Short Proof of Dirichlet's Principle

TL;DR

This paper presents a simplified Hilbert-space proof of Dirichlet's principle, demonstrating the existence and properties of solutions to Dirichlet's problem using a unique min-problem solution approach.

Contribution

It introduces a streamlined proof method for Dirichlet's principle, leveraging the uniqueness of a certain min-problem solution and recasting the problem in a functional-analytic framework.

Findings

Solution depends linearly and continuously on data

Solution is invariant under certain data changes

Applicable to regions with regular boundary traces

Abstract

A standard Hilbert-space proof of Dirichlet's principle is simplified, using an observation that a certain form of min-problem has unique solution, at a specified point. This solves Dirichlet's problem, after it is recast in the required form (using the Poincare/Friedrichs bound and Riesz representation). The solution's dependence on data is linear and continuous; and the solution is invariant under certain changes of data, away from the border of the region where Dirichlet's problem is given. If that region is regular enough for functions on it to have border-traces, then the problem can be stated and solved in terms of border-data.