부산대학교 도서관

This book is dedicated to the analysis of the asymptotic behavior ofdiscrete systems arising with large number of nodes that produce, inthe limit, interface energies. These systems apply in a lot ofsituations arising in science and technology. We cite, for instance,variational models in imaging (first-order models like Mumford-Shah,second-order models like Blake-Zisserman), and models in atomic physics(Lennard-Jones potentials) where the phenomenon of crystallizationappears, that is the ground states tend to arrange on a regularlattice. Further applications can be found in material science, forinstance the theory of brittle fractures, where the fracture has acorresponding microscopic interpretation.\par The general object of the analysis of the authors is the study ofenergy functionals whose domain is functions defined on lattices ineuclidean space and with values in a finite set, both in thedeterministic and in the random case. Precisely, this book is dedicatedmainly to the passage from a microscopic description to the macroscopiccorresponding model, whose behavior is typically driven by a surfaceenergy that appears from a suitable variational convergence procedure.\par In Chapter 2 some abstract preliminaries are introduced, like the$\Gamma$-convergence and a description of surface energies. Chapter 3is dedicated to the homogenization of pairwise systems with positivecoefficients, on both periodic lattices and aperiodic ones. The maingeneral mathematical results are investigated in Chapter 4 where theauthors treat compactness of equibounded configurations and integralrepresentation of the limit energy functionals. Chapter 5 deals withrandom lattices, while in Chapters 6, 7, and 8 other models are presented,like ferromagnetic systems, frustrated systems, dense graphs, and graphons.