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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

  1. elliptic if $\sum_{|\beta|=2m}a_\beta (x) \xi^{\beta} \neq 0$ whenever $\xi \neq 0$;

  2. 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 $\Omega$.

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