| 书目名称 | Geometric Flows on Planar Lattices | | 编辑 | Andrea Braides,Margherita Solci | | 视频video | | | 概述 | Introduces important concepts in modern Applied Analysis through prototypical problems, discussed in a natural way.Advanced research topics are introduced in a stimulating and constructive fashion.Cop | | 丛书名称 | Pathways in Mathematics | | 图书封面 |  | | 描述 | 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.. . . | | 出版日期 | Book 2021 | | 关键词 | Variational evolution; Gradient flows; Homogenization; Heterogeneous media; Geometric motions; Motion by | | 版次 | 1 | | doi | https://doi.org/10.1007/978-3-030-69917-8 | | isbn_softcover | 978-3-030-69919-2 | | isbn_ebook | 978-3-030-69917-8Series ISSN 2367-3451 Series E-ISSN 2367-346X | | issn_series | 2367-3451 | | copyright | The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerl |
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Front Matter |
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Abstract
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| 2 |
,Introduction: Motion on Lattices, |
Andrea Braides,Margherita Solci |
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Abstract
The scope of these notes is to present a model case –but already complex enough– of motion in heterogeneous media. Even though this analysis will be performed in the relatively simplified setting of a periodic lattice, where the heterogeneous structure is somewhat built-in in the environment itself, this must be thought of as a case study for a large class of inhomogeneous media. In such a lattice setting we consider the simplest order parameter –obtained by labelling the nodes of the lattice with zeros or ones–, and define an energy that favours constant values of such an order parameter and penalizes the creation of ‘discrete interfaces’.
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| 3 |
,Variational Evolution, |
Andrea Braides,Margherita Solci |
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Abstract
The first subject of our course will be the definition of a variational evolution that retains a gradient-flow type character, in that it is driven by an energy-minimizing .. While keeping in mind the application to a lattice environment, we consider the issue of the definition of a variational evolution in a general setting, not necessarily for a discrete energy. We first consider evolution for a single fixed energy; in the following we will treat the homogenization process characteristic of heterogeneous environments, which will make it necessary to take into account .-depending energies (where . is a typical length scale, in the case of a lattice its spacing) and will make it possible to overcome some (shallow) local minima in the evolution.
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,Discrete-to-Continuum Limits of Planar Lattice Energies, |
Andrea Braides,Margherita Solci |
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Abstract
Before dealing with their evolution, in this chapter we examine the ‘static’ limit of families of energies on lattices with vanishing spacing. This preliminary analysis is suggested by Theorem ., which implies that a proper environment for the study of minimizing movements along sequences of functionals .. may be provided by the computation of their Γ-limit ., which we may always suppose exists up to subsequences. With this scope in mind, this chapter is devoted to the study of the Γ-limit of energies defined on lattices at a space scaling which gives a surface energy in the limit. The corresponding minimizing movement for . will provide a reference geometric motion (motion by crystalline curvature), which will be defined and analyzed in the next chapter.
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,Evolution of Planar Lattices, |
Andrea Braides,Margherita Solci |
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Abstract
In this chapter we describe some evolutions in the plane using the method of minimizing movements along families of ferromagnetic energies as those studied in the previous chapter. As shown in Theorem . (and also Theorem .) the discrete-to-continuum Γ-limits of such energies are crystalline perimeters. In Sect. 4.1 we first describe motion by square crystalline curvature, which is obtained as a minimizing movement for the square perimeter (.) using a dissipation . introduced by Almgren and Taylor. This evolution justifies the definition of discrete dissipations .. introduced in Sect. . and provides a natural environment for a general evolution of lattices whose energies are asymptotically described by a square perimeter in the extreme regime described in Theorem .(a). In the final and most important section of this chapter we will examine a number of energies which have the square perimeter as a Γ-limit but whose minimizing movements are influenced in different ways by the local arrangements of interactions, showing how local minimization may influence the evolution through homogenization and pinning effects.
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,Perspectives: Evolutions with Microstructure, |
Andrea Braides,Margherita Solci |
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Abstract
In this chapter we proceed with some examples towards unexplored regions, where the description of the motion as a geometric evolution of sets is partial, and we have to take into account the evolution of some type of microstructure.
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Back Matter |
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Abstract
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发表于 2025-3-21 17:05:34
