Calculus of Variations and Geometric Measure Theory
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G. E. Comi - K. R. Payne

On locally essentially bounded divergence measure fields and sets of locally finite perimeter

created by comi on 11 Jan 2017

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Submitted Paper

Inserted: 11 jan 2017
Last Updated: 11 jan 2017

Year: 2017

Abstract:

Chen, Torres and Ziemer (2009) proved the validity of generalized Gauss-Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres (2005) and exploiting ideas of Vol'pert (1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss-Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.

Keywords: Sets of finite perimeter, divergence measure fields, Gauss-Green theorem, normal traces


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