Geöcze area and a convergence property

Abstract:

Suppose f is a continuous mapping with finite Lebesgue area from a polyhedral region $X \subset {R^k}$ into ${R^n},2 \leqq k \leqq n$. Let $f = l \circ m$ be the monotone-light factorization of f with middle space M. If f satisfies a “cylindrical condition” considered by T. Nishiura, then a current valued measure T over M can be associated with f by means of the Cesari-Weierstrass integral, and if $\{ {f_i}\}$ is any sequence of quasi-linear maps ${f_i}:X \to {R^n}$ converging uniformly to f with bounded areas, then \[ T(g)(\phi ) = \lim \limits _{i \to \infty } \int _X {(g \circ m)f_i^\# \phi } \] whenever $\phi$ is an infinitely differentiable k-form in ${R^n}$ and g is a continuous real valued function on M which vanishes on $m (\text {Bdry} \; X)$. The total variation measure of T, taken with respect to mass, coincides with the Geöcze area measure over M.