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.

Provides connections between topics of active current research Presents the subjects with examples from the main areas that have made Gamma-convergence so successful Proposes numerous examples of directions of further research Includes supplementary material: sn.pub/extras

Autorentext

Roberto Alicandro is professor of Mathematical Analysis at Università di Cassino e del Lazio meridionale. He is an expert in the Calculus of Variations and Homogenization and his results have applications in different fields, including atomistic-to-continuum limits for nonlinear models in material science, phase transition problems, topological singularities and defects in materials. He is the author of a monograph on Discrete Variational Problems with Andrea Braides and other co-authors. Nadia Ansini is professor of Mathematical Analysis atthe Department of Mathematics, Sapienza University of Rome. She is an expert in the Calculus of Variations, Homogenization and Multiple-scale models in mathematical materials science with subjects ranging from perforated domains, thin films, phase transitions, and variational evolution problems. She was awarded with two Marie Sklodowska-Curie Fellowships in 2000 and 2012. She is Lise Meitner visiting professor at Lund University (Sweden, 2022-2025).Andrea Braides is professor of Mathematical Analysis at SISSA, Trieste, on leave from the University of Rome Tor Vergata. He is an expert in the Calculus of Variations andHomogenization. He is the author of several monographs in the fields of Gamma-convergence and Discrete Variational Problems. He was an invited speaker at the 2014 International Congress of Mathematicians in Seoul in the section Mathematics in Science and Technology. Andrey Piatnitsky is an expert in the Calculus of Variations and in Partial Differential Equations, specializing in the homogenization of both deterministic and stochastic energies and operators, and singularly perturbed operators. He has been the invited speaker to major international conferences on these subjects. He and his co-authors produced a monograph on Homogenization.Antonio Tribuzio is a research fellow at the Institute for AppliedMathematics, Heidelberg University. His field of expertise is the Calculus of Variations. He worked, among others, on the relation between De Giorgi's Minimizing Movements and Gamma-convergence, discrete evolutions and scaling behaviour of energies related to Shape-Memory Alloys.

Inhalt

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.