Perron method

Remarks

The family $\mathcal{S}_f$ is not empty, since it contains the constant $\inf f$, and each $v \in \mathcal{S}_f$ is at most $\sup f$ by the maximum principle; so $H_f$ is finite. Perron's theorem says that $H_f$ is harmonic on $\Omega$. Its proof replaces a function of the family by its Poisson integral on a small ball, which keeps it in the family and makes it larger. Example: on the unit disc with $f(\xi) = \xi_1$, the function $v(x) = x_1$ belongs to $\mathcal{S}_f$, and the maximum principle gives $v \le x_1$ for every $v \in \mathcal{S}_f$, so $H_f(x) = x_1$. Non-example: $v(x) = x_1 + \frac12$ is subharmonic but not in $\mathcal{S}_f$, since its boundary values exceed $f$ by $\frac12$. A boundary point $\xi$ at which $H_f(x) \to f(\xi)$ for every continuous $f$ is called a regular boundary point.