Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 4 / b / iv

Fix $B⋐\Omega$ and $x_0\in B$. Choose $u_j$ in the Perron family with $u_j(x_0)\uparrow \overline{u}(x_0)$, replace successive terms by finite maxima using part (i), and take their harmonic lifts on $B$. The lifts remain in the family, are increasing, and are uniformly bounded. Interior estimates and the Arzela-Ascoli theorem give a harmonic limit $h$ on $B$ with $h\le \overline{u}$ and $h(x_0)=\overline{u}(x_0)$.

If $h(y)<\overline{u}(y)$ somewhere in $B$, take another family member larger than $h(y)$ and repeat the maximum-and-lift construction. Its harmonic limit $H$ satisfies $H\ge h$ and $H(x_0)=h(x_0)$. The strong minimum principle for elliptic operators forces $H=h$, contradicting the strict inequality at $y$. Thus $h=\overline{u}$ on $B$. Since $B$ was arbitrary, $\overline{u}$ is smooth and harmonic in $\Omega$. This is the Perron method for the Dirichlet problem.