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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 (Unpublished)

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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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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Sviridov, Alexander
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairManfredi, Juan Jmanfredi@pitt.eduMANFREDI
Committee MemberVainchtein, Annaannav@math.pitt.eduAAV4
Committee MemberChadam, Johnchadam@pitt.eduCHADAM
Committee MemberHajlasz, Piotrhajlasz@pitt.eduHAJLASZ
Committee MemberIyengar, Satishssi@pitt.eduSSI
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:, etd-11052010-120535
Date Deposited: 10 Nov 2011 20:04
Last Modified: 15 Nov 2016 13:51


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