Functions of bounded higher variation

R. L. Jerrard, H. M. Soner

Research output: Contribution to journalArticle

41 Scopus citations

Abstract

We say that a function u: ℝm → ℝn, with m ≥ n, has bounded n-variation if Det(uxα1,...,uxαn) is a measure for every 1 ≤ α1 <...< αn ≤ m. Here Det(v1,...,vn) 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(ℝm,ℝn) and |u| = 1 a.e., then the Jacobian of u is an m - n-dimensional rectifiable current.

Original languageEnglish (US)
Pages (from-to)645-677
Number of pages33
JournalIndiana University Mathematics Journal
Volume51
Issue number3
DOIs
StatePublished - 2002
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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