  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
domain, and the discontinuity set S of w is a regular curve connecting two
boundary points, then there exists a uniform neighbourhood U of S such that w
is a minimizer of the Mumford-Shah functional on U with respect to its own
boundary conditions. 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
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 the domain and the discontinuity set under which this kind of minimality
holds.
