| Title:
|
Monotone measures with bad tangential behavior in the plane (English) |
| Author:
|
Černý, Robert |
| Author:
|
Kolář, Jan |
| Author:
|
Rokyta, Mirko |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
317-339 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
monotone measure |
| Keyword:
|
monotonicity formula |
| Keyword:
|
tangent measure |
| MSC:
|
49J45 |
| idZBL:
|
Zbl 1249.49019 |
| idMR:
|
MR2843226 |
| . |
| Date available:
|
2011-08-15T19:06:39Z |
| Last updated:
|
2013-10-14 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141604 |
| . |
| Reference:
|
[1] Černý R.: Local monotonicity of measures supported by graphs of convex functions.Publ. Mat. 48 (2004), 369–380. MR 2091010 |
| Reference:
|
[2] Černý R., Kolář J., Rokyta M.: Concentrated monotone measures with non-unique tangential behaviour in $R^3$.Czechoslovak Math. J.(to appear). MR 2886262 |
| 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.Cambridge University Press, Cambridge, 1995. Zbl 0911.28005, 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., Australian National University Vol. 3, 1983. Zbl 0546.49019, MR 0756417 |
| . |
