Sobolev Maps to the Circle : From the Perspective of Analysis, Geometry, and Topology
1st ed. 2021
Book Details
Format
Hardback or Cased Book
ISBN-10
1071615106
ISBN-13
9781071615102
Edition
1st ed. 2021
Publisher
Springer-Verlag New York Inc.
Imprint
Springer-Verlag New York Inc.
Country of Manufacture
GB
Country of Publication
GB
Publication Date
Dec 2nd, 2021
Print length
530 Pages
Product Classification:
Calculus & mathematical analysisCalculus and mathematical analysisGeometryTopology
Ksh 23,400.00
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The theory of real-valued Sobolev functions is a classical part of analysis and has a wide range of applications in pure and applied mathematics. By contrast, the study of manifold-valued Sobolev maps is relatively new. The incentive to explore these spaces arose in the last forty years from geometry and physics. This monograph is the first to provide a unified, comprehensive treatment of Sobolev maps to the circle, presenting numerous results obtained by the authors and others. Many surprising connections to other areas of mathematics are explored, including the Monge-Kantorovich theory in optimal transport, items in geometric measure theory, Fourier series, and non-local functionals occurring, for example, as denoising filters in image processing. Numerous digressions provide a glimpse of the theory of sphere-valued Sobolev maps.
Each chapter focuses on a single topic and starts with a detailed overview, followed by the most significant results, and rather complete proofs. The "Complements and Open Problems" sections provide short introductions to various subsequent developments or related topics, and suggest new
directions of research. Historical perspectives and a comprehensive list of references close out each chapter. Topics covered include lifting, point and line singularities, minimal connections and minimal surfaces, uniqueness spaces, factorization, density, Dirichlet problems, trace theory, and gap phenomena.
Sobolev Maps to the Circle will appeal to mathematicians working in various areas, such as nonlinear analysis, PDEs, geometric analysis, minimal surfaces, optimal transport, and topology. It will also be of interest to physicists working on liquid crystals and the Ginzburg-Landau theory of superconductors.
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