A short proof of the existence of the solution to elliptic boundary problem
DOI:
https://doi.org/10.14419/gjma.v3i3.4731الكلمات المفتاحية:
Dynamical systems method (DSM)، Homeomorphism، Nonlinear equations، Surjectivityالملخص
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.
المراجع
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[4] T. Kato, Perturbation theory for linear operators. Springer- Verlag, New York, 1984.
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[8] J. Schauder, Uber lineare elliptische differentialgleichung zweiter Ordnung, Math. Zeitschr., 38, (1934), 251-282.
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