I'm just learning about the maximum principle in my PDE class. I thought I'd be able to apply it to something else I was learning about in electrostatics (namely, that the surface of a conductor is equipotential):
Let $B\subset\mathbb{R^3}$ (closed and bounded) be a conductor. It can be shown that the electric potential $V:B\to\mathbb{R}$ is constant on $int(B)$ and $\nabla^2V=0$ on $int(B)$. Since $V$ is a harmonic function, it attains its max and min values on $\partial B$. Also, since $V$ is constant on $int(B)$, I feel this should imply that $V$ is actually constant over all $B$ and not just $int(B)$, but I can't see why.
The conductor have to be a continuous function. By continuity the conductor will assume the same value in the whole B (the conductor can't jump when you go to the boundary). Then it have the same value over B(sorry for the english)