 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: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 meansBoth 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 $|A(x)\xi |\le \Lambda |\xi |$. For example, $A(x)=(1-|x{|}^2)I$ satisfies strict ellipticity in $B_1$ but has no uniform positive lower bound as $|x|\uparrow 1$.