For vector valued maps, convergence in W1,1 and of all minors of the Jacobian matrix in L1 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 S 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.
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Domenico Mucci, A characterization of graphs which can be approximated in area by smooth graphs. J. Eur. Math. Soc. 3 (2001), no. 1, pp. 1–38DOI 10.1007/PL00011301