# Please explain Evans 's PDE Liouville 's Theorem

Here is the proof :

d. Liouville's Theorem. We assert now that there are no nontrivial bounded harmonic functions on all of $\Bbb R^n$.

THEOREM 8 (Liouville's Theorem). Suppose $u:\Bbb R^n\to\Bbb R$ is harmonic and bounded. Then $u$ is constant.

Proof. Fix $x_0\in\Bbb R^n$, $r\gt0$, and apply Theorem 7 on $B(x_0,r)$: \eqalign{ |Du(x_0)|&\leq\dfrac{\sqrt nC_1}{r^{n+1}}\lVert u\rVert_{L^1(B(x_0,r))}\\&\leq\dfrac{\sqrt nC_1\alpha(n)}{r}\lVert u\rVert_{L^\infty(\Bbb R^n)}\to0,} as $r\to\infty$. Thus $Du\equiv0$, and so $u$ is constant.

I don't know why he can change L1 with the ball $B(x_0,r)$ to L infinity with Rn

If $u \in L^1(\Omega) \cap L^\infty(\Omega)$, then
$$\| u \|_1 = \int_\Omega |u| dx \leq \int_\Omega \| u \|_\infty dx = |\Omega| \| u \|_\infty$$
where $|\Omega|$ is the volume of $\Omega$. Now compute the volume of the ball.
He hasn't just replaced it by the $\infty$ norm, but $\alpha(n) r^n$ (the volume of the unit ball) times the infinity norm.