TY - RPRT
T1 - Stochastic homogenisation of free-discontinuity problems
Y1 - 2018
A1 - Filippo Cagnetti
A1 - Gianni Dal Maso
A1 - Lucia Scardia
A1 - Caterina Ida Zeppieri
AB - In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.
UR - http://preprints.sissa.it/handle/1963/35309
U1 - 35617
U2 - Mathematics
U4 - 1
U5 - MAT/05
ER -
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 -