Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T17:39:41.405Z Has data issue: false hasContentIssue false
  • English
  • Français

The Dirichlet Problem With Denjoy-Perron Integrable Boundary Condition

Published online by Cambridge University Press:  20 November 2018

M. Benedicks  and
W. F. Pfeffer
M. Benedicks
Affiliation:
Chalmers Institute of TechnologyandUniversity of GöteborgGöteborg, Sweden
W. F. Pfeffer
Affiliation:
University of Petroleum and MineralsDhahran, Saudi Arabia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Poisson integral of a Denjoy-Perron integrable function defined on the boundary of an open disc is harmonic in this disc. Moreover, almost everywhere on the boundary, the nontangential limits of the integral coincide with the boundary condition. This extends the classical result for Lebesgue integrable boundary conditions. By means of conformai maps, a generalization to domains bounded by a sufficiently smooth Jordan curve is also obtained.

Keywords

31A2026A39
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 1 , 01 March 1985 , pp. 113 - 119
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Dahlberg, D.E.J., Harmonic functions in Lipschitz domains, Proc. Symp. Pure Math., XXXV, Part 1, 313–322.Google Scholar
2. Dahlberg, D.E.J., On the Poisson integral for Lipschitz and domains, Studia Math. 66 (1979), pp. 13–24.Google Scholar
3. Duren, P.L., Theory of Hp spaces, Academic Press, New York, 1970.Google Scholar
4. Fabes, E.B., Jodeit, M., and Rivière, N. M., Potential techniques for boundary value problems on domains, Acta Math. 141 (1978), pp. 165–186.Google Scholar
5. Garnett, J.B., Bounded analytic functions, Academic Press, New York, 1981.Google Scholar
6. Henstock, R., A Riemann type integral of Lebesgue power, Canad. J. Math. 20 (1968), pp. 79—87.Google Scholar
7. Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslov. Math. J. 82 (1957), pp. 418–446.Google Scholar
8. Pfeffer, W.F., The Riemann-Stieltjes approach to integration, TWISK 187, CSIR: NRIMS, Pretoria, 1980.Google Scholar
9. Pfeffer, W.F., Integration by parts for the generalized Riemann-Stieltjes integral, J. Australian Math. Soc. 34(1983), pp. 229–233.Google Scholar
10. Saks, S., Theory of the integral, Dover Publications, New York, 1964.Google Scholar
11. Tsuji, M., Potential theory in modern function theory, Chelsea Co., New York, 1959.Google Scholar