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Advances in Calculus of Variations

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

Giovanni E. Comi / Kevin R. PayneORCID iD: http://orcid.org/0000-0002-7486-8940
Published Online: 2017-12-12 | DOI: https://doi.org/10.1515/acv-2017-0001

Abstract

Chen, Torres and Ziemer ([9], 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 ([7], 2005) and exploiting ideas of Vol’pert ([29], 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: Divergence measure fields; sets of finite perimeter; generalized Gauss–Green theorems; normal traces; extension and gluing theorems

MSC 2010: 26B20; 26B30; 28C05

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About the article


Received: 2017-01-04

Revised: 2017-10-23

Accepted: 2017-11-14

Published Online: 2017-12-12


The authors have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).


Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2017-0001.

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