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

Part of: Representations of solutions

Published online by Cambridge University Press:  26 February 2010

David Colton
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.

MSC classification

Secondary: 35C15: Integral representations of solutions
Type
Research Article
Information
Mathematika , Volume 18 , Issue 2 , December 1971 , pp. 283 - 290
Copyright
Copyright © University College London 1971

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References

1Bergman, S., Integral operators in the theory of linear partial differential equations (Springer, Berlin, 1961).CrossRefGoogle Scholar
2Browder, F. E., “Approximation by solutions of partial differential equations ”, Amer. J. Math., 84 (1962), 134–160.CrossRefGoogle Scholar
3Colton, D., “Integral operators for elliptic equations in three independent variables I “, Applicable Analysis (to appear).Google Scholar
4Colton, D., “Integral operators for elliptic equations in three independent variables II ” , Applicable Analysis (to appear).Google Scholar
5Colton, D., “Bergman operators for elliptic equations in three independent variables ”, Bull. Amer. Math. Soc., 77 (1971), 752–756.CrossRefGoogle Scholar
6Colton, D., “Bergman operators for elliptic equations in four independent variables ”, SIAM J. Math. Anal, (to appear).Google Scholar
7Colton, D., and Gilbert, R. P., “Integral operators and complete families of solutions for Arch. Rat. Mech. Anal., 43 (1971), 62–78.CrossRefGoogle Scholar
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9Gilbert, R. P., “The construction of solutions for boundary value problems by function theoretic methods”, SIAM J. Math. Anal., 1 (1970), 96–114.CrossRefGoogle Scholar
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12Nowacki, W., Dynamics of elastic systems (Chapman and Hall, London, 1963).Google Scholar
13Vekua, I. N., New methods for solving elliptic equations (John Wiley, New York, 1967).Google Scholar