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Remark on Hilbert's boundary value problem for Beltrami systems

Published online by Cambridge University Press:  14 November 2011

Heinrich Begehr
Heinrich Begehr
Affiliation:
Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, 1000 Berlin 33, Germany
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Synopsis

The Schauder continuation method for nonlinear problems is based on appropriate a priori estimates for related linear equations. Recently, in a paper by the present author and G. C. Hsiao, the Hilbert boundary value problem with positive index for nonlinear elliptic systems in the plane was solved by this method but the constructive derivation of the a priori estimate necessarily required a restriction on the ellipticity condition. This is because the norm of the generalized Hilbert transform in the case of positive index is too big. Here, as in a forthcoming paper by G.C. Wen, an indirect and therefore non-constructive proof of the a priori estimate is given which does not require any further restrictions and allows the Hilbert boundary value problem to be solved for nonlinear elliptic systems in general.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 3-4 , 1984 , pp. 305 - 310
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

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