Generation of semigroups associated to strongly coupled elliptic operator in $L^p({\mathbb{R}}^d;{\mathbb{R}}^m)$

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 $p\in (1,\mathrm{\infty })$, 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.