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We study the Γ-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 Γ-convergence, representation of the Γ-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.
Keywords: Free-discontinuity problems, Γ-convergence, Homogenisation
Cagnetti, Filippo 1 ; Dal Maso, Gianni 2 ; Scardia, Lucia 3 ; Zeppieri, Caterina Ida 4
@article{AIHPC_2019__36_4_1035_0,
author = {Cagnetti, Filippo and Dal Maso, Gianni and Scardia, Lucia and Zeppieri, Caterina Ida},
title = {$\Gamma$-convergence of free-discontinuity problems},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1035--1079},
publisher = {Elsevier},
volume = {36},
number = {4},
year = {2019},
doi = {10.1016/j.anihpc.2018.11.003},
mrnumber = {3955110},
zbl = {1417.49010},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.11.003/}
}
TY - JOUR AU - Cagnetti, Filippo AU - Dal Maso, Gianni AU - Scardia, Lucia AU - Zeppieri, Caterina Ida TI - $\Gamma$-convergence of free-discontinuity problems JO - Annales de l'I.H.P. Analyse non linéaire PY - 2019 SP - 1035 EP - 1079 VL - 36 IS - 4 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.11.003/ DO - 10.1016/j.anihpc.2018.11.003 LA - en ID - AIHPC_2019__36_4_1035_0 ER -
%0 Journal Article %A Cagnetti, Filippo %A Dal Maso, Gianni %A Scardia, Lucia %A Zeppieri, Caterina Ida %T $\Gamma$-convergence of free-discontinuity problems %J Annales de l'I.H.P. Analyse non linéaire %D 2019 %P 1035-1079 %V 36 %N 4 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2018.11.003/ %R 10.1016/j.anihpc.2018.11.003 %G en %F AIHPC_2019__36_4_1035_0
Cagnetti, Filippo; Dal Maso, Gianni; Scardia, Lucia; Zeppieri, Caterina Ida. $\Gamma$-convergence of free-discontinuity problems. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1035-1079. doi: 10.1016/j.anihpc.2018.11.003
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