On a regular bounded plane domain, Perron's method solves the Dirichlet problem

By [L1], $H_{\phi }$ is harmonic on $\mathrm{\Omega }$. By the hypothesis that every boundary point is regular and the definition packaged in [L2], for every $\zeta \in \mathrm{\partial }\mathrm{\Omega }$ one has [L1, L2, given] ${\lim }_{\begin{array}{c}z\to \zeta \\ z\in \mathrm{\Omega }\end{array}}{H}_{\phi }(z)=\phi (\zeta )\mathrm{.}$

Step 1.1 gives the boundary limits pointwise on $\mathrm{\partial }\mathrm{\Omega }$, and the continuity of $\phi$ turns those limits into a continuous extension of $H_{\phi }$ to $\bar{\mathrm{\Omega }}$ by setting the boundary values equal to $\phi$.

If $u$ is any other continuous harmonic function on $\bar{\mathrm{\Omega }}$ with $u=\phi$ on $\mathrm{\partial }\mathrm{\Omega }$, then [L3] applied to $u$ and the extension from step 2.1 gives $u=H_{\phi }$ on $\bar{\mathrm{\Omega }}$. Thus Perron's method solves the Dirichlet problem uniquely on regular bounded plane domains.