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Some examples concerning applicability of the Fredholm-Radon method in potential theory. (English) Zbl 0615.31005

This paper is concerned with a certain type of ”rectangular” open set \(D\subsetneqq {\mathbb{R}}^ 3:\) in particular, such a set must have a compact boundary which can be covered by a finite number of planes parallel to the co-ordinate axes. Examples are given to illustrate the difficulties involved in trying to use the Fredholm-Radon method to solve the Dirichlet problem in such a set D for given \(g\in {\mathfrak C}(D)\). These problems are then overcome by the introduction of a new norm on \({\mathfrak C}(D)\), and the following result is deduced.
Let D be as above, let \(G_ 1,...,G_ p\) be the bounded components of \(G={\mathbb{R}}^ 3\setminus \bar D\) and fix \(x_ j\in G_ j(j=1,...,p)\). Then there are unique constants \(c_ 1,...,c_ p\) and a function \(f\in {\mathfrak C}(D)\) such that a solution to the above Dirichlet problem is given by the sum of the double layer potential Wf(x) and a second term \(\Sigma c_ j | x-x_ j|^{-1}\). Further, f is uniquely determined if and only if D is bounded and G is connected (in which case the second term in the solution disappears). The Neumann problem is also treated.
Reviewer: S.J.Gardiner

MSC:

31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
47A53 (Semi-) Fredholm operators; index theories
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References:

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