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 $M$ and $N$ in the Sobolev space $W^{1, p} (M, N)$, 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 $\pi_{[p]}(N)$ of $N$ is not trivial, $[p]$ denoting the largest integer less then $p$, then smooth maps are not sequentially weakly dense in $W^{1, p} (\mM, \mN)$. On the other hand, in the case $p< m$ is an integer, examples of specific manifolds $\M$ and $N$ have been provided wheresmooth maps are sequentially weakly dense in $W^{1, p} (M, N)$ with $\pi_{p}(N)\not = 0$, although they are not dense for the \emph{strong convergence}. This is the case for instance for $M=\B^m$. Such a property does not holds for arbitrary manifolds $\mN$ and integers $p$. The counterexample deals with the case $p=3$, $m\geq 4$ and $N=S^2$, for which $\pi_3(S^2)=\Z$ is related to the Hopf fibration. We provide an explmicit map which is not weakly approximable in $W^{1,3}(M, S^2)$ 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.
报告人： Fabrice Bethuel为法国索邦大学LJLL实验室教授，主要从事领域为偏微分方程与变分理论。1989年在巴黎十一大获得博士学位，博士导师为Jean-Michel Coron。
他在调和映照、Ginzburg-Landau方程等方向作出了基础性的贡献。所获荣誉包括费马奖、Henri-Poincaré奖、法兰西学院Cours Peccot奖等，并于1998年受邀做ICM 45分钟报告。
