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

Sobolev mappings: from liquid crystals to irrigation via degree theory