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 M {\displaystyle M} and N {\displaystyle N} , which is assumed by Nash's smooth embedding theorem without loss of generality to be isometrically embedded into R ν {\displaystyle \mathbb {R} ^{\nu }} as
W s , p ( M , N ) := { u ∈ W s , p ( M , R ν ) | u ( x ) ∈ N for almost every x ∈ M } . {\displaystyle W^{s,p}(M,N):=\{u\in W^{s,p}(M,\mathbb {R} ^{\nu })\,\vert \,u(x)\in N{\text{ for almost every }}x\in M\}.}
First-order ( s = 1 {\displaystyle s=1} ) Sobolev mappings can also be defined in the context of metric spaces.
Approximation The strong approximation problem consists in determining whether smooth mappings from M {\displaystyle M} to N {\displaystyle N} are dense in W s , p ( M , N ) {\displaystyle W^{s,p}(M,N)} with respect to the norm topology. When s p > dim M {\displaystyle sp>\dim M} , Morrey's inequality implies that Sobolev mappings are continuous and can thus be strongly approximated by smooth maps. When s p = dim M {\displaystyle sp=\dim M} , Sobolev mappings have vanishing mean oscillation and can thus be approximated by smooth maps. When s p < dim M {\displaystyle sp<\dim M} , the question of density is related to obstruction theory:
C ∞ ( M , N ) {\displaystyle C^{\infty }(M,N)} is dense in W 1 , p ( M , N ) {\displaystyle W^{1,p}(M,N)} if and only if every continuous mapping on a from a ⌊ p ⌋ {\displaystyle \lfloor p\rfloor } –dimensional triangulation of M {\displaystyle M} into N {\displaystyle N} is the restriction of a continuous map from M {\displaystyle M} to N {\displaystyle N} . The problem of finding a sequence of weak approximation of maps in W 1 , p ( M , N ) {\displaystyle W^{1,p}(M,N)} is equivalent to the strong approximation when p {\displaystyle p} is not an integer. When p {\displaystyle p} is an integer, a necessary condition is that the restriction to a ⌊ p − 1 ⌋ {\displaystyle \lfloor p-1\rfloor } -dimensional triangulation of every continuous mapping from a ⌊ p ⌋ {\displaystyle \lfloor p\rfloor } –dimensional triangulation of M {\displaystyle M} into N {\displaystyle N} coincides with the restriction a continuous map from M {\displaystyle M} to N {\displaystyle N} . When p = 2 {\displaystyle p=2} , this condition is sufficient. For W 1 , 3 ( M , S 2 ) {\displaystyle W^{1,3}(M,\mathbb {S} ^{2})} with dim M ≥ 4 {\displaystyle \dim M\geq 4} , this condition is not sufficient.
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