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 $$\{\begin{array}{llc}\Delta u_{\beta }=\beta (u_{\beta }-g)& \text{in Ω∖Γ}{\partial }_{\nu }{u}_{\beta }=0& \text{on ∂Ω∪Γ}\end{array}$$ is in turn discontinuous along $\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $${\int }_{\Omega \setminus S_u}|\nabla u{|}^{2}dx+{\mathcal{H}}^{n-1}(S_u)+\beta {\int }_{\Omega \setminus S_u}(u-g{)}^{2}dx,$$ 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.

33 pages