Electron. J. Diff. Equ., Vol. 2011 (2011), No. 114, pp. 1-11.

p-harmonious functions with drift on graphs via games

Alexander P. Sviridov

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 6, 2011.
Math Subject Classifications: 35Q91, 35B51, 34A12, 31C20.
Key Words: Dirichlet problem; comparison principle; mean-value property; stochastic games; unique continuation.

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Alexander P. Sviridov
Department of Mathematics, University of Pittsburgh
Pittsburgh, PA 15260, USA
email:aps14@pitt.edu

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