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

Abstract

In this paper, we study vector--valued elliptic operators of the form $\mathcal{L}f:=\mathrm{div}(Q\nabla f)-F\cdot\nabla f+\mathrm{div}(Cf)-Vf$ 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 $1<p<\infty$, 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 $C=(C_{ij})_{1\le i,j\le m}$ are such that $F_{ij},C_{ij}:\mathbb{R}^d\to\mathbb{R}^d$ 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.