Divergence-Measure Fields: Gauss-Green Formulas and Normal Traces

TL;DR

This paper investigates the extension of the Gauss-Green formula to divergence-measure fields with discontinuities and irregular domains, addressing a fundamental question motivated by physical PDE solutions.

Contribution

It provides a comprehensive analysis and new results on Gauss-Green formulas for divergence-measure fields and rough domains, expanding classical theory.

Findings

Gauss-Green formulas hold for divergence-measure fields with discontinuities

Extension of Gauss-Green formulas to domains with rough boundaries

Historical review of developments in divergence-measure fields

Abstract

The classical Gauss-Green formula for the multidimensional case is generally stated for $C^{1}$ vector fields and domains with $C^{1}$ boundaries. However, motivated by the physical solutions with discontinuity/singularity for Partial Differential Equations (PDEs) and Calculus of Variations, such as nonlinear hyperbolic conservation laws and Euler-Lagrange equations, the following fundamental issue arises: Does the Gauss-Green formula still hold for vector fields with discontinuity/singularity (such as divergence-measure fields) and domains with rough boundaries? The objective of this paper is to provide an answer to this issue and to present a short historical review of the contributions by many mathematicians spanning more than two centuries, which have made the discovery of the Gauss-Green formula possible.