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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
##### Keywords:
Jacobian; total variation; relaxation
##### Citations:
Zbl 1023.49010; Zbl 1041.49016
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