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indecomposable sets

ESAIM: Control, Optimisation and Calculus of Variations (1)

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On indecomposable sets with applications
Andrew Lorent
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 20 / Issue 2 / April 2014

Published online by Cambridge University Press:

28 March 2014, pp. 612-631

Print publication:

April 2014
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In this note we show the characteristic function of every indecomposable setF in theplane is BVequivalent to the characteristic function a closed set $See Formula in PDF$ \hbox{See Formula in PDF} .We show by example this is false in dimension three and above. As a corollary to thisresult we show that for every ϵ > 0 a set of finite perimeter S can be approximated by aclosed subset $See Formula in PDF$ \hbox{See Formula in PDF} with finitely many indecomposablecomponents and with the property that $See Formula in PDF$ \hbox{See Formula in PDF} and $See Formula in PDF$ \hbox{See Formula in PDF} .We apply this corollary to give a short proof that locally quasiminimizing sets in theplane are BVlextension domains.