= Solution
Interpret the paper's unspecified $\Omega$ as containing $B_1$, the domain on which the coefficients are used. Write $A(x)=(a_{ij}(x))$, a real <symmetric matrix>. The <strict ellipticity> condition is pointwise positive definiteness:
$$
\boxed{\xi^TA(x)\xi>0\qquad(x\in B_1,\ \xi\in\mathbb R^n\setminus\{0\}).}
$$
The lower bound may depend on $x$. By continuity, <strict ellipticity> gives a positive lower bound on each <compact subset> of $B_1$, but does not require one bound throughout the whole open ball.
With the two-constant convention used in the next part, <uniform ellipticity> means
$$
\boxed{\lambda|\xi|^2\leq\xi^TA(x)\xi\leq\Lambda|\xi|^2
\qquad(x\in B_1,\ \xi\in\mathbb R^n),\quad0<\lambda\leq\Lambda<\infty.}
$$
Both constants are independent of $x$. Some terminology uses <uniform ellipticity> for just the lower bound and states coefficient boundedness separately; here the paper supplies both constants. The <spectral theorem> for the <symmetric matrix> gives $|A(x)\xi|\leq\Lambda|\xi|$. For example, $A(x)=(1-|x|^2)I$ satisfies <strict ellipticity> in $B_1$ but has no uniform positive lower bound as $|x|\uparrow1$.
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