  A class of vector-valued elliptic operators with unbounded coefficients,
coupled up to the second-order is investigated in the Lebesgue space
$L^p(\mathbb R^d;\mathbb R^m)$ with $p \in (1,\infty)$, providing sufficient
conditions for the generation of an analytic $C_0$-semigroup $T(t)$. Under
further assumptions, a characterization of the domain of the infinitesimal
generator is given.
