Sobolev mapping
In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations, including the theory of harmonic maps. Definition 

Given Riemannian manifolds  and , which is assumed by Nash's smooth embedding theorem without loss of generality to be isometrically embedded into  as 

First-order () Sobolev mappings can also be defined in the context of metric spaces. Approximation 

The strong approximation problem consists in determining whether smooth mappings from  to  are dense in  with respect to the norm topology. When , Morrey's inequality implies that Sobolev mappings are continuous and can thus be strongly approximated by smooth maps. When , Sobolev mappings have vanishing mean oscillation and can thus be approximated by smooth maps. When , the question of density is related to obstruction theory:
 is dense in  if and only if every continuous mapping on a from a –dimensional triangulation of  into  is the restriction of a continuous map from  to . The problem of finding a sequence of weak approximation of maps in  is equivalent to the strong approximation when  is not an integer. When  is an integer, a necessary condition is that the restriction to a -dimensional triangulation of every continuous mapping from a –dimensional triangulation of  into  coincides with the restriction a continuous map from  to . When , this condition is sufficient
For  with , this condition is not sufficient. Homotopy 

The homotopy problem consists in describing and classifying the path-connected components of the space  endowed with the norm topology. When  and , then the path-connected components of  are essentially the same as the path-connected components of : two maps in  are connected by a path in  if and only if they are connected by a path in , any path-connected component of  and any path-connected component of  intersects  non trivially. When , two maps in  are connected by a continuous path in  if and only if their restrictions to a generic -dimensional triangulation are homotopic. Extension of traces

The classical trace theory states that any Sobolev map  has a trace  and that when , the trace operator is onto. The proof of the surjectivity being based on an averaging argument, the result does not readily extend to Sobolev mappings. The trace operator is known to be onto when  or when ,  is finite and . The surjectivity of the trace operator fails if   or if  is infinite for some . Lifting

Given a covering map , the lifting problem asks whether any map  can be written as  for some , as it is the case for continuous or smooth  and  when  is simply-connected in the classical lifting theory. If the domain  is simply connected, any map  can be written as  for some 
when , when  and 
and when  is compact,  and . There is a topological obstruction to the lifting when  and an analytical obstruction when . References

Further reading 
https://mathoverflow.net/questions/108808/differential-of-a-sobolev-map-between-manifolds

Manifolds
Maps of manifolds
Sobolev spaces
Homotopy theory