On indecomposable sets with applications

Andrew Lorent

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 2, page 612-631
  • ISSN: 1292-8119

Abstract

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In this note we show the characteristic function of every indecomposable set F in the plane is BVequivalent to the characteristic function a closed set See Formula in PDF See Formula in PDF . We show by example this is false in dimension three and above. As a corollary to this result we show that for everyϵ > 0 a set of finite perimeter Scan be approximated by a closed subset See Formula in PDF See Formula in PDF with finitely many indecomposable components and with the property that See Formula in PDF See Formula in PDF and See Formula in PDF See Formula in PDF . We apply this corollary to give a short proof that locally quasiminimizing sets in the plane areBVl extension domains.

How to cite

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Lorent, Andrew. "On indecomposable sets with applications." ESAIM: Control, Optimisation and Calculus of Variations 20.2 (2014): 612-631. <http://eudml.org/doc/272893>.

@article{Lorent2014,
abstract = {In this note we show the characteristic function of every indecomposable set F in the plane is BVequivalent to the characteristic function a closed set See Formula in PDF See Formula in PDF . We show by example this is false in dimension three and above. As a corollary to this result we show that for everyϵ &gt; 0 a set of finite perimeter Scan be approximated by a closed subset See Formula in PDF See Formula in PDF with finitely many indecomposable components and with the property that See Formula in PDF See Formula in PDF and See Formula in PDF See Formula in PDF . We apply this corollary to give a short proof that locally quasiminimizing sets in the plane areBVl extension domains.},
author = {Lorent, Andrew},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {sets of finite perimeter; indecomposable sets},
language = {eng},
number = {2},
pages = {612-631},
publisher = {EDP-Sciences},
title = {On indecomposable sets with applications},
url = {http://eudml.org/doc/272893},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Lorent, Andrew
TI - On indecomposable sets with applications
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 2
SP - 612
EP - 631
AB - In this note we show the characteristic function of every indecomposable set F in the plane is BVequivalent to the characteristic function a closed set See Formula in PDF See Formula in PDF . We show by example this is false in dimension three and above. As a corollary to this result we show that for everyϵ &gt; 0 a set of finite perimeter Scan be approximated by a closed subset See Formula in PDF See Formula in PDF with finitely many indecomposable components and with the property that See Formula in PDF See Formula in PDF and See Formula in PDF See Formula in PDF . We apply this corollary to give a short proof that locally quasiminimizing sets in the plane areBVl extension domains.
LA - eng
KW - sets of finite perimeter; indecomposable sets
UR - http://eudml.org/doc/272893
ER -

References

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  6. [6] J. Kinnunenm, R. Korte, A. Lorent and N. Shanmugalingam, Regularity of sets with quasiminimal boundary surfaces in metric spaces. J. Geom. Anal.23 (2013) 1607–1640. Zbl1311.49116MR3107671
  7. [7] B. Kirchheim, Lipschitz minimizers of the 3-well problem having gradients of bounded variation. MIS. MPg. Preprint (12/1998). 
  8. [8] P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Vol. 44. of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). Zbl0819.28004MR1333890

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