Suppose \(\Omega\subseteq \mathbb{R}^d\) is open and bounded, and \(g\in C(\partial\Omega)\). The Perron solution \(u\colon\overline{\Omega}\to \mathbb{R}\) is defined by
\[ u(x)=\sup\{v(x)\mid v\in S_g\} \]where \(S_g\) is the set of \(C^0\)-subharmonic functions with \(v\le g\) on \(\partial\Omega\).
Remarks
- The Perron solution is well-defined and harmonic.
- It is not clear, if \(u = g\) on \(\partial\Omega\).