%0 Report
%D 2018
%T Stochastic homogenisation of free-discontinuity problems
%A Filippo Cagnetti
%A Gianni Dal Maso
%A Lucia Scardia
%A Caterina Ida Zeppieri
%X 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.
%G en
%U http://preprints.sissa.it/handle/1963/35309
%1 35617
%2 Mathematics
%4 1
%# MAT/05
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-03-19T09:16:27Z
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%0 Journal Article
%J Calc. Var. Partial Differential Equations 33 (2008) 37-74
%D 2008
%T A second order minimality condition for the Mumford-Shah functional
%A Filippo Cagnetti
%A Maria Giovanna Mora
%A Massimiliano Morini
%X 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.
%B Calc. Var. Partial Differential Equations 33 (2008) 37-74
%G en_US
%U http://hdl.handle.net/1963/1955
%1 2318
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-03-02T12:54:00Z\\nNo. of bitstreams: 1\\nCMM.pdf: 358759 bytes, checksum: c414c0080a17971ecba1251a5890b94f (MD5)
%R 10.1007/s00526-007-0152-3