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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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1 Answer

You should have $\xi^\beta$ instead of $|\xi|^\beta$ in both 1 and 2 (and also take care of the sign in 2). More importantly, allowing $C$ to depend on $x$ in 2 defeats the purpose of having $C$ at all: this form of "strong" ellipticity is no different from ellipticity. Indeed, for each fixed $x$ the sum $\sum_{|\beta|=2m} a_\beta(x) \xi^\beta$ is a continuous function of $\xi$ and is homogeneous of degree $2m$. If it does not vanish on the unit sphere $S$, then its absolute value attains a positive minimum $\mu$ there. The homogeneity then implies that $\left|\sum_{|\beta|=2m} a_\beta(x) \xi^\beta\right| \ge \mu |\xi|^{2m}$ for all $\xi$.

If $C$ is independent of $x$, as it should be, then you have easy examples like $(1-|x|^2)\,\Delta $ on the domain $|x|<1$.

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