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Constructive and Computational Methods for Differential and Integral Equations pp 184–236Cite as

On Dirichlet's problem for quasi-linear elliptic equations

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 430))

Abstract

This paper concerns the existence and uniqueness as well as the approximations of the solutions to the Dirichlet problem for the second-order quasi-linear elliptic equation of n variables in a simply connected domain with Lyapunov boundary. Sufficient conditions on the co-efficients of the equation are established for the existence proofs of the solution so that Vekua's function theoretic method for treating Dirichlet's problems for equations in two variables can be employed for the case n ≥ 3 variables. Based on Warschawski's work, an iterative scheme is also presented for constructing an approximating solution in the three-dimensional case. It is shown that the approximants converge to the actual solution geometrically.

An invited address at Constructive and Computational Methods for Differential and Integral Equations, Systems Analysis Institute, Research Center for Applied Science, Indiana University, February 17–20, 1974. This research was supported in part by the Air Force Office of Scientific Research through AF-AFOSR Grant No. 74-2592.

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References

  1. D. Colton and R. P. Gilbert, Rapidly convergent approximations to Dirichlet's problem for semilinear elliptic equations, Applicable Analysis, 1972, Vol. 2, pp. 229–240.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Indiana University, 47401, Bloomington, Indiana

    Robert P. Gilbert

  2. Department of Mathematics, University of Delaware, 19711, Newark, Delaware

    George C. Hsiao

Authors
  1. Robert P. Gilbert
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  2. George C. Hsiao
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David Lem Colton Robert Pertsch Gilbert

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© 1974 Springer-Verlag

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Gilbert, R.P., Hsiao, G.C. (1974). On Dirichlet's problem for quasi-linear elliptic equations. In: Colton, D.L., Gilbert, R.P. (eds) Constructive and Computational Methods for Differential and Integral Equations. Lecture Notes in Mathematics, vol 430. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066270

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  • DOI: https://doi.org/10.1007/BFb0066270

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07021-4

  • Online ISBN: 978-3-540-37302-5

  • eBook Packages: Springer Book Archive

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