Calculus of Variations and Geometric Measure Theory

D. Mucci

A Characterization of Graphs which can be Approximated in Area by Smooth Graphs

created on 18 May 1999
modified on 05 Dec 2002


Published Paper

Inserted: 18 may 1999
Last Updated: 5 dec 2002

Journal: J.Eur.Math.Soc.
Volume: 3
Pages: 1-38
Year: 2001


For vector valued maps, convergence in $W^{1,1}$ and of all minors of the Jacobian matrix in $L^{1}$ is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension $n\geq 3$ can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. $2$-dimensional plane intersecting the domain.