Calculus of Variations and Geometric Measure Theory
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D. Mucci

Remarks on the total variation of the Jacobian

created on 13 Nov 2002
modified by mucci on 22 Nov 2006


Published Paper

Inserted: 13 nov 2002
Last Updated: 22 nov 2006

Journal: NoDEA
Volume: 13
Pages: 223-233
Year: 2006


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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