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Vortices, Liouville's equation and the Bergman kernel function

Published online by Cambridge University Press:  26 February 2010

S. Richardson
Affiliation:
Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ.
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Extract

It is the purpose of this note to draw attention to some connections between the topics mentioned in the title. These allow familiar results from one area of mathematics to be usefully exploited in another, to the mutual benefit of both. Moreover, the basic ideas suggest possible generalizations whose examination should prove worthwhile. In this introduction we give a brief account of the results to be derived and discussed in more detail in the later sections.

Type
Research Article
Copyright
Copyright © University College London 1980

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References

1.Bergman, S.. The kernel function and conformal mapping (Amer. Math. Soc, 2nd ed., 1970).Google Scholar
2.Liouville, J.. “Sur 1'équation aux différences partielles , J. Math., 18 (1853), 7172.Google Scholar
3.Gustafsson, B.. “On the motion of a vortex in two-dimensional flow of an ideal fluid in simply and multiply connected domains”, Research Report, Dept. of Mathematics, Royal Institute of Technology, Stockholm (1979), 109 pp.Google Scholar
4.Milne-Thomson, L. M.. Theoretical Hydrodynamics (Macmillan, 5th ed., 1968).CrossRefGoogle Scholar
5.Nehari, Z.. Conformal Mapping (McGraw-Hill, 1952).Google Scholar
6.Pommerenke, Chr.. Univalent Functions (Vandenhoeck & Ruprecht, 1975).Google Scholar
7.Pôlya, G. and Szegö, G.. Isoperimetric Inequalities in Mathematical Physics (Princeton Univ. Press, 1951).Google Scholar
8.Hille, E.. Analytic Function Theory, Vol II (Ginn & Co., 1962).Google Scholar
9.Kober, H.. Dictionary of Conformal Representations (Dover, 1957).Google Scholar
10.Szegö, G.. “Conformal mapping of the interior of an ellipse onto a circle”, Amer. Math. Monthly, 57 (1950), 474478.CrossRefGoogle Scholar
11.Wittich, H.. “Ganze Losungen der Differentialgleichung Δu = eu”' Math. Z., 49 (1944), 579582.CrossRefGoogle Scholar
12.Garabedian, P. R.. “A partial differential equation arising in conformal mapping”, Pacific J. Math., 1 (1951), 485524.CrossRefGoogle Scholar
13.Keller, J. B.. “On solutions of Δu =f(u)”, Comm. Pure Appl. Math., 10 (1957), 503510.CrossRefGoogle Scholar
14.Stuart, J. T.. “On finite amplitude oscillations in laminar mixing layers”, J. Fluid Meek, 29 (1967), 417440.CrossRefGoogle Scholar
15.Joseph, D. D. and Lundgren, T. S.. “Quasilinear Dirichlet problems driven by positive sources”, Arch. Rational Mech. Anal., 49 (1973), 241269.CrossRefGoogle Scholar
16.Adler, J.. “Asymmetric self-heating of a circular cylinder”, Combust. Flame, 32 (1978), 163170.CrossRefGoogle Scholar