Concentrated monotone measures with non-unique tangential behavior in ℝ 3

Robert Černý; Jan Kolář; Mirko Rokyta

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 4, page 1141-1167
  • ISSN: 0011-4642

Abstract

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We show that for every ε > 0 there is a set A ⊂ ℝ 3 such that ℋ 1 ⌞ A is a monotone measure, the corresponding tangent measures at the origin are non-conical and non-unique and ℋ 1 ⌞ A has the 1 -dimensional density between 1 and 2 + ε everywhere in the support.

How to cite

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Černý, Robert, Kolář, Jan, and Rokyta, Mirko. "Concentrated monotone measures with non-unique tangential behavior in $\mathbb {R}^3$." Czechoslovak Mathematical Journal 61.4 (2011): 1141-1167. <http://eudml.org/doc/196308>.

@article{Černý2011,
abstract = {We show that for every $\varepsilon >0$ there is a set $A\subset \mathbb \{R\}^3$ such that $\{\mathcal \{H\}\}^1\llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are non-conical and non-unique and $\{\mathcal \{H\}\}^1\llcorner A$ has the $1$-dimensional density between $1$ and $2+\varepsilon $ everywhere in the support.},
author = {Černý, Robert, Kolář, Jan, Rokyta, Mirko},
journal = {Czechoslovak Mathematical Journal},
keywords = {monotone measure; monotonicity formula; tangent measure; monotone measure; monotonicity formula; tangent measure},
language = {eng},
number = {4},
pages = {1141-1167},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Concentrated monotone measures with non-unique tangential behavior in $\mathbb \{R\}^3$},
url = {http://eudml.org/doc/196308},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Černý, Robert
AU - Kolář, Jan
AU - Rokyta, Mirko
TI - Concentrated monotone measures with non-unique tangential behavior in $\mathbb {R}^3$
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 1141
EP - 1167
AB - We show that for every $\varepsilon >0$ there is a set $A\subset \mathbb {R}^3$ such that ${\mathcal {H}}^1\llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are non-conical and non-unique and ${\mathcal {H}}^1\llcorner A$ has the $1$-dimensional density between $1$ and $2+\varepsilon $ everywhere in the support.
LA - eng
KW - monotone measure; monotonicity formula; tangent measure; monotone measure; monotonicity formula; tangent measure
UR - http://eudml.org/doc/196308
ER -

References

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  1. Černý, R., Local monotonicity of Hausdorff measures restricted to curves in ℝ n , Commentat. Math. Univ. Carol. 50 (2009), 89-101. (2009) MR2562806
  2. Černý, R., Kolář, J., Rokyta, M., Monotone measures with bad tangential behavior in the plane, Commentat. Math. Univ. Carol. 52 (2011), 317-339. (2011) MR2843226
  3. Kolář, J., 10.1112/S0024609306018637, Bull. London Math. Soc. 38 (2006), 657-666. (2006) MR2250758DOI10.1112/S0024609306018637
  4. Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Camridge Studies in Advanced Mathematics 44, Cambridge University Press Cambridge (1995). (1995) MR1333890
  5. Preiss, D., 10.2307/1971410, Ann. Math. 125 (1987), 537-643. (1987) MR0890162DOI10.2307/1971410
  6. Simon, L., Lectures on Geometric Measure Theory. Proc. C. M. A., Vol. 3, Australian National University Canberra (1983). (1983) MR0756417

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