Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets

Abstract

Using a calibration method, we prove that, if $w$ is a function which satisfies all Euler conditions for the Mumford-Shah functional on a two-dimensional open set $Ω$, and the discontinuity set of $w$ is a segment connecting two boundary points, then for every point $(x_0, y_0)$ of $Ω$ there exists a neighbourhood $U$ of $(x_0, y_0)$ such that $w$ is a minimizer of the Mumford-Shah functional on $U$ with respect to its own boundary values on $\partial U$.