From: "lester l. helms" <l-helms@math.uiuc.edu>
Subject: Re: Green Function with Neumann condition
Date: Thu, 19 Oct 2000 20:41:59 -0700
Newsgroups: sci.math
Summary: [missing]

The Green function for the Neumann problem on a planar disk can be derived
using Fourier series as in the R.V. Churchill and J.W.Brown book Complex
Variables and Applications, 4th Edition, 1984, and in the S.G. Mikhlin book
Mathematical Physics, An Advanced Course, 1970. Integral representations of
solutions of the Neumann problem for half-spaces, quadrants, circular
annuli, etc., can be found in these two sources.

The Green function for the Neumann problem on a ball in 3-space can be
found in Exercise 6, p.247, of O.D. Kellogg's book Foundations of Potential
Theory, 1953. According to my notes, details of a solution to this
"exercise" can be found in the Mikhlin book cited above. I don't know if
this result can be extended to the n > 3 case.

Apparently, no integral representation is known in the n > 3 case. It
appears that solutions to the Neumann problem in this case require the use
spherical harmonics as in the book Harmonic Function Theory by S. Axler, P.
Bourdon, and W. Ramey, 1992.

If anyone knows of an integral representation for the n > 3 case, I would
appreciate hearing about it.

L. Helms

"A. Novruzi" wrote:

> Hi all,
> Does anybody have references for Green function (bounded or exterior of
> bounded domain) with Neuman conditions on boundary?
>
> Thanks!
> Arian.
>
> Sent via Deja.com http://www.deja.com/
> Before you buy.

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