Consider $\Omega$ an open, convex and bounded set of $R^n$. Let $g: \partial \Omega \rightarrow R$ a function. Supose that $g$ is continuous except in one point. By the convexity of $\Omega $ we can use Perron method to construct (the Perron solution)a function $u:{U} \rightarrow R$ with
i) $\Delta u = 0 \ on \ \partial \Omega$ and $u \in C^{2}(U)$
ii)$\displaystyle\lim_{y \rightarrow x} u(y) = g(x)$ if $g$ is continuous in $x \in \partial \Omega$
My question is :
$u: {U} \rightarrow R$ is the unique function defined in $U$ that satisfies the properties above?
Someone can say to me a reference that answer my question?