Y$, and 
 for all $x\in \mathfrak{X}\setminus Y$,
 $$
 u(x)=\alpha \max_{y \in S(x)}u(y)+\beta \min_{y \in S(x)}u(y)
 +\gamma \Big( \frac{\sum_{y \in S(x)}u(y)}{\#(S(x))}\Big).
 $$
 Here  $\alpha, \beta,  \gamma $ are non-negative constants such that
 $\alpha+\beta + \gamma =1$,  the set $S(x)$ is the collection of vertices 
 connected to $x$ by an edge, and $\#(S(x))$ denotes its cardinality. 
 We prove the existence and uniqueness of  a solution of the above 
 Dirichlet problem and study the qualitative properties of the solution.

Submitted October 26, 2010. Published September 06, 2011.
Math Subject Classifications: 35Q91, 35B51, 34A12, 31C20.
Key Words: Dirichlet problem; comparison principle; mean-value property;  
           stochastic games; unique continuation.