Orateur
Description
Manifold-valued Sobolev maps naturally arise in variational problems and models of partially ordered media, where the topology of the target can enforce the formation of singularities. These singularities may act as obstructions to familiar constructions, such as approximation by smooth maps. In this talk, given a positive integer $p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $N$. The target manifold is required to satisfy suitable topological conditions; however, in contrast with previous works in this area, we do not assume that $N$ is $(p-1)$-connected. Using tools from geometric measure theory --- namely, flat chains with coefficients in an appropriate homotopy group of $N$ --- we associate to each map $u$ in the weak sequential closure of smooth maps an object that captures its point singularities. The vanishing of this object characterizes local strong approximability by smooth maps. This talk is based on joint work with G. Orlandi (Verona).