I'm working on a homework assignment in PDE, and I'm required to use the maximum principle to demonstrate that when $\Delta u(x)=0$ and periodic boundary conditions are applied, $u(x)$ is a constant.
The EXACT wording of the question is: "Let u be harmonic with periodic boundary conditions. Use the maximum principle to show that u is constant."
The maximum principle, as written in my textbook, comes in three parts:
1) Strong max: Let $u$ be harmonic in $\Omega$. If there exists $x_0$ $\epsilon$ $\Omega$ with $u(x_0)=\sup(u(x):x$ $\epsilon$ $\Omega)$ or $u(x_0)=\inf(u(x):x$ $\epsilon$ $\Omega)$, then $u$ is constant on $\Omega$.
Alternatively, using the ball mean property, $u(x)=constant$ iff $u(x_o)=\frac{1}{\omega_d r^d}\int_{B(x_o,r)}u(x)dx = sup(u(x)),x\in \Omega$
Where B is the ball: $B(x,r):={y\in R^d:|x-y|\le r}$
2) Weak max: Let $\Omega$ be bounded and $u$ $\epsilon$ $C^0(\Omega \cup \partial\Omega)$ be harmonic. Then for all $x$ $\epsilon$ $\Omega$, $\min(u(y):y$ $\epsilon$ $\partial\Omega) \le u(x)\le \max(u(y):y$ $\epsilon$ $\partial\Omega)$
3) Translational Corollary: Let $x_0$ $\epsilon$ $\Omega\subset R^d(d\ge 2),$ $u:\Omega\backslash {x_0}\rightarrow R$ be harmonic and bounded. Then u can be extended as a harmonic function on all of $\Omega$; i.e., there exists a harmonic function $\tilde{u}:\Omega\rightarrow R$ that coincides with u on $\Omega\backslash {x_0}$
Periodic boundary conditions are defined as follows:
$\Omega=(0,L_1)\times ...\times (0,L_n)\subset R^n$ and, for $u:\bar{\Omega}\rightarrow R$ that: $u(x_1,...,x_{i-1},L_i,x_{i+1},...,x_n)=u(x_1,...,x_{i-1},0,x_{i+1},...,x_n)$ for all $x=(x_1,...x_n)\in\Omega,i=1,...,n$
So far, I have written the following "true" (as best as I can tell) statements...but I can't see why they require $u(x)$ to be constant:
i) $\Delta u(x)=0$ iff $u(x_0)=\frac{1}{\omega_d r^r}\int_{B(x_0,r)}u(x)dx$
ii) $u(x)=constant$ iff $u(x_0)=\sup_{\Omega}(u(x))$
iii) if $\frac{1}{\omega_d r^d}\int_{B(x_0,r)}u(x)dx=\sup_{\Omega}(u(x))$ then $u(x)=constant$
iv) By periodic boundary conditions, (and using the domain for the un-extended $\Omega$ from earlier), $u(x_0)=\frac{1}{\omega_d r^d}\int_{B(x_0 + nL,r)}u(x+nL)dx$
Where $n\in Z^d$, and $nL=(n_1*L_1,...,n_d*L_d)$
**Note: $\omega_d$ is the volume of the unit sphere in $d$-dimensions