Calculus of Variations and Geometric Measure Theory

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