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Functions of bounded higher variation. (English) Zbl 1057.49036
This paper deals with various properties of functions with bounded \(n\)-variation, that is, functions \(u:\mathbb R^m\rightarrow \mathbb R^n\) (with \(m\geq n\)) such that Det\(\,(u_{x_{\alpha_1}},\dots ,u_{x_{\alpha_n}})\) is a measure for every \(1\leq\alpha_1 <\dots <\alpha_n\leq m\).
The main results of the present paper are the following: (i) several versions of the chain rule and the coarea formula; (ii) an arbitrary function with bounded \(n\)-variation cannot be strongly approximated by smooth functions; (iii) if \(u:\mathbb R^m\rightarrow \mathbb R^n\) is a function with bounded \(n\)-variation such that \(| u| =1\) a.e., then the Jacobian of \(u\) is an \(m-n\) dimensional rectifiable current. The proofs of the main results rely upon refined techniques from geometric measure theory.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
35J20 Variational methods for second-order elliptic equations
49Q10 Optimization of shapes other than minimal surfaces
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