Geometric Flows on Planar Lattices (Pathways in Mathematics)

✍ Scribed by Andrea Braides, Margherita Solci

• Publisher: Birkhäuser
• Year: 2021
• Tongue: English
• Leaves: 138
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
Contents
1 Introduction: Motion on Lattices
References
2 Variational Evolution
2.1 Discrete Orbits
2.1.1 Discrete Orbits at a Given Time Scale τ
2.1.2 Passage to the Limit as τ→0 in Discrete Orbits
2.2 The Minimizing-Movement Approach
2.2.1 Discrete-to-Continuum Limit for Lattice Energies
2.2.2 Minimizing Movements Along a Sequence
2.3 Some Notes on Minimizing Movements on Metric Spaces
2.3.1 An Existence Result
2.3.2 Minimizing Movements and Curves of Maximal Slope
2.3.3 The Colombo-Gobbino Condition
References
3 Discrete-to-Continuum Limits of Planar Lattice Energies
3.1 Energies on Sets of Finite Perimeter
3.2 Limits of Homogeneous Energies in a Square Lattice
3.2.1 The Prototype: Homogeneous Nearest Neighbours
3.2.2 Next-to-Nearest Neighbour Interactions
3.2.3 Directional Nearest-Neighbour Interactions
3.2.4 General Form of the Limits of Homogeneous Ferromagnetic Energies
3.3 Limits of Inhomogeneous Energies in a Square Lattice
3.3.1 Layered Interactions
3.3.2 Alternating Nearest Neighbours (Hard Inclusions') 3.3.3 Homogenization and Design of Networks 3.4 Limits in General Planar Lattices by Reduction to the Square Lattice References 4 Evolution of Planar Lattices 4.1 Flat Flows 4.1.1 Flat Flow for the Square Perimeter 4.1.2 Motion of a Rectangle 4.1.3 Motion of a General Set 4.1.4 An Example with Varying Initial Data 4.1.5 Flat Flow for anOctagonal' Perimeter
4.2 Discrete-to-Continuum Geometric Evolutionon the Square Lattice
4.2.1 A Model Case: Nearest-Neighbour Homogeneous Energies
4.2.2 Next-to-Nearest-Neighbour Homogeneous Energies
4.2.3 Evolutions Avoiding Hard Inclusions
4.2.4 Asymmetric Motion
4.2.5 Homogenized Motion
4.2.6 Motions with an Oscillating Forcing Term
4.3 Conclusions
References
5 Perspectives: Evolutions with Microstructure
5.1 High-Contrast Ferromagnetic Media: Mushy Layers
5.2 Some Evolutions for Antiferromagnetic Systems
5.2.1 Nearest-Neighbour Antiferromagnetic Interactions: Nucleation
5.2.2 Next-to-Nearest Neighbour Antiferromagnetic Interactions: The Effect of Corner Defects
5.3 More Conclusions
References
A -Limits in General Lattices
B A Non-trivial Example with Trivial Minimizing Movements
Index