Sviridov, Alexander Petrovich
(2011)
Elliptic equations in graphs via stochastic games.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
Consider a connected ﬁnite graph E with set of vertices X. Choose a nonempty subset Y ⊂ X, not equal to the whole X, and call it the boundary Y = ∂ X. We are given a real valued function F : Y → R. Our objective is to ﬁnd function u on X, such that u = F on Y and u satisﬁes the following equation for all x ∈ X Y u(x) = α max_(y ∈ S(x)) u(y) + β min_(y ∈ S(x))u(y) + γ( ∑_(y ∈ S(x)) u(y) ⁄ #(S(x) ) , (1)where α, β, and γ are some predetermined nonnegative constants such that α + β + γ = 1, for x ∈ X, #S(x) is the set of vertices connected to x by an edge, and #(S (x)) denotes the cardinality of S (x). We prove existence and uniqueness of the solution of the above Dirichlet problem and study qualitative properties of the solutions.
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Details
Item Type: 
University of Pittsburgh ETD

Status: 
Unpublished 
Creators/Authors: 

ETD Committee: 

Date: 
21 July 2011 
Date Type: 
Completion 
Defense Date: 
18 June 2010 
Approval Date: 
21 July 2011 
Submission Date: 
5 November 2010 
Access Restriction: 
No restriction; Release the ETD for access worldwide immediately. 
Institution: 
University of Pittsburgh 
Schools and Programs: 
Dietrich School of Arts and Sciences > Mathematics 
Degree: 
PhD  Doctor of Philosophy 
Thesis Type: 
Doctoral Dissertation 
Refereed: 
Yes 
Uncontrolled Keywords: 
inﬁnity harmonic function; pharmonic function; pharmonious function; stochastic games; unique continuation 
Other ID: 
http://etd.library.pitt.edu/ETD/available/etd11052010120535/, etd11052010120535 
Date Deposited: 
10 Nov 2011 20:04 
Last Modified: 
15 Nov 2016 13:51 
URI: 
http://dscholarship.pitt.edu/id/eprint/9581 
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