An Introduction to Gamma-Convergence

by Gianni Dal Maso

Overview

The last twentyfive years have seen an increasing interest for variational convergences and for their applications to different fields, like homogenization theory, phase transitions, singular perturbations, boundary value problems in wildly perturbed domains, approximation of variatonal problems, and non- smooth analysis. Among variational convergences, De Giorgi's r-convergence plays a cen- tral role for its compactness properties and for the large number of results concerning r -limits of integral functionals. Moreover, almost all other varia- tional convergences can be easily expressed in the language of r -convergence. This text originates from the notes of the courses on r -convergence held by the author in Trieste at the International School for Advanced Studies (S. I. S. S. A. ) during the academic years 1983-84,1986-87, 1990-91, and in Rome at the Istituto Nazionale di Alta Matematica (I. N. D. A. M. ) during the spring of 1987. This text is far from being a treatise on r -convergence and its appli- cations.

Product Details

ISBN-13:	9780817636791
Publisher:	Birkhäuser Boston
Publication date:	01/01/1993
Series:	Progress in Nonlinear Differential Equations and Their Applications , #8
Edition description:	1993
Pages:	341
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. The direct method in the calculus of variations.- 2. Minimum problems for integral functionals.- 3. Relaxation.- 4.—-convergence and K-convergence.- 5. Comparison with pointwise convergence.- 6. Some properties of—-limits.- 7. Convergence of minima and of minimizers.- 8. Sequential characterization of—-limits.- 9.—-convergence in metric spaces.- 10. The topology of—-convergence.- 11.—-convergence in topological vector spaces.- 12. Quadratic forms and linear operators.- 13. Convergence of resolvents and G-convergence.- 14. Increasing set functions.- 15. Lower semicontinuous increasing functionals.- 16. $$ \bar{\Gamma } $$-convergence of increasing set functional.- 17. The topology of $$ \bar{\Gamma } $$-convergence.- 18. The fundamental estimate.- 19. Local functionals and the fundamental estimate.- 20. Integral representation of—-limits.- 21. Boundary conditions.- 22. G-convergence of elliptic operators.- 23. Translation invariant functional.- 24. Homogenization.- 25. Some examples in homogenization.- Guide to the literature.

Editorial Reviews

Provides a self-contained systematic presentation of a notion of "variational convergence," called [gamma] convergence, that has been developed in the last 25 years in connection with the study of homogenization problems in the mathematical theory of composite materials, and has been applied to the study of phase transitions, singular perturbations, boundary value problems in widely perturbed domains, and other variational problems characterized by severe perturbations of the data. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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