*Published Paper*

**Inserted:** 18 sep 2001

**Last Updated:** 17 dec 2006

**Journal:** Boll. UMI

**Volume:** 6-B

**Number:** 8

**Pages:** 339-351

**Year:** 2003

**Abstract:**

Given $\Omega$ subset of $*R*^n$, which is open, connected and bounded, with Lipschitz boundary and volume $

\Omega

$, we prove that a
sequence $F_k$ of Dirichlet-type functionals defined on
$H^1(\Omega;*R*^d)$, with volume constraints $*v*^k$ on $m$ fixed level-sets, and such that the sum of $v^k_i$ over $i=1..m$ is less than $

\Omega

$ for all $k$, gamma-converges, as $*v*^k$ tends to $*v*$ with $v_1+...+v_m =

\Omega

$, to the squared *total variation* on $BV(\Omega;*R*^d)$, with $*v*$ as volume constraint on the same level-sets.