Summary: 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 \[ \begin{cases} \Delta u_\beta=\beta(u_\beta-g)\quad &\text{in }\Omega\backslash\Gamma\\ \partial_\nu u_\beta=0\quad & \text{on }\partial\Omega\cup\Gamma\end{cases} \] is in turn discontinuous along \(\Gamma\) and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional \[ \int_{\Omega \backslash S_u}|\nabla u|^2dx +{\mathcal H}^{n-1}(S_u)+\beta \int_{\Omega\backslash S_u}(u-g)^2dx, \] 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.