Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set

Using a calibration method, we prove that, if w is a function which satisfi es all Euler conditions for the Mumford-Shah functional on a two-dimensiona l open set Omega, and the discontinuity set S-w of w is a regular curve con necting two boundary points, then there exists a uniform neighbourhood U of S-w such that w is a minimizer of the Mumford-Shah functional on U with re spect to its own boundary conditions on partial derivativeU. We show that E uler conditions do not guarantee in general the minimality of w in the clas s of functions with the same boundary value of w on partial derivative Omeg a and whose extended graph is contained in a neighbourhood of the extended graph of w, and we give a sufficient condition in terms of the geometrical properties of Omega and S-w under which this kind of minimality holds, (C) 200 Editions scientifiques et medicales Elsevier SAS.