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Title:  Continuity Properties and Variational Problems Involving the Determinant of the Hessian 
Author(s):  Jung, Nara 
Doctoral Committee Chair(s):  Jerrard, Robert L. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  This thesis investigates continuity properties of the distributional determinant of the Hessian of a scalar function u in various function spaces. It is well known that when D is a bounded, open subset of ndimensional Euclidean space Rn, the distributional determinant of the Hessian is weakly continuous in the Sobolev space of functions whose second derivatives are Lp functions on D, denoted by W(2,p)(D), for p greater than (nn/n+2) when n is greater than or equal to 3, and for p greater than or equal to 1 when n = 2. I show that it not strongly continuous in the norm topology in W(2,p)(D), when D is a subset of Rn for n greater than or equal to 3 and p is less than (nn/n+2) and similarly that it fails to be strongly continuous in W(1,p)(D), when D is a subset of R2 and p is less than 2. As my main result, I prove that when D is a subset of R3 then the map from u to Det(Hessian of u) is a continuous function from the intersection of BV2 and W(1, infty)(D), with a suitable topology, into the space of distributions. Here BV2 is the space of functions whose first derivatives are functions of bounded variation. This function space is naturally chosen to investigate a variational problem involving integral of the absolute value of det(Hessian of u). 
Issue Date:  2003 
Type:  Text 
Language:  English 
Description:  56 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2003. 
URI:  http://hdl.handle.net/2142/86811 
Other Identifier(s):  (MiAaPQ)AAI3086092 
Date Available in IDEALS:  20150928 
Date Deposited:  2003 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois