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

There are several methods for proving the existence of the solution to the elliptic boundary problem $Lu=f\text{ in}D,u{\mathrm{\mid }}_S=0,(\ast )$. 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 and technically is easy. It requires some known a priori estimates and a continuation in a parameter method.