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

humectant

取之不竭

lobster

衰老

Cerebrovascular

Estimates of phases: positive and negative results,While the previous chapters were concerned with the . and . of a lifting, this chapter investigates the matter of . for the phase . of .. We exhibit two unusual aspects: “existence without estimates,” and “estimates without existence.”

小丑

vasculitis

Traces,We present a complete trace theory for .-valued maps. When . is not an integer and ., standard trace theory for . maps asserts that .where we identify . with .. When . is replaced by a manifold ., in general we only have the inclusion .and equality may fail.

休息

Dirichlet problems. Gaps. Infinite energies,We investigate minimization problems of the form .where . and . is a given boundary condition (satisfying also . when .).

使高興

Domains with topology,We investigate the impact of the topology of . on the main topics presented above (existence of lifting, relaxed energy, density of smooth maps, etc.).

和平主義者

Appendices,We recall standard properties of Sobolev spaces used throughout the book, and we gather the most technical parts of some of the proofs presented in the main text.