### Abstract

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(v_{1},...,v_{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.

Original language | English (US) |
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Pages (from-to) | 645-677 |

Number of pages | 33 |

Journal | Indiana University Mathematics Journal |

Volume | 51 |

Issue number | 3 |

DOIs | |

State | Published - 2002 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Indiana University Mathematics Journal*,

*51*(3), 645-677. https://doi.org/10.1512/iumj.2002.51.2229