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Remarks on the total variation of the Jacobian. (English) Zbl 1117.49017
The first remark is the result obtained independently by E. Paolini [Manuscr. Math. 111, No. 4, 499–512 (2003; Zbl 1023.49010)] for the case \(n=2\), \(4/3 < p <2 \) on the subclass \(u(x)= \varphi ({ x \over {| x |}})\) . In the example related to the form of \( \varphi\) the author explicitely computes the gap between the determinant \( | Du | \) and the total variation [see I. Fonseca, N. Fusco and P. Marcellini, J. Funct. Anal. 207, No. 1, 1–32 (2004; Zbl 1041.49016)] \(TV_p (u, B_r^2) \) for every radius \( 0 < r < 1\) . Also for the same case, the second remark is devoted to providing examples of functions \(u \in W^{1,p}(B^2;R^2)\) such that the determinant \(| Du | = 0 \) in \(B^2\) and \(TV_p (u, B^2) = + \infty\). Finally, in the third one the author shows that this gap phenomenon does not occur in case of dimension \(n \geq 3\).

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
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