LEADER 05088nam 22006495 450 001 9910482954103321 005 20251113190128.0 010 $a3-030-69917-X 024 7 $a10.1007/978-3-030-69917-8 035 $a(CKB)4100000011807162 035 $a(MiAaPQ)EBC6525578 035 $a(Au-PeEL)EBL6525578 035 $a(OCoLC)1247676802 035 $a(PPN)254723179 035 $a(DE-He213)978-3-030-69917-8 035 $a(EXLCZ)994100000011807162 100 $a20210323d2021 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aGeometric Flows on Planar Lattices /$fby Andrea Braides, Margherita Solci 205 $a1st ed. 2021. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2021. 215 $a1 online resource (138 pages) $cillustrations 225 1 $aPathways in Mathematics,$x2367-346X 311 08$a3-030-69916-1 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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 ? -- 2.1.2 Passage to the Limit as ??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. 327 $aReferences -- 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. 330 $aThis book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations. 410 0$aPathways in Mathematics,$x2367-346X 606 $aGeometry, Differential 606 $aMathematical optimization 606 $aCalculus of variations 606 $aMathematical analysis 606 $aDifferential Geometry 606 $aCalculus of Variations and Optimization 606 $aAnalysis 615 0$aGeometry, Differential. 615 0$aMathematical optimization. 615 0$aCalculus of variations. 615 0$aMathematical analysis. 615 14$aDifferential Geometry. 615 24$aCalculus of Variations and Optimization. 615 24$aAnalysis. 676 $a516.36 700 $aBraides$b Andrea$062002 702 $aSolci$b Margherita 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910482954103321 996 $aGeometric flows on planar lattices$91901682 997 $aUNINA