TY - JOUR T1 - A second order minimality condition for the Mumford-Shah functional JF - Calc. Var. Partial Differential Equations 33 (2008) 37-74 Y1 - 2008 A1 - Filippo Cagnetti A1 - Maria Giovanna Mora A1 - Massimiliano Morini AB - 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. UR - http://hdl.handle.net/1963/1955 U1 - 2318 U2 - Mathematics U3 - Functional Analysis and Applications ER -