Theorem 2.12 (Dirichlet Problem) . [033A]

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Theorem 2.12 (Dirichlet Problem).

Let $\varphi\in PSH(X,\omega)\cap L^{\infty}(X)$ . Let $B$ be a small ball in $X$ . Then there exists $\hat{\varphi}\in PSH(X,\omega)$ such that $\hat{\varphi}=\varphi$ in $X\setminus B$ , $\hat{\varphi}\geq\varphi$ and $(\omega_{\hat{\varphi}})^{n}=0$ in $B$ . Moreover if $\varphi_{1}\leq\varphi_{2}$ then $\hat{\varphi_{1}}\leq\hat{\varphi_{2}}$ .