The Perron solution is well-defined and harmonic .
Proof
We have \(S_g \neq \emptyset\), since \(x \to \min g \in S_g\). Furthermore, we have
\[u \leq \max g \quad \text{in } \Omega\]because of the strong maximum principle .
For \(y \in \Omega\), there exists a sequence \(v_n(y) \to u(y)\) with \(v_n \leq v_{n+1}\) and each \(v_n\) is subharmonic.
Consider the harmonic replacement $w_n = P_{B(x)} v_n
\[. Then $w_n$ is bounded and \]u = \lim w_n,$$ which is harmonic.