Solution
= Solution
Take
$$
L=\frac{d^4}{dx^4}-1.
$$
Its <principal symbol> is $\xi^4$, so it is elliptic, and $a_4a_0=-1<0$. Yet $u(x)=\sin x$ satisfies $Lu=0$ on $(0,\pi)$ and vanishes at both boundary points while remaining positive inside. Thus second-order ellipticity is essential to the usual <weak maximum principle for elliptic operators>.