Chapter 4 presents several divergence theorems or, more generally, several Gauss-Green formulas. In Sect. 4.1 we start with Theorem 4.1 that provides the Gauss-Green formula for any \(F\in \mathcal {D}\mathcal {M}^1(U)\) and any Borel set \(\Omega \subset U\) and also covers specializations for the important cases (G), (L), and (C). Later other special cases are considered. Typical examples show the variability and applicability of the results. In particular, the treatment of singular vector fields, concentrations on the boundaries of open and closed sets, inner boundaries and other degeneracies is demonstrated. The special case of normal measures is considered in Sect. 4.2. The definition and construction of normal measures is followed by some general integrability condition. This leads to Gauss-Green formulas where F and partially also the normal field \(\nu ^\Omega \) are explicitly contained in the boundary term and where, in addition, some weight on \(\partial \Omega \) can be included. Several examples illustrate the variety of applications. Some comprehensive discussion of the results also includes the relation to previous results from the literature. In Sect. 4.3 we briefly transfer the former results for vector fields to Sobolev and BV functions for completeness, but also for the convenience of the reader, since it might be not completely straightforward to do that. In addition we study a Sobolev function on a set \(\Omega \) of finite perimeter where the trace functional is not related to a Radon measure on \(\partial \Omega \) and where both boundary integrals are needed for a general Gauss-Green formula. For a bounded open \(\Omega \) with Lipschitz boundary, we supplement the classical Gauss-Green formula with a new version that contains a normal measure and does not require a trace function on the boundary. Finally, for any bounded open set \(\Omega \) the existence of a weak solution for a general boundary value problem is shown.

Divergence Theorems

章节
发表日期：

2025年

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Friedemann Schuricht,
Moritz Schönherr

摘要

Chapter 4 presents several divergence theorems or, more generally, several Gauss-Green formulas. In Sect. 4.1 we start with Theorem 4.1 that provides the Gauss-Green formula for any \(F\in \mathcal {D}\mathcal {M}^1(U)\) and any Borel set \(\Omega \subset U\) and also covers specializations for the important cases (G), (L), and (C). Later other special cases are considered. Typical examples show the variability and applicability of the results. In particular, the treatment of singular vector fields, concentrations on the boundaries of open and closed sets, inner boundaries and other degeneracies is demonstrated. The special case of normal measures is considered in Sect. 4.2. The definition and construction of normal measures is followed by some general integrability condition. This leads to Gauss-Green formulas where F and partially also the normal field \(\nu ^\Omega \) are explicitly contained in the boundary term and where, in addition, some weight on \(\partial \Omega \) can be included. Several examples illustrate the variety of applications. Some comprehensive discussion of the results also includes the relation to previous results from the literature. In Sect. 4.3 we briefly transfer the former results for vector fields to Sobolev and BV functions for completeness, but also for the convenience of the reader, since it might be not completely straightforward to do that. In addition we study a Sobolev function on a set \(\Omega \) of finite perimeter where the trace functional is not related to a Radon measure on \(\partial \Omega \) and where both boundary integrals are needed for a general Gauss-Green formula. For a bounded open \(\Omega \) with Lipschitz boundary, we supplement the classical Gauss-Green formula with a new version that contains a normal measure and does not require a trace function on the boundary. Finally, for any bounded open set \(\Omega \) the existence of a weak solution for a general boundary value problem is shown.