| Summary: | We study the problem of finding functions, defined within and on an ellipse,
whose Laplacian is -1 and which satisfy a homogeneous Robin boundary condition
on the ellipse. The parameter in the Robin condition is denoted by beta. The
general solution and various asymptotic approximations are obtained. To find
the general solution, the boundary value problem is formulated in elliptic
cylindrical coordinates. A Fourier series solution is then derived. The Fourier
coefficients satisfy a 3-term recurrence relation which can be solved. The
integral of the solution over the ellipse, denoted by Q, is a quantity of
interest in some physical applications. The dependence of Q on beta and the
ellipse geometry is found. Finding asymptotics directly from the pde
formulations is easier than from our series solution. We use the asymptotic
approximations to Q as checks on the series solution. Several other
inequalities are also used to check the solution. It is intended that this
arXiv preprint will be referenced by the journal version, which will be
submitted soon, as the arXiv contains material, e.g. codes for calculating Q,
not in the much shorter journal version. Maple codes used in deriving or
checking results in this paper are in the process of being tidied prior to
being made available via links given at the URL given in the pdf version.
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