*Published Paper*

**Inserted:** 13 nov 2002

**Last Updated:** 22 nov 2006

**Journal:** NoDEA

**Volume:** 13

**Pages:** 223-233

**Year:** 2006

**Abstract:**

The total variation \,$TV(u)$\, of the Jacobian determinant of non-smooth vector
fields \,$u$\, has recently been studied in:
FONSECA I., FUSCO N., MARCELLINI P., *On the Total Variation of the Jacobian*.
We focus on the subclass \,$u(x)=\phi(x/\vert x\vert)$\, of homogeneous
extensions of smooth functions \,$\phi :\partial B^n\to{\bf{R}^n}$. In the case
\,$n=2$, we explicitely compute \,$TV(u)$\, for some relevant
examples exhibiting a gap with respect to the total variation
\,$\vert{\mbox{\rm Det}}\,Du\vert$\, of the distributional determinant.
We then provide examples of functions with \,$\vert{\mbox{\rm Det}}\,Du\vert=0$\,
and \,$TV(u)=+\infty$. We finally show that this gap
phenomenon doesn't occur if \,$n\geq 3$.

**Keywords:**
relaxation, Jacobian, Total variation

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