I've seen a uniqueness argument come up a few times but I don't really understand it.
The argument is that if two harmonic functions $f$ and $g$ agree on the boundary of some domain, then $f-g$ or $g-f$ is $0$ on the boundary, and from here the proofs usually state that "by the maximum principle the result follows", i.e., $f$ is unique.
How is this a result of the maximum principle?
Also, if $f$ harmonic has boundary value = $0$, can't it have negative values inside the domain? This wouldn't contradict the maximum principle.
Thanks,
The maximum principle has a dual principle: the minimum principle. This can be seen by replacing $f$ with $-f$, then $-f$ is also harmonic and its maximum occurs on the boundary by the maximum principle.
Let $D$ be the domain (with boundary). $\max_{x\in D} f(x) = -\min_{x\in D}(-f)(x)$. If the minimum of $-f$ occurred at some point $y\in D$, then $\max_{x\in D}f(x) = -(-f)(y) = f(y)$. By applying the maximum principle to $f$, you know that $y$ must be on the boundary and so the minimum of $-f$ occurs on the boundary as well. Replacing $-f$ with $f$, you have the result.
This can also be verified with the Poisson integral formula and can be interpreted as saying that a harmonic function which is zero on the boundary is zero everywhere.