Let $u_1$ and $u_2$ be two Dirichlet functions; hence both attain their maximum and minimum values on the boundary of the domain $D$ (let us call the boundary $B$).
My book says the following:
Let $v=u_1-u_2$. Then $V$ also attains its minimum and maximum values on $B$ as a consequence of the minimum-maximum principle.
I don't see why this has to be correct. I feel $v$ may not attain its minimum and maximum values on $B$- it may attain them in the interior of $D$ itself.