Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-23T14:21:54.086Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Begehr, H. and Hsiao, G. C.. The Hilbert boundary value problem for nonlinear elliptic systems. Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), 97–112.CrossRefGoogle Scholar
2Bers, L. and Nirenberg, L.. On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications. Convego Intemazionale sulle Equazioni Lineari alle Derivate Partiali, Trieste 1954, pp. 111–140 (Roma: Edizioni Cremonense, 1955).Google Scholar
3Haack, W. and Wendland, W.. Lectures on partial and Pfaffian differential equations (Oxford: Pergamon, 1972).Google Scholar
4Vekua, I. N.. Generalized analytic functions (Oxford: Pergamon, 1962).Google Scholar
5Wen, G. C.. Some nonlinear boundary value problems for nonlinear elliptic equations of second order in the plane. Complex Variables Theory Appl. to appear.Google Scholar
6Wendland, W.. Elliptic systems in the plane (London: Pitman, 1978).Google Scholar