Geodesic
Concentrated monotone measures with non-unique tangential behavior in $\mathbb{R}^3$
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1141-1167
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10587-011-0054-6
Classification : 28A75, 49Q15, 53A10
Keywords: monotone measure; monotonicity formula; tangent measure 
@article{10_1007_s10587_011_0054_6,
     author = {\v{C}ern\'y, Robert and Kol\'a\v{r}, Jan and Rokyta, Mirko},
     title = {Concentrated monotone measures with non-unique tangential behavior in $\mathbb{R}^3$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1141--1167},
     year = {2011},
     volume = {61},
     number = {4},
     doi = {10.1007/s10587-011-0054-6},
     mrnumber = {2886262},
     zbl = {1249.53006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0054-6/}
}
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
SP  - 1141
EP  - 1167
VL  - 61
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0054-6/
DO  - 10.1007/s10587-011-0054-6
LA  - en
ID  - 10_1007_s10587_011_0054_6
ER  - 
%0 Journal Article
%A Černý, Robert
%A Kolář, Jan
%A Rokyta, Mirko
%T Concentrated monotone measures with non-unique tangential behavior in $\mathbb{R}^3$
%J Czechoslovak Mathematical Journal
%D 2011
%P 1141-1167
%V 61
%N 4
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0054-6/
%R 10.1007/s10587-011-0054-6
%G en
%F 10_1007_s10587_011_0054_6
Černý, Robert; Kolář, Jan; Rokyta, Mirko. Concentrated monotone measures with non-unique tangential behavior in $\mathbb{R}^3$. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1141-1167. doi: 10.1007/s10587-011-0054-6

[1] Černý, R.: Local monotonicity of Hausdorff measures restricted to curves in $\mathbb R^n$. Commentat. Math. Univ. Carol. 50 (2009), 89-101. | MR

[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

[3] Kolář, J.: Non-regular tangential behaviour of a monotone measure. Bull. London Math. Soc. 38 (2006), 657-666. | DOI | MR

[4] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Camridge Studies in Advanced Mathematics 44. Cambridge University Press Cambridge (1995). | MR

[5] Preiss, D.: Geometry of measures in $\mathbb R^n$: Distribution, rectifiability and densities. Ann. Math. 125 (1987), 537-643. | DOI | MR

[6] Simon, L.: Lectures on Geometric Measure Theory. Proc. C. M. A., Vol. 3. Australian National University Canberra (1983). | MR

Cité par Sources :