Calculus of Variations and Geometric Measure Theory

E. Paolini

On the Relaxed Total Variation of Singular Maps

created on 12 Nov 2002
modified by paolini on 30 Nov 2016


Published Paper

Inserted: 12 nov 2002
Last Updated: 30 nov 2016

Journal: manuscripta mathematica
Volume: 111
Pages: 499-512
Year: 2003
Doi: 10.1007/s00229-003-0381-5


We consider the Total Variation functional $TV(u) = \int \vert \det Du\vert$ which is defined on $W^{1,n}(\Omega,{\mathbf R}^n)$ for $\Omega\subset {\mathbf R}^n$. An extension $TV^p$ is defined by relaxation in the weak topology of $W^{1,p}$ for $p<n$; so the relaxed functional is defined also on maps which may have singularities. In this paper we study the relaxed total variation and find many useful tools to compute the functional on maps which have a singularity in one point.

Keywords: relaxation, Jacobian, Total variation