dc.contributor.author Sitar, Scott dc.date.accessioned 2007-08-02 17:33:19 (GMT) dc.date.available 2007-08-02 17:33:19 (GMT) dc.date.issued 2007-08-02T17:33:19Z dc.date.submitted 2007 dc.identifier.uri http://hdl.handle.net/10012/3147 dc.description.abstract In this thesis, we examined some Dirichlet type problems of en the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems. dc.format.extent 430377 bytes dc.format.mimetype application/pdf dc.language.iso en en dc.publisher University of Waterloo en dc.subject partial differential equations en dc.subject harmonic functions en dc.subject entire functions en dc.subject potential theory en dc.title Entire Solutions to Dirichlet Type Problems en dc.type Master Thesis en dc.pending false en dc.subject.program Applied Mathematics en uws-etd.degree.department Applied Mathematics en uws-etd.degree Master of Mathematics en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
﻿

### This item appears in the following Collection(s)

UWSpace

University of Waterloo Library
200 University Avenue West