Gamma$-Convergence of Free-discontinuity Problems

• Main Authors: Cagnetti, Filippo, Maso, Gianni Dal, Scardia, Lucia, Zeppieri, Caterina Ida
• Format: Journal Article
• Language: English
• Published: 19-12-2017
• Subjects: Mathematics - Analysis of PDEs
• DOI: 10.48550/arxiv.1712.07093

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.