  Sobolev spaces are a natural framework for the analysis of problems in
partial differential equations and calculus of variations. Some physical and
geometric contexts, such as liquid crystals models and harmonic maps, lead to
consider Sobolev maps, that is, Sobolev vector functions whose range is
constrained in a surface or submanifold of the space. This additional nonlinear
constraint provokes the appearance of finite-energy topological singularities.
These singularities are characterized by a nontrivial topological invariant
such as the topological degree, they represent an obstruction to the strong
approximation by smooth maps and they become source and sink terms in an
optimal transportation or irrigation problem of topological charges arising in
the study of the weak approximation and of the relaxed energy.

Analysis of PDEs