%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 Report
%D 2017
%T Gamma-Convergence of Free-discontinuity problems
%A Filippo Cagnetti
%A Gianni Dal Maso
%A Lucia Scardia
%A Caterina Ida Zeppieri
%X We study the Gamma-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Gamma-convergence, representation of the Gamma-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.
%I SISSA
%G en
%U http://preprints.sissa.it/handle/1963/35276
%1 35583
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-03-20T13:20:54Z
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%0 Journal Article
%J ESAIM: COCV 17 (2011) 1-27
%D 2011
%T Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach
%A Filippo Cagnetti
%A Rodica Toader
%X A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [6] is recovered. In this case, the convergence of the discrete time approximations is improved.
%B ESAIM: COCV 17 (2011) 1-27
%I Cambridge University Press / EDP Sciences
%G en_US
%U http://hdl.handle.net/1963/2355
%1 1662
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-11-07T13:38:16Z\\r\\nNo. of bitstreams: 1\\r\\nQuasCrack.pdf: 298737 bytes, checksum: 127034af65efce0ab97317eccb47e76c (MD5)
%R 10.1051/cocv/2009037
%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
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%R 10.1007/s00526-007-0152-3