Hostname: page-component-54dcc4c588-xh45t Total loading time: 0 Render date: 2025-09-11T15:02:36.974Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of approximate solutions of a quasilinear partial differential equation

Published online by Cambridge University Press:  17 April 2009

T.R. Cranny
T.R. Cranny
Affiliation:
Department of MathematicsThe University of QueenslandQueensland 4072Australia
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.

This article is a sequel to a paper in which a quasilinear partial differential equation with nonlinear boundary condition was approximated using mollifiers, and the existence of solutions to the approximating problem shown under quite general conditions. In this paper we show that standard a priori Hölder estimates ensure the convergence of these solutions to a classical solution of the original problem. Some partial results giving such estimates for special cases are described.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 50 , Issue 3 , December 1994 , pp. 425 - 433
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Cranny, T.R., Leray-Schauder degree theory and partial differential equations under nonlinear boundary conditions, Doctoral Thesis (Department of Mathematics, University of Queensland, 1992).Google Scholar
[2]Cranny, T.R., ‘Approximation of a quasilinear elliptic equation with nonlinear boundary condition’, Bull. Austral. Math. Soc. 50 (1994), 405–424.CrossRefGoogle Scholar
[3]Gilbarg, D. and Hörmander, L., ‘Intermediate Schauder estimates’, Arch. Rational Mech. Anal. 74 (1980), 297–318.CrossRefGoogle Scholar
[4]Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order (Springer-Verlag, Berlin, Heidelberg, New York, 1983).Google Scholar
[5]Leiberman, G.M., ‘Solvability of quasilinear elliptic equations with nonlinear boundary conditions’, Trans. Amer. Math. Soc. 273 (1982), 753–765.CrossRefGoogle Scholar
[6]Lieberman, G.M., ‘Intermediate Schauder estimates for oblique derivative problems’, Arch. Rational Mech. Anal. 93 (1985), 129–134.CrossRefGoogle Scholar
[7]Stein, E.M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, New Jersey, 1970).Google Scholar
[8]Trudinger, N.S., ‘Fully nonlinear, uniformly elliptic equations under natural structure conditions’, Trans. Amer. Math. Soc. 278 (1983), 751–769.CrossRefGoogle Scholar