Continuity Properties and Variational Problems Involving the Determinant of the Hessian

Jung, Nara

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Description

Title

Continuity Properties and Variational Problems Involving the Determinant of the Hessian

Author(s)

Jung, Nara

Issue Date

2003

Doctoral Committee Chair(s)

Jerrard, Robert L.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

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 n-dimensional 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).

Graduation Semester

2003

Type of Resource

text

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http://hdl.handle.net/2142/86811

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