Calculus of Variations and Geometric Measure Theory
home | mail | papers | authors | news | seminars | events | open positions | login

F. Cagnetti - G. Dal Maso - L. Scardia - C. I. Zeppieri

$\Gamma$-convergence of free-discontinuity problems

created by zeppieri1 on 20 Mar 2017


Submitted Paper

Inserted: 20 mar 2017
Last Updated: 20 mar 2017

Year: 2017


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.

Keywords: $\Gamma$-convergence, Free-discontinuity problems, homogenisation


Credits | Cookie policy | HTML 4.0.1 strict | CSS 2.1