*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

**Abstract:**

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

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