Calculus of Variations and Geometric Measure Theory

M. Morini

Global calibrations for the non-homogeneous Mumford-Shah functional

created on 11 May 2001
modified on 09 Jun 2004


Published Paper

Inserted: 11 may 2001
Last Updated: 9 jun 2004

Journal: Annali dela Scuola Normale Superiore
Pages: 34
Year: 2002


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 $$\Delta u{\beta}=\beta(u{\beta}-g) $$ in $\Omega\setminus\Gamma$ with homogeneous Neumann boundary conditions, is in turn discontinuous along $\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $$\int{\Omega\setminus Su}
2\, dx +{\cal H}{n-1}(Su)+\beta\int{\Omega\setminus Su}(u-g)2\, dx,$$ 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.

Keywords: minimizing movements, calibration method