Titlebook: Sobolev Maps to the Circle; From the Perspective Haim Brezis,Petru Mironescu Book 2021 Springer Science+Business Media, LLC, part of Spring

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Lifting in fractional Sobolev spaces and in ,s smooth as . permits.” The difference with Chapter . is that . belongs either to a fractional Sobolev space . (whose definition is recalled in Section .), or to the space . (functions of vanishing mean oscillation, whose definition is recalled in Section . below).

变异

Uniqueness of lifting and beyond,we have two liftings, ., .. Then, . for some .. Therefore, we are led to the question of finding minimal assumptions on a measurable function . implying that . must be constant. Clearly, continuity is sufficient, but, as we are going to see, constancy holds for a surprisingly large class of function

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Degree,w many times . covers ., taking into account algebraic multiplicity.” One can still give a robust definition for . when . belongs merely to ., and thus, by the Sobolev embeddings, for maps in the critical spaces ., with .. We establish some basic properties of this degree.

机密

The geometry of , and , in 2D; point singularities and minimal connections, generally, if . is a “nice” map, i.e., continuous on . except at a finite number of distinct points ., we have .Next, we turn to the geometric interpretation of . as a “minimal length” required to “connect the singularities.”

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1421-1750 . 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 comprehens978-1-0716-1512-6Series ISSN 1421-1750 Series E-ISSN 2374-0280

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Book 2021 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 comprehens