Let $\Omega \in \Re^2$ be open and bounded. Let $u \in C^2(\Omega) \cap C(\bar \Omega)$ satisfy
$$-\Delta u = f$$ with $f>0$.
Find an example of $\Omega \in \Re^2$ , $u \in C^2(\Omega) \cap C(\bar \Omega)$, and $f \in C(\Omega)$ such that
$$\max\limits_{\bar \Omega} u \neq \max\limits_{\delta\Omega} u$$
Ok, so from this I have established that my function $u$ is super-harmonic as $f$ is greater than 0. So I know that it obeys the minimum principles.
The issue with this is I don't know how to find an example to show this.
I just need to show that $u$ is non constant and attains it's max in $\Omega$.
I feel like I just need a small amount of guidance in the right direction to solve this.
Edit: What I was thinking is to set $u = -x^2 + 2x$ and $\Omega = B_2(0)$ and hence $f=2$. But I am not sure whether this is just a load of rubbish
Your idea (in the edit) works. Here is a slight modification of it.
Superharmonic is kind of like concave (not the same concept but similar), so it's natural to take a simple concave function... upside-down paraboloid comes to mind. And indeed, $$ u(x, y) = -x^2-y^2 $$ has $-\Delta u = 4$. On the unit disk, the maximum of $u$ is $0$ but the maximum of its boundary values is $-1$.