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G-Convergence of free discontinuity problems

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posted on 2023-06-09, 15:57 authored by Filippo Cagnetti, Gianni Dal Maso, Lucia Scardia, Caterina Ida Zeppieri
We study the G-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 G-convergence, representation of the G-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.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire

ISSN

0294-1449

Publisher

Elsevier

Issue

4

Volume

36

Page range

1035-1079

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Analysis and Partial Differential Equations Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-11-20

First Open Access (FOA) Date

2019-11-15

First Compliant Deposit (FCD) Date

2018-11-19

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