A short proof of the existence of the solution to elliptic boundary problem
About this article
Abstract
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.
Keywords
- Dynamical systems method (DSM)
- Homeomorphism
- Nonlinear equations
- Surjectivity
References
[1] S. Agmon, Lectures on elliptic boundary value problems. Van Nostrand, Princeton, 1965.
[2] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 1983.
[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.