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oai:www.ideals.illinois.edu:2142/50350 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Hinkkanen, Aimo Miles, Joseph B. Loeb, Peter A. Hinkkanen, Aimo Merenkov, Sergiy A. Suwannaphichat, Sineenuch 2014-09-16T17:11:56Z 2014-09-16T17:11:56Z 2016-09-22T20:59:30Z 2014-08 2014-09-16 2014-08 We consider the problem of finding the extremal function in the class of real-valued biconvex functions satisfying a boundary condition on a product of the unit ball with itself, with a suitable norm in the plane. We want to maximize the biconvex function at a point in the domain where the second component is fixed and therefore we can consider the biconvex function as a convex function of the first component. We then find a representation for the convex function in terms of some functions of a suitable quotient of second order partial derivatives of the convex function, where these functions will satisfy certain conditions so that the biconvex function will have the given boundary values. From the quotient of second order partial derivatives of the convex function, we obtain a relation leading to the Hopf differential equation, whose solution involves a parameter function. With a given boundary function, we perform a variation of the parameter function by a small real-valued function. Then we find the change of the representation of the convex function. If the convex function is an extremal function, then the rate of change with respect to the variation made to the parameter function is zero. This will be the condition that we are looking for in an extremal function. Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-07-02T15:03:46Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Suwannaphichat_Sineenuch.pdf: 540281 bytes, checksum: 83b60b6adef81152b10b56797933a3be (MD5) Made available in DSpace on 2014-09-16T17:11:56Z (GMT). No. of bitstreams: 2 Sineenuch_Suwannaphichat.pdf: 540276 bytes, checksum: 30910f8452149b9d20612c29a6314f5d (MD5) license.txt: 4074 bytes, checksum: 8e1020ca4452e9419375cd880af19db8 (MD5) Embargo set by: Seth Robbins for item 50461 Lift date: 2016-09-16T17:13:01Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Limited Restriction Lifted for Item 50461 on 2016-09-22T20:59:30Z. http://hdl.handle.net/2142/50350 en Copyright 2014 Sineenuch Suwannaphichat Extremal functions Convex functions Extremal functions related to convexity and martingales text University of Illinois at Urbana-Champaign Dissertation Mathematics 1257 Mathematics 0439 Ph.D. PHD:Mathematics -UIUC 10KS0439PHD