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

Silvano Delladio[1]

A Whitney-type result about rectifiability of graphs

Pages: 387-397
Received: 24 June 2013   
Accepted: 3 September 2013
Mathematics Subject Classification (2010): Primary 49Q15, 28A75, 28A78; Secondary 28A12.

Keywords: Rectifiable sets, geometric measure theory, Whitney extension theorem.
Author address:
[1] : Department of Mathematics, UniversitÓ 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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