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.
, which is assumed by Nash's smooth embedding theorem without loss of generality to be isometrically embedded into
) Sobolev mappings can also be defined in the context of metric spaces.
[6][7] The strong approximation problem consists in determining whether smooth mappings from
with respect to the norm topology.
, Morrey's inequality implies that Sobolev mappings are continuous and can thus be strongly approximated by smooth maps.
, Sobolev mappings have vanishing mean oscillation[8] and can thus be approximated by smooth maps.
, the question of density is related to obstruction theory:
is the restriction of a continuous map from
[10][5] The problem of finding a sequence of weak approximation of maps in
is equivalent to the strong approximation when
is an integer, a necessary condition is that the restriction to a
-dimensional triangulation of every continuous mapping from a
coincides with the restriction a continuous map from
, this condition is sufficient.
, this condition is not sufficient.
[12] The homotopy problem consists in describing and classifying the path-connected components of the space
endowed with the norm topology.
are essentially the same as the path-connected components of
are connected by a continuous path in
1.1 The classical trace theory states that any Sobolev map
The proof of the surjectivity being based on an averaging argument, the result does not readily extend to Sobolev mappings.
[17] The surjectivity of the trace operator fails if
, the lifting problem asks whether any map
, as it is the case for continuous or smooth
is simply-connected in the classical lifting theory.
is simply connected, any map
[23] There is a topological obstruction to the lifting when
and an analytical obstruction when