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Birkhäuser

Sobolev Maps to the Circle

From the Perspective of Analysis, Geometry, and Topology

  • Book
  • © 2021

Overview

Authors:
  1. Haim Brezis

    1. Visiting Distinguished Professor, Department of Mathematics, Rutgers University, Piscataway, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Petru Mironescu

    1. Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, France

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • First monograph to offer a unified and comprehensive treatment of Sobolev maps to the circle
  • Explores surprising connections with other areas of mathematics, such as optimal transport, geometric measure theory, and image processing
  • Open problems are presented throughout to suggest and encourage new research directions

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications (PNLDE, volume 96)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xxxi
    Download chapter PDF

  2. Lifting in \(W^{1,p}\)

    • Haïm Brezis, Petru Mironescu
    Pages 1-63

  3. The geometry of Ju and \(\varSigma (u)\) in 2D; point singularities and minimal connections

    • Haïm Brezis, Petru Mironescu
    Pages 65-132

  4. The geometry of Ju and \(\varSigma (u)\) in 3D (and higher); line singularities and minimal surfaces

    • Haïm Brezis, Petru Mironescu
    Pages 133-171

  5. A digression: sphere-valued maps

    • Haïm Brezis, Petru Mironescu
    Pages 173-204

  6. Lifting in fractional Sobolev spaces and in \(\textit{VMO}\, \)

    • Haïm Brezis, Petru Mironescu
    Pages 205-226

  7. Uniqueness of lifting and beyond

    • Haïm Brezis, Petru Mironescu
    Pages 227-252

  8. Factorization

    • Haïm Brezis, Petru Mironescu
    Pages 253-278

  9. Applications of the factorization

    • Haïm Brezis, Petru Mironescu
    Pages 279-298

  10. Estimates of phases: positive and negative results

    • Haïm Brezis, Petru Mironescu
    Pages 299-309

  11. Density

    • Haïm Brezis, Petru Mironescu
    Pages 311-329

  12. Traces

    • Haïm Brezis, Petru Mironescu
    Pages 331-337

  13. Degree

    • Haïm Brezis, Petru Mironescu
    Pages 339-380

  14. Dirichlet problems. Gaps. Infinite energies

    • Haïm Brezis, Petru Mironescu
    Pages 381-401

  15. Domains with topology

    • Haïm Brezis, Petru Mironescu
    Pages 403-430

  16. Appendices

    • Haïm Brezis, Petru Mironescu
    Pages 431-506

  17. Back Matter

    Pages 507-530
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Keywords

About this book

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.


Reviews

“A nice and unique feature of the book that every chapter contains a section on ‘Complements and open problems’. All in all, I find this text very well written and it would definitely be recommended to graduate students and researchers who would like to learn about the fine properties of Sobolev functions valued in the circle.” (Alpár R. Mészáros, zbMATH 1501.46001, 2023)

“This monograph offers a rigorous discussion to a fascinating topic of mathematics with multiple relevant applications to various fields. Also for this reason it is highly recommended both for mathematicians and physicists working in the various fields involved in the theory of Sobolev maps to the circle … . the book may also be useful for Ph.D students who are interested in the topic, and the content of the individual chapters could be used in advanced courses and seminars.” (Antonia Chinnì, Mathematical Reviews, February, 2023)

Authors and Affiliations

  • Visiting Distinguished Professor, Department of Mathematics, Rutgers University, Piscataway, USA

    Haim Brezis

  • Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, France

    Petru Mironescu

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