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 $\Omega$, and the discontinuity set $S_w$ of $w$ is a regular curve connecting 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 respect to its own boundary conditions on $\partial U$. 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 $\partial \Omega$ 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.