Gamma-Convergence for Beginners

Abstract

This book introduces the main concepts of the theory of De Giorgi's Gamma-convergence and gives a description of its main applications to the study of asymptotic variational problems. The content is based on results obtained during thirty years of research. The book is divided into sixteen short chapters, an Introduction, and an Appendix. After explaining how a notion of variational convergence arises naturally from the study of the asymptotic behaviour of variational problems, the Introduction presents a number of examples that show how diversified the applications of this notion may be. The first chapter covers the abstract theory of Gamma-convergence, including its links with lower semicontinuity and relaxation, and the fundamental results on the convergence of minimum problems. The following ten chapters are all set in a one-dimensional framework to illustrate the main issues of convergence without the burden of high-dimensional technicalities. These include variational problems in Sobolev spaces, in particular homogenization theory, limits of discrete systems, segmentation and phase-transition problems, free-discontinuity problems and their approximation, etc. Chapters 12-15 are devoted to problems in a higher-dimensional setting, showing how some one-dimensional reasoning may be extended, if properly formulated, to a more general setting, and how some concepts already introduced can be integrated with vectorial issues. The final chapter introduces the more general and abstract localization methods of Gamma-convergence. All chapters are complemented by a guide to the literature, and a short description of extensions and developments.