This morning a collegue of mine came to me with the following question: does there exist any elliptic operator of order $2m$ with real (variable) coefficients that is not strongly elliptic?
After some investigation, I wasn't able to find an answer. I found out some classical examples, but they all use the complex unit somewhere.
I recall that $L=\sum_{|\beta|=2m}a_\beta (x) D^\beta$ is called
elliptic if $\sum_{|\beta|=2m}a_\beta (x) \xi^{\beta} \neq 0$ whenever $\xi \neq 0$;
strongly elliptic if $\sum_{|\beta|=2m}a_\beta (x) \xi^{\beta} \geq C(x) |\xi|^{2m}$ for some $C(x)>0$ and any $\xi$.
The coefficients of $L$ may be taken smooth "enough" on a bounded domain $\Omega$.