In the sequel, let $D\subset{\mathbb R}^m(m=2,3)$ be a bounded domain of class $C^2$. And assume that the boundary $\partial D$ is connected.
Exterior Dirichlet Problem. Find a function $u$ that is harmonic in ${\mathbb R}^m\setminus \bar D$, is continuous in ${\mathbb R}^m\setminus D$, and satisfies the boundary condition $ u = f\qquad\text{on}~\partial D, $ where $f$ is a given continuous function. For $|x|\to\infty$ it is required that $ u(x)=O(1),\quad m=2,\qquad\text{and}\quad u(x)=o(1),\quad m=3, $ uniformly for all directions.
I don't fully understand how the uniqueness of the solution (if exists) to this problem is proved.
For the $m=3$ case, the book only said that from the maximum-minimum principle, observing that $u(x)=o(1),~|x|\to\infty$, we obtain $u=0$ in ${\mathbb R}^3\setminus D$ when $f=0$.
Question: How is the maximum-minimum principle used here?