A solution of the Dirichlet problem for the Poisson’s equation

\[ \left\{ \begin{aligned} -\Delta u &= f &&\text{ in } \Omega\\ u&=g &&\text{ on } \partial \Omega, \end{aligned} \right. \]

is unique.

Proof
Let \(u\) and \(v\) be two solutions. The difference \(u-v\) is harmonic and by the weak maximum principle we obtain \(u\le v\). By switching the roles of \(u\) and \(v\) we get the other inequality.
Remarks