| Title:
|
Concentrated monotone measures with non-unique tangential behavior in $\mathbb{R}^3$ (English) |
| Author:
|
Černý, Robert |
| Author:
|
Kolář, Jan |
| Author:
|
Rokyta, Mirko |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
1141-1167 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that for every $\varepsilon >0$ there is a set $A\subset \mathbb{R}^3$ such that ${\Cal H}^1\llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are non-conical and non-unique and ${\Cal H}^1\llcorner A$ has the $1$-dimensional density between $1$ and $2+\varepsilon $ everywhere in the support. (English) |
| Keyword:
|
monotone measure |
| Keyword:
|
monotonicity formula |
| Keyword:
|
tangent measure |
| MSC:
|
28A75 |
| MSC:
|
49Q15 |
| MSC:
|
53A10 |
| idZBL:
|
Zbl 1249.53006 |
| idMR:
|
MR2886262 |
| DOI:
|
10.1007/s10587-011-0054-6 |
| . |
| Date available:
|
2011-12-16T15:55:23Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141812 |
| . |
| Reference:
|
[1] Černý, R.: Local monotonicity of Hausdorff measures restricted to curves in $\mathbb R^n$.Commentat. Math. Univ. Carol. 50 (2009), 89-101. MR 2562806 |
| Reference:
|
[2] Černý, R., Kolář, J., Rokyta, M.: Monotone measures with bad tangential behavior in the plane.Commentat. Math. Univ. Carol. 52 (2011), 317-339. MR 2843226 |
| Reference:
|
[3] Kolář, J.: Non-regular tangential behaviour of a monotone measure.Bull. London Math. Soc. 38 (2006), 657-666. MR 2250758, 10.1112/S0024609306018637 |
| Reference:
|
[4] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Camridge Studies in Advanced Mathematics 44.Cambridge University Press Cambridge (1995). MR 1333890 |
| Reference:
|
[5] Preiss, D.: Geometry of measures in $\mathbb R^n$: Distribution, rectifiability and densities.Ann. Math. 125 (1987), 537-643. MR 0890162, 10.2307/1971410 |
| Reference:
|
[6] Simon, L.: Lectures on Geometric Measure Theory. Proc. C. M. A., Vol. 3.Australian National University Canberra (1983). MR 0756417 |
| . |
