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

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 Sw of w is a regular curve connecting two boundary points, then there exists a uniform neighbourhood U of Sw such that w is a minimizer of the Mumford–Shah functional on U with respect to its own boundary conditions on ∂U. We show that Euler conditions do not guarantee in general the minimality of w in the class of functions with the same boundary value of w on ∂Ω 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 Ω and Sw under which this kind of minimality holds.