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

• Autore: Braides Andrea
• Pubblicazione: Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2021
• Edizione: 1st ed. 2021.
• Descrizione fisica: 1 online resource (138 pages) : illustrations
• Disciplina: 516.36
• Soggetto topico: Geometry, Differential; Mathematical optimization; Calculus of variations; Mathematical analysis; Differential Geometry; Calculus of Variations and Optimization; Analysis
• Persona (resp. second.): SolciMargherita
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 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 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.
• Sommario/riassunto: 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.
• Titolo autorizzato: Geometric flows on planar lattices
• ISBN: 3-030-69917-X
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910482954103321
• Lo trovi qui: Univ. Federico II
• Serie: Pathways in Mathematics, . 2367-346X