Abstract

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 $$ \left\lbrace \begin{array}{ll}\Delta u_{\beta }=\beta & \text{in $\Omega \setminus \Gamma $}\\ \partial _{\nu } u_{\beta }=0 & \text{on $\partial \Omega \cup \Gamma $} \end{array}\right.$$ is in turn discontinuous along $\Gamma $ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $$ \int _{\Omega \setminus S_u}|\nabla u|^2\, dx +{\mathcal{H}}^{n-1}+\beta \int _{\Omega \setminus S_u}^2\, dx, $$ over $SBV$, for $\beta $ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown