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Perrons solutions coincide with the boundary conditions on the boundary if the boundary point is regular.

Suppose \(\Omega \subseteq \mathbb{R}^d\) is open and bounded, \(g \in C^0(\partial \Omega)\), and \(u\) is the Perron solution of \(g\). If \(x_0 \in \partial \Omega\) is regular, \(u\) is continuous in \(x_0\) and \(u(x_0) = g(x_0)\).