Neumann functions for second order elliptic systems with measurable coefficients

TL;DR

This paper investigates Neumann functions for second order elliptic systems with measurable coefficients, establishing their existence, uniqueness, estimates, and bounds under certain regularity assumptions on solutions.

Contribution

It provides a unified framework for scalar and vector elliptic systems, linking local boundedness of solutions to global bounds of Neumann functions.

Findings

Existence and uniqueness of Neumann functions established.

Global pointwise bounds derived under local boundedness assumptions.

Equivalence between local boundedness of solutions and Neumann function bounds proved.

Abstract

We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the Neumann functions under the assumption that weak solutions of the system enjoy interior H\"older continuity. Also, we establish global pointwise bounds for the Neumann functions under the assumption that weak solutions of the system satisfy a certain natural local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is in fact equivalent to the global pointwise bound for the Neumann function. We present a unified approach valid for both the scalar and the vectorial cases.