  Using a calibration method we prove that, if $\Gamma\subset \Omega$ is a
closed regular hypersurface and if the function $g$ is discontinuous along
$\Gamma$ and regular outside, then the function $u_{\beta}$ which solves $$
\begin{cases} \Delta u_{\beta}=\beta(u_{\beta}-g)& \text{in
$\Omega\setminus\Gamma$} \partial_{\nu} u_{\beta}=0 & \text{on
$\partial\Omega\cup\Gamma$} \end{cases} $$ is in turn discontinuous along
$\Gamma$ and it is the unique absolute minimizer of the non-homogeneous
Mumford-Shah functional $$ \int_{\Omega\setminus S_u}|\nabla u|^2 dx +{\cal
H}^{n-1}(S_u)+\beta\int_{\Omega\setminus S_u}(u-g)^2 dx, $$ over $SBV(\Omega)$,
for $\beta$ large enough. Applications of the result to the study of the
gradient flow by the method of minimizing movements are shown.
