A note on the Gauss-Green theorem

Abstract

A very general measuretheoretic version of the Gauss-Green formula (for n-space) has recently been studied by De Giorgi in [DG 1 ] and [DG 2]. Independently Fleming and Young have obtained related results (for 3-space) in [FY], employing the technique of generalized surfaces. Here the work of De Giorgi will be supplemented through use of the geometric concept of exterior normal introduced in [F 1]. Assuming that A is a subset of Euclidean n-space E, measurable with respect to Lebesgue measure L, consider the following two conditions: (1) There exist finite real valued (signed) Borel measures (D, * ,n over E. such that