Previous |  Up |  Next

Article

Full entry | PDF   (0.3 MB)
Keywords:
monotone measure; monotonicity formula; tangent measure
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.
References:
[1] Černý R.: Local monotonicity of measures supported by graphs of convex functions. Publ. Mat. 48 (2004), 369–380. MR 2091010
[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
[3] Kolář J.: Non-regular tangential behaviour of a monotone measure. Bull. London Math. Soc. 38 (2006), 657–666. DOI 10.1112/S0024609306018637 | MR 2250758
[4] Mattila P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge, 1995. MR 1333890 | Zbl 0911.28005
[5] Preiss D.: Geometry of measures in $\mathbb R^n$: Distribution, rectifiability and densities. Ann. Math. 125 (1987), 537–643. DOI 10.2307/1971410 | MR 0890162
[6] Simon L.: Lectures on geometric measure theory. Proc. C.M.A., Australian National University Vol. 3, 1983. MR 0756417 | Zbl 0546.49019

Partner of