Short Proof of Dirichlet's Principle

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.