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Integral representations of solutions to a class of fourth order elliptic equations in three independent variables

Published online by Cambridge University Press:  26 February 2010

David Colton
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana Department of Mathematics, University of Glasgow, Glasgow, Scotland.
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Extract

Both S. Bergman [1] and I. N. Vekua [13] have constructed integral operators which map ordered pairs of analytic functions of one complex variable onto solutions of fourth order elliptic equations in two independent variables. Such operators play an important role in the investigation of the analytic properties of solutions to higher order elliptic equations and in the approximation of solutions to boundary value problems associated with these equations. Unfortunately, little progress has been made in developing an analogous theory for elliptic equations in more than two independent variables. Recently, however, Colton and Gilbert [7] constructed integral operators for a class of fourth order elliptic equations with spherically symmetric coefficients in p + 2 (p ≥ 0) independent variables, and at present Dean Kukral [11], a student of R. P. Gilbert, is in the process of trying to extend some recent results of Colton [3, 4, 5] for second order equations in three independent variables to the fourth order case.

Type
Research Article
Copyright
Copyright © University College London 1971

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References

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