On Sobolev maps between manifolds and branched transportation

Release time：2023-10-17Views：420

Title：On Sobolev maps between manifolds and branched transportation

Speaker：Fabcrice Béthuel(Sorbonne Université)

Time：Friday, October 27, 15:00-16:00

Location：Mingde Building, B201-1

Abstract：In the talk, I wish to stress the link between branched transportation theory, and some issues in the study of Sobolev maps between manifold. In particular, I will present a counterexample to the sequential weak density of smooth maps between two manifolds ? and ? in the Sobolev space ?1,? (?, ?), in the case ? is an integer. It has been shown quite a while ago that, if ? < ? = ???(?) is not an integer and the [?]-th homotopy group ?[?] (?) of ? is not trivial, [?] denoting the largest integer less than ?, then smooth maps are not sequentially weakly dense in ?1,? (?, ?). On the other hand, in the case ? < ? is an integer, examples of specific manifolds ? and ? have been provided where smooth maps are sequentially weakly dense in ?1,? (?, ?)with ?[?] (?) ≠ 0, although they are not dense for the strong convergence. This is the case for instance for ? = ? ?. Such a property does not hold for arbitrary manifolds ? and integers ?.

The counterexample deals with the case ?=3, ? ≥ 4 and ? = ? 2 , for which ?3 (? 2 ) = ? is related to the Hopf fibration. We provide an explicit map which is not weakly approximable in ?1,3 (?, ? 2 ), by smooth. One of the

central ingredients in our argument is related to issues in branched transportation and irrigation theory in the critical exponent case.

About the speaker：

Fabrice Béthuel现任法国索邦大学教授，数学硕士研究生负责人。Fabrice Béthuel是世界著名数值分析和偏微分方程专家，法国大学研究院院士，世界数学家大会特邀报告人，获得过众多国际大奖，比如法国科学院Mergier-Bourdeix Prize， FERMAT Prize和IBM Prize，担任或曾担任国际重要数学杂志编委，比如欧洲数学会杂志(Journal of European Mathematical Societty)和泛函分析杂志(Journal of Functional Analysis).

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