{"product_id":"9781071615102","title":"Progress in Nonlinear Differential Equations and Their Applications: From the Perspective of Analysis, Geometry, and Topology","description":"\u003ch1\u003eProgress in Nonlinear Differential Equations and Their Applications: From the Perspective of Analysis, Geometry, and Topology\u003c\/h1\u003e \u003ch2\u003eBrezis, Haim; Mironescu, Petru\u003c\/h2\u003e \u003cp\u003e\u003c\/p\u003e\u003cdiv\u003eThe 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. \u003c\/div\u003e\u003cbr\u003e\u003cdiv\u003eEach 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\u003c\/div\u003e\u003cdiv\u003edirections 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.\u003c\/div\u003e\u003cdiv\u003e\u003cbr\u003e\u003c\/div\u003e\u003cdiv\u003e\n\u003ci\u003eSobolev Maps to the Circle\u003c\/i\u003e 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.\u003c\/div\u003e\u003cdiv\u003e\u003cbr\u003e\u003c\/div\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Birkhäuser\u003c\/p\u003e \u003cp\u003ePublication Date: 2021-12-02\u003c\/p\u003e \u003cp\u003eFormat: Hardcover\u003c\/p\u003e \u003cp\u003eISBN-13: 9781071615102\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-1-0716-1512-6\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 530\u003c\/p\u003e ","brand":"Springer US","offers":[{"title":"Default Title","offer_id":45378504196236,"sku":"9781071615102","price":152.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9781071615102.jpg?v=1775703897","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9781071615102","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}