p-HARMONIOUS FUNCTIONS WITH DRIFT ON GRAPHS VIA GAMES

In a connected finite graph E with set of vertices X, choose a nonempty subset, not equal to the whole set, Y ⊂ X, and call it the boundary Y = ∂X. Given a real-valued function F : Y → R, our objective is to find a function u, such that u = F on Y , and for all x ∈ X \ Y , u(x) = α max y∈S(x) u(y) + β min y∈S(x) u(y) + γ “P y∈S(x) u(y) #(S(x)) ” . Here α, β, γ are non-negative constants such that α+β+γ = 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.