Elliptic equations in graphs via stochastic games

Sviridov, Alexander Petrovich (2011) Elliptic equations in graphs via stochastic games. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

Consider a connected finite graph E with set of vertices X. Choose a nonempty subset Y &#8834 X, not equal to the whole X, and call it the boundary Y = ∂ X. We are given a real valued function F : Y &#8594 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 &#8712 X Y u(x) = &#945 max_(y &#8712 S(x)) u(y) + &#946 min_(y &#8712 S(x))u(y) + &#947( &#8721_(y &#8712 S(x)) u(y) &#8260 #(S(x) ) , (1)where &#945, &#946, and &#947 are some predetermined non-negative constants such that &#945 + &#946 + &#947 = 1, for x &#8712 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Status:	Unpublished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:	Creators Email Pitt Username ORCID Sviridov, Alexander Petrovich alexsviridov@gmail.com
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Manfredi, Juan J manfredi@pitt.edu MANFREDI Committee Member Vainchtein, Anna annav@math.pitt.edu AAV4 Committee Member Chadam, John chadam@pitt.edu CHADAM Committee Member Hajlasz, Piotr hajlasz@pitt.edu HAJLASZ Committee Member Iyengar, Satish ssi@pitt.edu SSI
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; p-harmonic function; p-harmonious function; stochastic games; unique continuation
Other ID:	http://etd.library.pitt.edu/ETD/available/etd-11052010-120535/, etd-11052010-120535
Date Deposited:	10 Nov 2011 20:04
Last Modified:	15 Nov 2016 13:51
URI:	http://d-scholarship.pitt.edu/id/eprint/9581

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