Functions of bounded higher variation
Robert L. Jerrard, Halil Mete Soner · Indiana University Mathematics Journal · 2002
We say that a function u: R m → R n , with m ≥ n, has bounded n-variation if Det(u xα1 ,...,u xαn ) is a measure for every 1 ≤ α 1 <... < α n ≤ m. Here Det(ν 1, ...,ν n ) denotes the distributional determinant of the matrix whose columns are the given vectors, arranged in the given order. In this paper we establish a number of properties of BnV functions and related functions. We establish general (and rather weak) versions of the chain rule and the coarea formula; we show that stronger forms of the chain rule can fail, and we also demonstrate that BnV functions cannot, in general, be strongly approximated by smooth functions; and we prove that if u ∈ BnV(R m ,R n ) and |u| = 1 a.e., then the Jacobian of u is an m - n-dimensional rectifiable current.