On the $L^p$-theory of vector-valued elliptic operators

In this paper, we study vector–valued elliptic operators of the form $\mathcal{L}f:=\mathrm{d}\mathrm{i}\mathrm{v}(Q\mathrm{\nabla }f)-F\cdot \mathrm{\nabla }f+\mathrm{d}\mathrm{i}\mathrm{v}(Cf)-Vf$ acting on vector-valued functions $f:{\mathbb{R}}^d\to {\mathbb{R}}^m$ and involving coupling at zero and first order terms. We prove that $\mathcal{L}$ admits realizations in $L^p({\mathbb{R}}^d,{\mathbb{R}}^m)$, for $1<p<\mathrm{\infty }$, that generate analytic strongly continuous semigroups provided that $V=(v_{ij}{)}_{1\le i,j\le m}$ is a matrix potential with locally integrable entries satisfying a sectoriality condition, the diffusion matrix $Q$ is symmetric and uniformly elliptic and the drift coefficients $F=(F_{ij}){1\le i,j\le m}$and$C=(C{ij}){1\le i,j\le m}$aresuchthat$F{ij},C_{ij}:$\mathbb{R}$^d\to$\mathbb{R}$^d$ are bounded. We also establish a result of local elliptic regularity for the operator $\mathcal{L}$, we investigate on the $L^p$-maximal domain of $\mathcal{L}$ and we characterize the positivity of the associated semigroup.