题目：On Sobolev maps between manifolds and branched transportation
报告人：Fabcrice Béthuel(巴黎索邦大学)
时间：10月27日（星期五），15:00-16:00
地点：明德楼B201-1报告厅
摘要：In the talk, I wish to stress the link between branched transportationtheory, and some issues in the study of Sobolev maps between manifold. Inparticular, 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.
报告人简介：
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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