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NON-REGULAR TANGENTIAL BEHAVIOUR OF A MONOTONE MEASURE

Published online by Cambridge University Press:  24 July 2006

JAN KOLÁŘ
Affiliation:
Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00 Prague 8, Czech Republic Department of Mathematics, University College London, Gower street, London WC1E 6BT, United Kingdom Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republickolar@math.cas.cz
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Abstract

A Radon measure $\mu$ on ${mathbb R}^n$ is said to be $k$-monotone if $r\mapsto{\mu(B(x,r))}/{r^k}$ is a non-decreasing function on $(0,\infty)$ for every $x\in {\mathbb R}^n$. (If $\mu$ is the $k$-dimensional Hausdorff measure restricted to a $k$-dimensional minimal surface then this important property is expressed by the monotonicity formula.) We give an example of a 1 -monotone measure $\mu$ in ${\mathbb R}^2$ with non-unique and non-conical tangent measures at a point. Furthermore, we show that $\mu$ can be the one-dimensional Hausdorff measure restricted to a closed set $A\subset {\mathbb R}^2$.

Type
Papers
Copyright
The London Mathematical Society 2006

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