Monotone measures with bad tangential behavior in the plane

Robert Černý; Jan Kolář; Mirko Rokyta

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 3, page 317-339
  • ISSN: 0010-2628

Abstract

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We show that for every ε > 0 , there is a set A ⊂ ℝ 2 such that ℋ 1 ⌞ A is a monotone measure, the corresponding tangent measures at the origin are not unique and ℋ 1 ⌞ A has the 1 -dimensional density between 1 and 3 + ε everywhere on the support.

How to cite

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Černý, Robert, Kolář, Jan, and Rokyta, Mirko. "Monotone measures with bad tangential behavior in the plane." Commentationes Mathematicae Universitatis Carolinae 52.3 (2011): 317-339. <http://eudml.org/doc/246158>.

@article{Černý2011,
abstract = {We show that for every $\varepsilon > 0$, there is a set $A\subset \mathbb \{R\}^2$ such that $\mathcal \{H\}^1 \llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are not unique and $\mathcal \{H\}^1 \llcorner A$ has the $1$-dimensional density between $1$ and $3+\varepsilon $ everywhere on the support.},
author = {Černý, Robert, Kolář, Jan, Rokyta, Mirko},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {monotone measure; monotonicity formula; tangent measure; monotone measure; monotonicity formula; tangent measure},
language = {eng},
number = {3},
pages = {317-339},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Monotone measures with bad tangential behavior in the plane},
url = {http://eudml.org/doc/246158},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Černý, Robert
AU - Kolář, Jan
AU - Rokyta, Mirko
TI - Monotone measures with bad tangential behavior in the plane
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 3
SP - 317
EP - 339
AB - We show that for every $\varepsilon > 0$, there is a set $A\subset \mathbb {R}^2$ such that $\mathcal {H}^1 \llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are not unique and $\mathcal {H}^1 \llcorner A$ has the $1$-dimensional density between $1$ and $3+\varepsilon $ everywhere on the support.
LA - eng
KW - monotone measure; monotonicity formula; tangent measure; monotone measure; monotonicity formula; tangent measure
UR - http://eudml.org/doc/246158
ER -

References

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  1. Černý R., Local monotonicity of measures supported by graphs of convex functions, Publ. Mat. 48 (2004), 369–380. MR2091010
  2. Černý R., Kolář J., Rokyta M., Concentrated monotone measures with non-unique tangential behaviour in R 3 , Czechoslovak Math. J.(to appear). MR2886262
  3. Kolář J., 10.1112/S0024609306018637, Bull. London Math. Soc. 38 (2006), 657–666. MR2250758DOI10.1112/S0024609306018637
  4. Mattila P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge, 1995. Zbl0911.28005MR1333890
  5. Preiss D., 10.2307/1971410, Ann. Math. 125 (1987), 537–643. MR0890162DOI10.2307/1971410
  6. Simon L., Lectures on geometric measure theory, Proc. C.M.A., Australian National University Vol. 3, 1983. Zbl0546.49019MR0756417

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