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

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.