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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 sufficientconditions for the generation of an analytic $C_0$-semigroup $T(t)$. Underfurther assumptions, a characterization of the domain of the infinitesimalgenerator is given.

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