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

TL;DR

This paper presents a straightforward functional analysis approach to prove the existence of solutions for elliptic boundary problems, simplifying previous methods by relying on known estimates and continuation techniques.

Contribution

The paper introduces a simple, technically easy functional analysis method for establishing existence of solutions to elliptic boundary problems, avoiding more complex traditional approaches.

Findings

Method is conceptually simple and easy to implement.

Requires only known a priori estimates and continuation in a parameter.

Provides an alternative proof of existence for elliptic boundary problems.

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.