We consider partial differential operators
H
=
−
div
(
C
∇
)
H=-\operatorname {div} (C\nabla )
in divergence form on
R
d
\mathbf {R}^d
with a positive-semidefinite, symmetric, matrix
C
C
of real
L
∞
L_\infty
-coefficients, and establish that
H
H
is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.