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 Title: Extremal functions related to convexity and martingales Author(s): Suwannaphichat, Sineenuch Director of Research: Hinkkanen, Aimo Doctoral Committee Chair(s): Miles, Joseph B.; Loeb, Peter A. Doctoral Committee Member(s): Hinkkanen, Aimo; Merenkov, Sergiy A. Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Extremal functions Convex functions Abstract: 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. Issue Date: 2014-09-16 URI: http://hdl.handle.net/2142/50350 Rights Information: Copyright 2014 Sineenuch Suwannaphichat Date Available in IDEALS: 2014-09-162016-09-22 Date Deposited: 2014-08
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