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arXiv · 1503.00642

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

A. G. Ramm
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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 and technically is easy. It requires some known a priori estimates and a continuation in a parameter method.

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A. G. Ramm. 2015-03-02. A short proof of the existence of the solution to elliptic boundary problem. https://arxiv.org/abs/1503.00642

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