Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-14T07:37:39.956Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of a quasilinear elliptic equation with nonlinear boundary condition

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.

We consider a quasilinear elliptic partial differential equation with nonlinear boundary condition under assumptions which do not allow the application of standard degree theory results or techniques such as the method of continuity. An approximation using mollifiers is introduced, allowing the application of Leray-Schauder degree theory, and homotopy arguments are then used to prove the existence of solutions to the approximating problems. A subsequent paper will discuss the question of the convergence of these approximate solutions to a classical solution of the original problem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 50 , Issue 3 , December 1994 , pp. 405 - 424
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., ‘Convergence of approximate solutions of a quasilinear partial differential equation’, Bull. Austral. Math. Soc. 50 (1994), 425–433.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]Krylov, N.V., ‘On estimates for the derivatives of solutions of nonlinear parabolic equations’, Soviet Math. Dokl. 29 (1984), 14–17.Google Scholar
[6]Lieberman, G.M., ‘The nonlinear oblique derivative problem for quasilinear elliptic equations’, Nonlinear Anal. 8 (1984), 49–65.CrossRefGoogle Scholar
[7]Lieberman, G.M., ‘Intermediate Schauder estimates for oblique derivative problems’, Arch. Rational Mech. Anal. 93 (1985), 129–134.CrossRefGoogle Scholar
[8]Lieberman, G.M. and Trudinger, N.S., ‘Nonlinear oblique boundary value problems for nonlinear elliptic equations’, Trans. Amer. Math. Soc. 295 (1986), 509–546.CrossRefGoogle Scholar
[9]Stein, E.M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, New Jersey, 1970).Google Scholar
[10]Thompson, H.B., ‘Second order ordinary differential equations with fully nonlinear two point boundary conditions’, Pacific J. Math, (to appear).Google Scholar
[11]Thompson, H.B., ‘Second order ordinary differential equations with fully nonlinear two point boundary conditions II’, Pacific J. Math, (to appear).Google Scholar
[12]Trudinger, N.S., ‘Boundary value problems for fully nonlinear elliptic equations’, Proc. Centre Math. Anal. Austral. Nat. Univ. (1984), 65–83.Google Scholar