finite perimeter sets
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2011 ◽  
Vol 53 (3) ◽  
pp. 631-635 ◽  
Author(s):  
SILVANO DELLADIO

AbstractWe give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at a.e. (x1, x2) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana).


2003 ◽  
Vol 13 (3) ◽  
pp. 421-466 ◽  
Author(s):  
Bruno Franchi ◽  
Raul Serapioni ◽  
Francesco Serra Cassano

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