A direct proof of the equivalence between Dirichlet's principle and Perron's method
Tsogtgerel Gantumur
TL;DR
This paper proves that for a bounded domain $\Omega$ and continuous boundary data $g$ with a continuous finite-energy extension $\phi\in H^{1}(\Omega)\cap C(\bar{\Omega})$, the Dirichlet-energy minimizer $u$ (with $u-\phi\in H^{1}_{0}(\Omega)$) coincides almost everywhere with the Perron solution $h_g$ of the Laplace Dirichlet problem. The argument stays entirely within $H^{1}(\Omega)$ and uses a direct energy comparison on an increasing exhaustion by Wiener-regular subdomains $\Omega_k$, together with strict convexity of the Dirichlet energy and Friedrichs' inequality, plus Weyl's lemma and Wiener's exhaustion to connect the variational and Perron constructions. The proof avoids weak convergence and distributional Poisson problems, offering a concise, elementary Hilbert-space route to the equivalence in the finite-energy setting. It sharpens prior results by showing that, under these hypotheses, the Dirichlet principle yields exactly the Wiener-exhaustion (Perron–Wiener) solution, complementing and simplifying earlier approaches by Simader, Hildebrandt, and Arendt–Daners.
Abstract
We give a short proof that for a bounded domain $Ω\subset\mathbb{R}^n$ and continuous boundary data $g\in C(\partialΩ)$ admitting a continuous finite-energy extension $φ\in H^{1}(Ω)\cap C(\barΩ)$, the minimizer of the Dirichlet energy \[ E(v) = \int_Ω |\nabla v|^{2}\,dx, \qquad v-φ\in H^{1}_{0}(Ω), \] coincides with the Perron solution $h_g$ of the Dirichlet problem $Δu = 0$ in $Ω$ with boundary data $g$. The argument stays entirely in $H^{1}(Ω)$ 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.