Local Minimization, Variational Evolution and Γ-Convergence

Andrea Braides

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English

Springer International Publishing AG
12 November 2013

Calculus & mathematical analysis; Functional analysis & transforms; Differential calculus & equations; Calculus of variations

Series: Lecture Notes in Mathematics

This book addresses new questions related to the asymptotic description of converging energies from the standpoint of local minimization and variational evolution. It explores the links between Gamma-limits, quasistatic evolution, gradient flows and stable points, raising new questions and proposing new techniques. These include the definition of effective energies that maintain the pattern of local minima, the introduction of notions of convergence of energies compatible with stable points, the computation of homogenized motions at critical time-scales through the definition of minimizing movement along a sequence of energies, the use of scaled energies to study long-term behavior or backward motion for variational evolutions. The notions explored in the book are linked to existing findings for gradient flows, energetic solutions and local minimizers, for which some generalizations are also proposed.

By: Andrea Braides
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 2014 ed.
Volume: 2094
Dimensions: Height: 235mm, Width: 155mm, Spine: 10mm
Weight: 2.934kg
ISBN: 9783319019819
ISBN 10: 3319019813
Series: Lecture Notes in Mathematics
Pages: 174
Publication Date: 12 November 2013
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Introduction.- Global minimization.- Parameterized motion driven by global minimization.- Local minimization as a selection criterion.- Convergence of local minimizers.- Small-scale stability.- Minimizing movements.- Minimizing movements along a sequence of functionals.- Geometric minimizing movements.- Different time scales.- Stability theorems.- Index.

Reviews for Local Minimization, Variational Evolution and Γ-Convergence

From the book reviews: The volume is carefully written and the material is organized in such a way that a Ph.D. student can gradually become familiar with GAMMA-convergence analysis and related tools. When possible, one-dimensional examples are chosen to illustrate the topics and several figures help the reader follow the presentation. The volume is very suitable for a Ph.D. course devoted to an audience with a good background in functional analysis, function spaces, and variational problems. (Giuseppe Buttazzo, Mathematical Reviews, August, 2014)