Extremal functions related to convexity and martingales
type: CreativeWork
description: 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.
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.
author: Suwannaphichat, Sineenuch
language: en
keywords: Extremal functions, Convex functions
date published: 2014-09-16T17:11:56Z
date modified: 2026-07-22T22:25:40Z
provider: University of Illinois - Urbana-Champaign
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Illinois at Urbana-Champaign

PropertyValue
property ID: Handle
value: http://hdl.handle.net/2142/50350