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 $$f:\mathbb {R}^d\rightarrow \mathbb {R}^m$$ 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 $$F_{ij},C_{ij}:\mathbb {R}^d\rightarrow \mathbb {R}^d$$ 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. Moreover, we prove $$(L^p-L^q)$$ –estimates and Gaussian upper bounds for kernels associated to the operator $$ {\mathcal {L}} $$ .