Geometric variational problems with cross-sectional constraints

We study here some geometric variational problems motivated by the modeling of plants' growth. In Chapter 2, we conclude the general existence of an area minimizing surface with given boundary on two parallel hyperplanes and the constraint that the intersection of the surface with each hyperplane parallel to these of the boundary encloses the same area. In Chapter 3, we study the area minimizing surface in R 3 whose intersection with each of the hyperplanes R 2 x {h}, h ∈ [0, 1] encloses a prescribed area. We conclude that, up to a translation, the minimizer exists and is invariant under revolution. In Chapter 4, as a specific case of the problem in Chapter 2, the minimizing surface bounded by two parallel circles of the same size is studied carefully. We conclude that such an area minimizing surface is the skewed cylinder determined by the two circles. In Chapter 5, we study an analogous energy minimizing problem in PDE with a boundary constraint and a cross-sectional constraint on the L1 norm over a rectangular region. The even terms and the conditions for the odd terms in the Fourier expansion of the energy minimizer are given.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Natural Sciences

Grantor

Rice University

Year dc:date.issued

2005

Author dc:creator

• Meng, Zheng

Advisor dc:contributor.advisor

• Hardt, Robert M.

• Mathematics

Statement dc:rights

• Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1911/18846

OAI identifier oai:identifier

oai:repository.rice.edu:1911/18846