Calculus of Variations and Geometric Measure Theory
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S. Delladio

A Whitney-type result about rectifiability of graphs

created by delladio on 07 Feb 2010



Inserted: 7 feb 2010

Year: 2010


Given $m>0$ and a measurable set $E\subset R^n$, $E^{(m)}$ denotes the set of $m$-density points of $E$, namely the points $x$ in $R^n$ at which $L^n(B(x,r)\backslash E)$ is an infinitesimal of order greater than $r^{m}$ (as $r$ goes to $0$). We investigate the size of $E^{(m)}$ in the particular case when $E$ is a generalized Cantor set in $R$. Moreover we prove the following result. Let $\varphi$ be in $C^h(\Omega)$ and $\Phi$ be in $C^h(\Omega;R^n))$, where $\Omega$ is an open subset of $R^n$ and $h\geq 1$. If $K:=\{ x\in\Omega\,\vert\,\nabla\varphi(x)=\Phi(x)\}$ then the graph of $\varphi_{\vert\Omega\cap K^{(n+h)}}$ is a $n$-dimensional $C^{h+1}$-rectifiable set.

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