Geometric Flows on Planar Lattices / by Andrea Braides, Margherita Solci.

Format:

Book

Author/Creator:

Braides, Andrea, author.

Contributor:

Solci, Margherita, editor.

Series:

Pathways in Mathematics, 2367-346X

Language:

English

Subjects (All):

Geometry, Differential.

Mathematical optimization.

Calculus of variations.

Mathematical analysis.

Differential Geometry.

Calculus of Variations and Optimization.

Analysis.

Local Subjects:

Differential Geometry.

Calculus of Variations and Optimization.

Analysis.

Physical Description:

1 online resource (138 pages) : illustrations

Edition:

1st ed. 2021.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2021.

System Details:

Mode of access: World Wide Web.

Summary:

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.

Contents:

Intro

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 τ0 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

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

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 an `Octagonal' 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

A -Limits in General Lattices

B A Non-trivial Example with Trivial Minimizing Movements

Index.

Notes:

Includes bibliographical references and index.

ISBN:

3-030-69917-X

OCLC:

1247676802