Optimization problems in partial differential equations

Liu, Yichen Optimization problems in partial differential equations. PhD thesis, University of Liverpool.

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Abstract

The primary objective of this research is to investigate various optimization problems connected with partial differential equations (PDE). In chapter 2, we utilize the tool of tangent cones from convex analysis to prove the existence and uniqueness of a minimization problem. Since the admissible set considered in chapter 2 is a suitable convex set in $L^\infty(D)$, we can make use of tangent cones to derive the optimality condition for the problem. However, if we let the admissible set to be a rearrangement class generated by a general function (not a characteristic function), the method of tangent cones may not be applied. The central part of this research is Chapter 3, and it is conducted based on the foundation work mainly clarified by Geoffrey R. Burton with his collaborators near 90s, see [7, 8, 9, 10]. Usually, we consider a rearrangement class (a set comprising all rearrangements of a prescribed function) and then optimize some energy functional related to partial differential equations on this class or part of it. So, we call it rearrangement optimization problem (ROP). In recent years this area of research has become increasingly popular amongst mathematicians for several reasons. One reason is that many physical phenomena can be naturally formulated as ROPs. Another reason is that ROPs have natural links with other branches of mathematics such as geometry, free boundary problems, convex analysis, differential equations, and more. Lastly, such optimization problems also offer very challenging questions that are fascinating for researchers, see for example [2]. More specifically, Chapter 2 and Chapter 3 are prepared based on four papers [24, 40, 41, 42], mainly in collaboration with Behrouz Emamizadeh. Chapter 4 is inspired by [5]. In [5], the existence and uniqueness of solutions of various PDEs involving Radon measures are presented. In order to establish a connection between rearrangements and PDEs involving Radon measures, the author try to investigate a way to extend the notion of rearrangement of functions to rearrangement of Radon measures in Chapter 4.

Item Type:	Thesis (PhD)
Additional Information:	Date: 2015-07 (completed)
Subjects:	?? QA ??
Depositing User:	Symplectic Admin
Date Deposited:	24 Aug 2015 10:13
Last Modified:	16 Dec 2022 04:43
DOI:	10.17638/02015545
URI:	https://livrepository.liverpool.ac.uk/id/eprint/2015545