Gamma-Convergence for Beginners

Andrea Braides (, Department of Mathematics, Universita di Roma 'Tor Vergata')

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English

Oxford University Press
01 October 2002

Calculus of variations; Applied mathematics; Physics; Computer vision

Series: Oxford Lecture Series in Mathematics and Its Applications

The theory of Gamma-convergence is commonly recognized as an ideal and flexible tool for the description of the asymptotic behaviour of variational problems. Its applications range from the mathematical analysis of composites to the theory of phase transitions, from Image Processing to Fracture Mechanics. This text, written by an expert in the field, provides a brief and simple introduction to this subject, based on the treatment of a series of fundamental problems that illustrate the main features and techniques of Gamma-convergence and at the same time provide a stimulating starting point for further studies. The main part is set in a one-dimensional framework that highlights the main issues of Gamma-convergence without the burden of higher-dimensional technicalities. The text deals in sequence with increasingly complex problems, first treating integral functionals, then homogenisation, segmentation problems, phase transitions, free-discontinuity problems and their discrete and continuous approximation, making stimulating connections among those problems and with applications. The final part is devoted to an introduction to higher-dimensional problems, where more technical tools are usually needed, but the main techniques of Gamma-convergence illustrated in the previous section may be applied unchanged.

The book and its structure originate from the author's experience in teaching courses on this subject to students at PhD level in all fields of Applied Analysis, and from the interaction with many specialists in Mechanics and Computer Vision, which have helped in making the text addressed also to a non-mathematical audience. The material of the book is almost self-contained, requiring only some basic notion of Measure Theory and Functional Analysis.

By: Andrea Braides ( Department of Mathematics Universita di Roma 'Tor Vergata')
Imprint: Oxford University Press
Country of Publication: United Kingdom
Volume: 22
Dimensions: Height: 237mm, Width: 157mm, Spine: 12mm
Weight: 467g
ISBN: 9780198507840
ISBN 10: 0198507844
Series: Oxford Lecture Series in Mathematics and Its Applications
Pages: 230
Publication Date: 01 October 2002
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Preface Introduction 1: Gamma-convergence by numbers 2: Integral problems 3: Some homogenization problems 4: From discrete systems to integral functionals 5: Segmentation problems 6: Phase-transition problems 7: Free-discontinuity problems 8: Approximation of free-discontinuity problems 9: More homogenization problems 10: Interaction between elliptic problems and partition problems 11: Discrete systems and free-discontinuity problems 12: *Some comments on vectorial problems 13: *Dirichlet problems in perforated domains 14: *Dimension-reduction problems 15: *The 'slicing' method 16: *An introduction to the localization method of Gamma-convergence AppendicesA: Some quick recalls B: Characterization of Gamma-convergence for 1D(italic 'D') integral problems List of symbols References Index

Prof. Andrea Braides Address Via Balilla 22, 00185 Roma, ITALY Tel (+39)0670452392 (home) (+39)0672594688 (office) Fax (+39)0672594699 Email braides@mat.uniroma2.it Italian, Udine (Italy), April 12,1961

Reviews for Gamma-Convergence for Beginners

The presentation is overall quite clear, and the style is often captivating. Many figures, examples and exercises complete the monograph. Finally, it is worth adding a mention on the bibiography, which is at present a truly complete account of papers in this area. Mathematical Reviews