Fundamental matrices and Green matrices for non-homogeneous elliptic systems

TL;DR

This paper develops a theoretical framework for the existence, uniqueness, and estimates of fundamental and Green matrices for non-homogeneous elliptic systems with measurable coefficients, extending classical results to more general settings.

Contribution

It introduces non-homogeneous versions of classical bounds and analyzes conditions on lower order terms to ensure fundamental solution properties for elliptic systems.

Findings

Established existence and uniqueness of fundamental solutions and Green functions.

Derived scale-invariant estimates for these solutions.

Identified conditions on lower order terms for bounds to hold.

Abstract

In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions.