Entire Solutions to Dirichlet Type Problems
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Date
2007-08-02T17:33:19Z
Authors
Sitar, Scott
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
In this thesis, we examined some Dirichlet type problems of
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.
Description
Keywords
partial differential equations, harmonic functions, entire functions, potential theory