Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1141-1167, 2011

# Concentrated monotone measures with non-unique tangential behavior in $\Bbb R^3$

## Robert Černý, Jan Kolář, Mirko Rokyta

R. Černý, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: rcerny@karlin.mff.cuni.cz; J. Kolář, Institute of Mathematics, Czech Academy of Sciences, ®itná 25, 115 67 Praha 1, Czech Republic, e-mail: kolar@math.cas.cz; M. Rokyta, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: rokyta@karlin.mff.cuni.cz

Abstract: We show that for every $\varepsilon>0$ there is a set $A\subset\Bbb R^3$ 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.

Keywords: monotone measure, monotonicity formula, tangent measure

Classification (MSC 2010): 53A10, 49Q15, 28A75

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