Inserted: 19 feb 2007
Last Updated: 19 sep 2008
Journal: Calc. Var.
A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\Gamma)$, $\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.
Keywords: free discontinuity problems, Mumford-Shah functional, necessary and sufficient condition for minimality, second variation, shape derivative