 Solution

Positivity and continuity of $g$ give $\mu ={\min }_{[1/2,2]}g>0$ and a finite maximum $G$. Hence, for every $x$ in the shell and every $\xi \in {\mathbb{R}}^3$,This is the uniformly elliptic operator condition. Positivity at separate points without a compactness/continuity lower bound would not alone supply a common ellipticity constant.