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 -