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

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