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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
##### Keywords:
bounded $$n$$-variation; coarea formula; Jacobian
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