Real Analysis

Maximum Principle

$\Omega$ bounded open in $\mathbb{R}^n$, $u\in C\left(\overline{\Omega}\right)\cap C^2(\Omega)$ harmonic in $\Omega$. Then $\displaystyle\max_\Omega u=\max_{\partial\Omega}u$,

Proof: let $\varepsilon>0$, and let $u_\varepsilon(x):=u(x)+\varepsilon|x|^2$. Then $\Delta u_\varepsilon=2\varepsilon n>0$. So $\displaystyle\max_\Omega u_\varepsilon=\max_{\partial\Omega}u_\varepsilon$, and the claim follows if we let $\varepsilon\rightarrow 0$.

Mean value principle

Let $B_r$ be the open ball of radius $r$ in $\mathbb{R}^n$, and let $u\in C^2\left(\overline{B_r}\right)$ be harmonic in $B_r$. Then
$$
u(0)=\mbox{mean of }u\mbox{ over }\partial B_r.
$$

Proof: set $\Theta u:=u(\Theta^{-1})$ for $\Theta\in O(n)$, and
$$
Au(x):=\mbox{mean of the function }\Theta\in  O(n)\mapsto \Theta u(x)\mbox{ over }O(n),\qquad x\in \overline{B_r}.
$$
Then $\Delta Au=0$ since $\Delta$ is $O(n)$-invariant, and $Au(x)=\mbox{mean of }u\mbox{ over }\partial B_r$ for $|x|=r$. So $Au$ is constant by the maximum principle. But $Au(0)=u(0)$.