Geometric Flows on Planar Lattices (Pathways in Mathematics)

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 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
	References
A -Limits in General Lattices
B A Non-trivial Example with Trivial Minimizing Movements
Index