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An elliptic boundary-value problem with a discontinuous nonlinearity

Published online by Cambridge University Press:  14 November 2011

G. Keady
G. Keady
Affiliation:
Mathematics Department, University of Western Australia
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Synopsis

We study the boundary-value problem, for (λ/k,ψ),

Here ∆ denotes the Laplacian, H is the Heaviside step function and one of A or k is a given positive constant. We define

and usually omit the subscript. Throughout we are interested in solutions with ψ>0 in Ω and hence with λ/=0.

In the special case Ω = B(0, R), denoting the explicit exact solutions by ℑe, the following statements are true, (a) The set Aψ, issimply-connected, (b) Along ℑe, the diameter of Aψ tendsto zero when the area of Aψ, tends to zero.

For doubly-symmetrised solutions in domains Ω such as rectangles, it is shown that the statements (a) and (b) above remain true.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 91 , Issue 1-2 , 1981 , pp. 161 - 174
Copyright
Copyright © Royal Society of Edinburgh 1981

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