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# Boundary perturbation of the Laplace eigenvalues and applications to electron bubbles and polygons

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 dc.contributor.advisor Gilbert Strang. en_US dc.contributor.author Greenfield, Pavel, 1974- en_US dc.contributor.other Massachusetts Institute of Technology. Dept. of Mathematics. en_US dc.date.accessioned 2005-10-14T19:58:49Z dc.date.available 2005-10-14T19:58:49Z dc.date.copyright 2003 en_US dc.date.issued 2003 en_US dc.identifier.uri http://hdl.handle.net/1721.1/29345 dc.description Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. en_US dc.description Includes bibliographical references (p. 89-91). en_US dc.description.abstract We analyze the evolution of Laplace eigenvalues on a domain induced by the motion of the boundary. We apply our analysis to two problems: 1. We study the equilibrium and stability of electron bubbles. Electron bubbles are cavities formed around electrons injected into liquid helium. They can be treated as simple mathematical systems that minimize the energy with three terms: the energy of the electron proportional to a Laplace eigenvalue, the surface energy proportional to the surface area of the cavity, and the hydrostatic pressure proportional to its volume. This system possesses a surprising result: an instability of the 2S electron bubbles. 2. We compute the simple eigenvalues on a regular polygon with N sides. The polygon is treated as a perturbation of the unit circle and its eigenvalues are approximated by a Taylor series. The accuracy of our approach is measured by comparison with finite element estimates. For the lowest eigenvalue, the first Taylor term provides an estimate within 10-5 of the true value. The second term reduces the error to 10-7. We discuss how to utilize the available symmetry to obtain better finite element estimates. Finally, we briefly discuss the expansion of simple eigenvalues on regular polygons in powers of 1/N. en_US dc.description.statementofresponsibility by Pavel Greenfield. en_US dc.format.extent 91 p. en_US dc.format.extent 3847929 bytes dc.format.extent 3847738 bytes dc.format.mimetype application/pdf dc.format.mimetype application/pdf dc.language.iso eng en_US dc.publisher Massachusetts Institute of Technology en_US dc.rights M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. en_US dc.rights.uri http://dspace.mit.edu/handle/1721.1/7582 dc.subject Mathematics. en_US dc.title Boundary perturbation of the Laplace eigenvalues and applications to electron bubbles and polygons en_US dc.type Thesis en_US dc.description.degree Ph.D. en_US dc.contributor.department Massachusetts Institute of Technology. Dept. of Mathematics. en_US dc.identifier.oclc 52767086 en_US
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