Generation results for vector-valued elliptic operators with unbounded coefficients in L^p spaces

TL;DR

This paper investigates vector-valued elliptic operators with unbounded coefficients in L^p spaces, establishing conditions for the generation of analytic semigroups and exploring properties like hypercontractivity.

Contribution

It provides new sufficient conditions for the generation of analytic semigroups by such operators and characterizes their domains, advancing understanding of their functional analysis properties.

Findings

Proved generation of analytic C_0-semigroups under new conditions

Characterized the domain of the semigroup generators

Established results on hypercontractivity and ultraboundedness

Abstract

We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first-order, in the Lebesgue space L^p(R^d;R^m) with p in (1,\infty). Sufficient conditions to prove generation results of an analytic C_0-semigroup T(t), together with a characterization of the domain of its generator, are given. Some results related to the hypercontractivity and the ultraboundedness of the semigroup are also established.