Riv. Mat. Univ. Parma, Vol. 5, No. 2, 2014

A Whitney-type result about rectifiability of graphs

Pages: 387-397
Accepted: 3 September 2013
Mathematics Subject Classification (2010): Primary 49Q15, 28A75, 28A78; Secondary 28A12.

Keywords: Rectifiable sets, geometric measure theory, Whitney extension theorem.
[1] : Department of Mathematics, Università degli Studi di Trento, via Sommarive 14, Povo, Trento, Italy

Abstract: Given $$m>0$$ and a measurable set $$E\subset\mathbb{R}^n$$, $$E^{(m)}$$ denotes the set of $$m$$-density points of $$E$$, namely the points $$x\in\mathbb{R}^n$$ at which $${\cal L}^n(B(x,r)\backslash E)$$ is an infinitesimal of order greater than $$r^{m}$$ (as $$r\to 0$$). We investigate the size of $$E^{(m)}$$ in the particular case when $$E$$ is a generalized Cantor set in $$\mathbb{R}$$. Moreover we prove the following result. Let $$\varphi\in C^h(\Omega)$$ and $$\Phi\in C^h(\Omega;\mathbb{R}^n))$$, where $$\Omega$$ is an open subset of $$\mathbb{R}^n$$ 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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