Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

Abstract

AbstractWe analyze a class of weakly differentiable vector fieldsF: ℝn→ ℝnwith the property thatF∈L∞and divFis a (signed) Radon measure. These fields are calledbounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure fieldFover the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ℋ︁N− 1on ℝN, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace ofFon the boundary of any set of finite perimeterEas the limit of the normal traces ofFon the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem forFholds onE. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N− 1)‐dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.