Petru Mironescu: Sobolev Maps to the Circle

Cover: 9781071615102 | Sobolev Maps to the Circle | Petru Mironescu (u. a.) | Buch | xxxi

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Beschreibung

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.

Zusammenfassung

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

Inhaltsverzeichnis

Lifting in $W^{1,p}$.- The Geometry of $J(u)$ and $\Sigma(u)$ in 2D; Point Singularities and Minimal Connections.- The Geometry of $J(u)$ and $\Sigma(u)$ in 3D (and higher); Line Singularities and Minimal Surfaces.- A Digression: Sphere-Valued Maps.- Lifting in Fractional Sobolev Spaces and in $VMO$.- Uniqueness of Lifting and Beyond.- Factorization.- Applications of the Factorization.- Estimates of Phases: Positive and Negative Results.- Density.- Traces.- Degree.- Dirichlet Problems, Gaps, Infinite Energies.- Domains with Topology.- Appendices.

Details

Erscheinungsjahr:	2021
Fachbereich:	Analysis
Genre:	Importe , Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Inhalt:	xxxi 530 S. 17 s/w Illustr. 1 farbige Illustr. 530 p. 18 illus. 1 illus. in color.
ISBN-13:	9781071615102
ISBN-10:	1071615106
Sprache:	Englisch
Einband:	Gebunden
Autor:	Mironescu, Petru Brezis, Haim
Auflage:	1st edition 2021
Hersteller:	Springer US Springer New York
Verantwortliche Person für die EU:	Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, D-14197 Berlin, juergen.hartmann@springer.com
Maße:	241 x 160 x 36 mm
Von/Mit:	Petru Mironescu (u. a.)
Erscheinungsdatum:	02.12.2021
Gewicht:	1,004 kg