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

TL;DR

This paper studies a class of coupled vector-valued elliptic operators with unbounded coefficients in Lebesgue spaces, establishing conditions for generating analytic semigroups and characterizing their generators' domains.

Contribution

It provides new sufficient conditions for the generation of analytic semigroups by strongly coupled elliptic operators with unbounded coefficients in $L^p$ spaces.

Findings

Established conditions for semigroup generation.

Characterized the domain of the generator under additional assumptions.

Extended the theory to coupled elliptic operators with unbounded coefficients.

Abstract

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 sufficient conditions for the generation of an analytic $C_0$-semigroup $T(t)$. Under further assumptions, a characterization of the domain of the infinitesimal generator is given.