The introduction first summarizes the development of the Gauss-Green formula \(\displaystyle \int _{\Omega } \varphi \mathrm {div}F \, d \mathcal {L}^{\,n} + \int _{\Omega } F\cdot D\varphi \, d \mathcal {L}^{\,n}= \int _{\partial {\Omega }} \varphi F\cdot \nu \, d \mathcal {H}^{\,n-1} \, \) during the last decades. Here the extension to integrable vector fields \(F\in \mathcal {D}\mathcal {M}^1(\Omega )\) , where the distributional divergence is a Radon measure, and to sets \(\Omega \) having finite perimeter is essential, while \(\varphi \) typically belongs to \(C(\overline {\Omega })\) . It turns out that the left hand side, often considered as linear functional \(T_F\) in \(\varphi \) , depends only on the values of \(\varphi \) on \(\partial \Omega \) . Thus the right hand side is in fact a Radon measure on \(\partial \Omega \) , mostly related to some pointwise trace of F on \(\partial \Omega \) . To overcome several limitations of this important formula, the extension of \(T_F\) to a functional on \(\mathcal {W}^{1,\infty }(\Omega )\) is discussed. This leads to finitely additive measures, such as those contained in the dual of \(\mathcal {L}^{\infty }(\Omega )\) , which are often not considered very useful. It turns out that the density of a set at a point, as used in geometric measure theory, is a typical example of such a (merely) finitely additive measure. The integrals related to that kind of measures have some property that is widely unknown and essentially differs from the usual known ones, but which makes them a natural tool for the treatment of traces as e.g. needed in the Gauss-Green formula. At the end of the introduction the content of each chapter is outlined.

Introduction

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发表日期：

2025年

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Friedemann Schuricht,
Moritz Schönherr

摘要

The introduction first summarizes the development of the Gauss-Green formula \(\displaystyle \int _{\Omega } \varphi \mathrm {div}F \, d \mathcal {L}^{\,n} + \int _{\Omega } F\cdot D\varphi \, d \mathcal {L}^{\,n}= \int _{\partial {\Omega }} \varphi F\cdot \nu \, d \mathcal {H}^{\,n-1} \, \) during the last decades. Here the extension to integrable vector fields \(F\in \mathcal {D}\mathcal {M}^1(\Omega )\) , where the distributional divergence is a Radon measure, and to sets \(\Omega \) having finite perimeter is essential, while \(\varphi \) typically belongs to \(C(\overline {\Omega })\) . It turns out that the left hand side, often considered as linear functional \(T_F\) in \(\varphi \) , depends only on the values of \(\varphi \) on \(\partial \Omega \) . Thus the right hand side is in fact a Radon measure on \(\partial \Omega \) , mostly related to some pointwise trace of F on \(\partial \Omega \) . To overcome several limitations of this important formula, the extension of \(T_F\) to a functional on \(\mathcal {W}^{1,\infty }(\Omega )\) is discussed. This leads to finitely additive measures, such as those contained in the dual of \(\mathcal {L}^{\infty }(\Omega )\) , which are often not considered very useful. It turns out that the density of a set at a point, as used in geometric measure theory, is a typical example of such a (merely) finitely additive measure. The integrals related to that kind of measures have some property that is widely unknown and essentially differs from the usual known ones, but which makes them a natural tool for the treatment of traces as e.g. needed in the Gauss-Green formula. At the end of the introduction the content of each chapter is outlined.