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

• Published in: Advances in calculus of variations Vol. 13; no. 2; pp. 179 - 217
• Main Authors: Comi, Giovanni E., Payne, Kevin R.
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.04.2020; Walter de Gruyter GmbH
• Subjects: 26B20; 26B30; 28C05; Approximation; Divergence measure fields; extension and gluing theorems; generalized Gauss–Green theorems; Mathematical analysis; normal traces; PERT; sets of finite perimeter; Smooth boundaries
• ISSN: 1864-8258; 1864-8266
• DOI: 10.1515/acv-2017-0001

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.