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
Best Answer
The way I see it, the book asks you to prove the following statement:
This statement is true. The 'periodic boundary condition' seems to implicitly assume that $u$ can be periodically extended to $\mathbb R^d$, preserving a certain amount of regularity. And not only continuity, but such that the extended function is still $C^2$, I would guess.
The function $u(x,y) = (e^x + e^{-x})\sin(y)$ on $\Omega = (-1,1)\times (0,2\pi)$ admits a continuous extension to $\mathbb R^2$ and is harmonic on $\Omega$, but is not constant. So the extension needs to be smooth enough for the statement to hold. This is where my guess regarding the precise meaning of 'periodic boundary conditions' comes from (also this would describe the situation on a torus).