Global calibrations for the non-homogeneous Mumford-Shah functional

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 $$ \begin{cases} \Delta u_{\beta}=\beta(u_{\beta}-g)& \text{in $\Omega\setminus\Gamma$} \partial_{\nu} u_{\beta}=0 & \text{on $\partial\Omega\cup\Gamma$} \end{cases} $$ 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.