A direct proof of the equivalence between Dirichlet's principle and Perron's method

TL;DR

This paper provides a concise proof demonstrating the equivalence of Dirichlet's principle and Perron's method for solving Laplace's equation, using only elementary tools within Sobolev spaces.

Contribution

It offers a new, straightforward proof of the equivalence between Dirichlet's principle and Perron's method, avoiding complex elliptic theory and weak convergence arguments.

Findings

The minimizer of Dirichlet energy matches Perron solution for continuous boundary data.

The proof relies solely on strong convergence and elementary inequalities.

No advanced elliptic theory or distributional analysis is required.

Abstract

We give a short proof that for a bounded domain $\Omega\subset\mathbb{R}^n$ and continuous boundary data $g\in C(\partial\Omega)$ admitting a continuous finite-energy extension $\phi\in H^{1}(\Omega)\cap C(\bar\Omega)$, the minimizer of the Dirichlet energy \[ E(v) = \int_{\Omega} |\nabla v|^{2}\,dx, \qquad v-\phi\in H^{1}_{0}(\Omega), \] coincides with the Perron solution $h_g$ of the Dirichlet problem $\Delta u = 0$ in $\Omega$ with boundary data $g$. The argument stays entirely in $H^{1}(\Omega)$ and uses only strong convergence via strict convexity of the Dirichlet energy, Friedrichs' inequality, Weyl's lemma, and Wiener's exhaustion by regular subdomains. No weak convergence, Poisson problems with distributional right hand sides, or general elliptic theory are needed.