The Hausdorff lower semicontinuous envelope of the length in the plane

Raphaël Cerf

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 1, page 33-71
  • ISSN: 0391-173X

Abstract

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We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.

How to cite

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Cerf, Raphaël. "The Hausdorff lower semicontinuous envelope of the length in the plane." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.1 (2002): 33-71. <http://eudml.org/doc/84467>.

@article{Cerf2002,
abstract = {We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.},
author = {Cerf, Raphaël},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {residual domain; Hausdorff measure; 1-set; surface energy; phase coexistence phenomena},
language = {eng},
number = {1},
pages = {33-71},
publisher = {Scuola normale superiore},
title = {The Hausdorff lower semicontinuous envelope of the length in the plane},
url = {http://eudml.org/doc/84467},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Cerf, Raphaël
TI - The Hausdorff lower semicontinuous envelope of the length in the plane
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 1
SP - 33
EP - 71
AB - We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.
LA - eng
KW - residual domain; Hausdorff measure; 1-set; surface energy; phase coexistence phenomena
UR - http://eudml.org/doc/84467
ER -

References

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  1. [1] K. S. Alexander – J. T. Chayes – L. Chayes, The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation, Comm. Math. Phys. 131 no.1 (1990), 1-50. Zbl0698.60098MR1062747
  2. [2] R. Caccioppoli, Misura e integrazione sugli insiemi dimensionalmente orientati I, II, Rend. Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Ser. VIII Vol. XII fasc. 1,2 (gennaio-febbraio 1952) 3-11, 137-146. Zbl0048.03704MR47118
  3. [3] R. Cerf, Large deviations for i.i.d. random compact sets, Proc. Amer. Math. Soc. 127 (1999), 2431-2436. Zbl0934.60017MR1487361
  4. [4] R. Cerf, Large deviations of the finite cluster shape for two-dimensional percolation in the Hausdorff and L 1 metric, J. Theoret. Probab. 12 no. 4 (1999), 1137-1163. Zbl0974.60089MR1777542
  5. [5] R. Cerf, Large deviations for three dimensional supercritical percolation, Astérisque 267 (2000). Zbl0962.60002MR1774341
  6. [6] E. De Giorgi, Nuovi teoremi relativi alle misure ( r - 1 ) -dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4 (1995), 95-113. Zbl0066.29903MR74499
  7. [7] R. L. Dobrushin – R. Kotecký – S. B. Shlosman, “Wulff construction: a global shape from local interaction”, AMS translations series, Providence (Rhode Island) 1992. Zbl0917.60103MR1181197
  8. [8] K. J. Falconer, “The geometry of fractal sets”, Cambridge studies in advanced mathematics, Cambridge University Press, 1985. Zbl0587.28004MR867284
  9. [9] E. Giusti, “Minimal surfaces and Functions of bounded variation”, Monographs in Mathematics, Birkhäuser, Basel, 1984. Zbl0545.49018MR775682
  10. [10] G. P. Leonardi, Optimal subdivisions of n -dimensional domains, PhD Thesis, Università di Trento, 1998. Zbl1053.49522
  11. [11] P. Mattila, “Geometry of sets and measures in Euclidean spaces”, Cambridge studies in advanced mathematics, Cambridge University Press, 1995. Zbl0819.28004MR1333890
  12. [12] M. H. A. Newman, “Elements of the topology of plane sets of points”, 2nd edition, Cambridge University Press, 1954. Zbl0045.44003MR132534JFM65.0873.04

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