Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients

TL;DR

This paper extends key analytical tools like the Caccioppoli inequality, Meyers's reverse Hölder inequality, and the fundamental solution to higher-order elliptic systems with rough coefficients, broadening their applicability.

Contribution

It develops new methods to construct fundamental solutions for higher-order elliptic systems without regularity assumptions on coefficients.

Findings

Established Caccioppoli inequality for higher-order systems

Proved Meyers's reverse Hölder inequality for gradients

Constructed fundamental solutions with minimal coefficient regularity

Abstract

We extend several well-known tools from the theory of second-order divergence-form elliptic equations to the case of higher-order equations. These tools are the Caccioppoli inequality, Meyers's reverse Holder inequality for gradients, and the fundamental solution. Our construction of the fundamental solution may also be of interest in the theory of second-order operators, as we impose no regularity assumptions on our elliptic operator beyond ellipticity and boundedness of coefficients.