On Sobolev maps between manifolds and branched transportation


题目:On Sobolev maps between manifolds and branched transportation

报告人:Fabcrice Béthuel(巴黎索邦大学)



摘要: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 𝑀 and 𝑁 in the Sobolev space 𝑊1,𝑝 (𝑀, 𝑁), in the case 𝑝 is an integer. It has been shown quite a while ago that, if 𝑝 < 𝑚 = 𝑑𝑖𝑚(𝑀) is not an integer and the [𝑝]-th homotopy group 𝜋[𝑝] (𝑁) of 𝑁 is not trivial, [𝑝] denoting the largest integer less than 𝑝, then smooth maps are not sequentially weakly dense in 𝑊1,𝑝 (𝑀, 𝑁). On the other hand, in the case 𝑝 < 𝑚 is an integer, examples of specific manifolds 𝑀 and 𝑁 have been provided where smooth maps are sequentially weakly dense in 𝑊1,𝑝 (𝑀, 𝑁)with 𝜋[𝑝] (𝑁) ≠ 0, although they are not dense for the strong convergence. This is the case for instance for 𝑀 = 𝐵 𝑚. Such a property does not hold for arbitrary manifolds 𝑁 and integers 𝑝.
      The counterexample deals with the case 𝑝=3, 𝑚 ≥ 4 and 𝑁 = 𝑆 2 , for which 𝜋3 (𝑆 2 ) = 𝑍 is related to the Hopf fibration. We provide an explicit map which is not weakly approximable in 𝑊1,3 (𝑀, 𝑆 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.


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 PrizeIBM 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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