A short proof of the existence of the solution to elliptic boundary problem

  • Authors

    • Alexander G. Ramm Mathematics Department, Kansas State University, CW 207, Manhattan, KS 66506-2602, USA
    2015-06-06
    https://doi.org/10.14419/gjma.v3i3.4731
  • Dynamical systems method (DSM), Homeomorphism, Nonlinear equations, Surjectivity

  • There are several methods for proving the existence of the solution to the elliptic boundary problem \(Lu=f \text{in} D,\quad u|_S=0,\quad    (*)\). Here L is an elliptic operator of second order, f is a given function, and uniqueness of the solution to problem (*) is assumed. The known methods for proving the existence of the solution to (*) include variational methods, integral equation methods, method of upper and lower solutions. In this paper a method based on functional analysis is proposed. This method is conceptually simple. It requires some a priori estimates and a continuation in a parameter method, which is well-known.

  • References

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      [3] L. Hormander, The analysis of linear partial differential operators, I-IV. Springer- Verlag, New York, 1983-1984.

      [4] T. Kato, Perturbation theory for linear operators. Springer- Verlag, New York, 1984.

      [5] O. Ladyzhenskaya, N. Uraltseva, Linear and quasilinear elliptic equations. Acad. Press, New York, 1968.

      [6] O. Ladyzhenskaya, The boundary value problems of mathematical physics. Springer-Verlag, Berlin, 1985.

      [7] C. Miranda, Partial differential equations of elliptic type. Springer-Verlag, Berlin, 1970.

      [8] J. Schauder, Uber lineare elliptische differentialgleichung zweiter Ordnung, Math. Zeitschr., 38, (1934), 251-282.

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  • How to Cite

    Ramm, A. G. (2015). A short proof of the existence of the solution to elliptic boundary problem. Global Journal of Mathematical Analysis, 3(3), 105-108. https://doi.org/10.14419/gjma.v3i3.4731