Energy minimizers, gradient flow solutions, and computational investigations in the theory of biharmonic maps

Simple item page
dc.contributor.advisorHardt, Robert M.en_US
dc.creatorLedbetter, Ashley Annen_US
dc.date.accessioned2009-06-04T08:02:21Zen_US
dc.date.available2009-06-04T08:02:21Zen_US
dc.date.issued1998en_US
dc.description.abstractWe present the definitions, derive the relevant Euler-Lagrange equations, and establish various properties concerning biharmonic maps. We investigate several classes of examples exhibiting singular behavior. Existence of a weak solution to the associated evolution equation is proved using a penalization argument and the Galerkin method. We prove higher integrability of order greater than two for derivatives of Laplacian energy minimizers contingent upon certain energy constraints. We initiate a numerical analysis of biharmonic maps using a discrete Laplacian energy, a finite difference scheme, and involving spherical coordinates in a variety of dimensions in order to understand isolated singularities. Since singularities of Laplacian energy minimizers first occur in dimension five, where both mathematical and numerical analysis is quite complicated, we also consider an analogous problem for a lower non-integer power, p, of the Laplacian. The Euler-Lagrange equations are now not only nonlinear and fourth-order, but also degenerate. Nevertheless, we are able to analyze a specific solution of the p-biharmonic map equation having a degree two singularity. We prove the non-minimality of the solution by a construction that splits this singularity. In this work, we have developed relevant computer code to expedite biharmonicity testing and energy computations. We have also developed computer code applicable to the initial study of the discretized biharmonic problem. We hope this code will be useful in future research aimed at a more comprehensive numerical analysis.en_US
dc.format.extent148 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS MATH. 1998 LEDBETTERen_US
dc.identifier.citationLedbetter, Ashley Ann. "Energy minimizers, gradient flow solutions, and computational investigations in the theory of biharmonic maps." (1998) Diss., Rice University. <a href="https://hdl.handle.net/1911/19279">https://hdl.handle.net/1911/19279</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/19279en_US
dc.language.isoengen_US
dc.rightsCopyright 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.en_US
dc.subjectMathematicsen_US
dc.titleEnergy minimizers, gradient flow solutions, and computational investigations in the theory of biharmonic mapsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineNatural Sciencesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
Files
Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
9827412.PDF
Size:
3.4 MB
Format:
Adobe Portable Document Format
Download
Collections
Rice University Theses and Dissertations