Γ-Convergence of free discontinuity problems

Cagnetti, Filippo, Dal Maso, Gianni, Scardia, Lucia and Ida Zeppieri, Caterina (2019) Γ-Convergence of free discontinuity problems. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 36 (4). pp. 1035-1079. ISSN 0294-1449

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Abstract

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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Depositing User: Alice Jackson
Date Deposited: 20 Nov 2018 10:44
Last Modified: 15 Nov 2019 02:00
URI: http://sro.sussex.ac.uk/id/eprint/80312

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