Geometric Flows on Planar Lattices

1. Front Matter

   Pages i-x

   PDF

2. Introduction: Motion on Lattices

   • Andrea Braides, Margherita Solci

   Pages 1-6

3. Variational Evolution

   • Andrea Braides, Margherita Solci

   Pages 7-29

4. Discrete-to-Continuum Limits of Planar Lattice Energies

   • Andrea Braides, Margherita Solci

   Pages 31-51

5. Evolution of Planar Lattices

   • Andrea Braides, Margherita Solci

   Pages 53-101

6. Perspectives: Evolutions with Microstructure

   • Andrea Braides, Margherita Solci

   Pages 103-118

7. Back Matter

   Pages 119-134

   PDF

This book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations.

• Variational evolution
• Gradient flows
• Homogenization
• Heterogeneous media
• Geometric motions
• Motion by crystalline curvature
• Discrete-to-continuum limits for lattice energies
• Evolution of microstructure
• Sets of finite perimeter
• Interfacial energies

“The book fairly self-contained. Many illuminating examples are worked out in detail, and a large number of extensions are included." (John Urbas, zbMATH 1486.53002, 2022)

• Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy

  Andrea Braides

• Dipartimento di Architettura, Università di Sassari, Alghero, Italy

  Margherita Solci

Andrea Braides is professor of Mathematical Analysis at the University of Rome Tor Vergata. He is the author among others of the books Gamma-convergence for Beginners and (with A.Defranceschi) Homogenization of Multiple Integrals. He was an invited speaker at the 2014 International Congress of Mathematicians in Seoul in the section Mathematics in Science and Technology.

Margherita Solci is professor of Mathematical Analysis at the University of Sassari at Alghero. She works on various topics involving variational convergence; in particular, static and dynamic passages from discrete to continuum.

Policies and ethics