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.