Unsure of why a certain condition implies uniqueness for the interior Dirichlet problem for the Laplace equation?

Question

In Folland's Introduction to Partial Differential Equations book he shows uniqueness for the interior Dirichlet problem for the Laplace equation on a domain $\Omega$ by stating that if

$$\cases{\Delta u = 0 \quad \text{in} \ \Omega \\ u = f \quad \text{on} \ \partial \Omega,}$$

with $f = 0$, then this implies $u = 0$.

I don't see how $u$ being zero everywhere inside $\Omega$ when $f$ is zero on the boundary implies a unique solution.

1. How does $u$ being zero imply uniqueness..normally to show uniqueness in some situation, we need to take two functions and show that they must be equal. But here we only use one function?
2. The equation above is specifically for the case when $f = 0$..what if $f$ wasn't zero? Why don't why have to explicitly show uniqueness for that case too?

Answer 1

Suppose two solutions $u_1$ and $u_2$ and let $u=u_1-u_2$. Then $u$ is a solution of the Dirichlet problem with boundary value $0$, so it is equal to $0$ on the domain.