Sobolev mapping

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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 [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math], which is assumed by Nash's smooth embedding theorem without loss of generality to be isometrically embedded into [math]\displaystyle{ \mathbb{R}^\nu }[/math] as [1][2] [math]\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\}. }[/math] First-order ([math]\displaystyle{ s=1 }[/math]) Sobolev mappings can also be defined in the context of metric spaces.[3][4]

Approximation

The strong approximation problem consists in determining whether smooth mappings from [math]\displaystyle{ M }[/math] to [math]\displaystyle{ N }[/math] are dense in [math]\displaystyle{ W^{s, p} (M, N) }[/math] with respect to the norm topology. When [math]\displaystyle{ sp \gt \dim M }[/math], Morrey's inequality implies that Sobolev mappings are continuous and can thus be strongly approximated by smooth maps. When [math]\displaystyle{ sp = \dim M }[/math], Sobolev mappings have vanishing mean oscillation[5] and can thus be approximated by smooth maps.[6]

When [math]\displaystyle{ sp \gt \dim M }[/math], the question of density is related to obstruction theory: [math]\displaystyle{ C^\infty (M, N) }[/math] is dense in [math]\displaystyle{ W^{1, p} (M, N) }[/math] if and only if every continuous mapping on a from a [math]\displaystyle{ \lfloor p\rfloor }[/math]–dimensional triangulation of [math]\displaystyle{ M }[/math] into [math]\displaystyle{ N }[/math] is the restriction of a continuous map from [math]\displaystyle{ M }[/math] to [math]\displaystyle{ N }[/math].[7][2]

The problem of finding a sequence of weak approximation of maps in [math]\displaystyle{ W^{1, p} (M, N) }[/math] is equivalent to the strong approximation when [math]\displaystyle{ p }[/math] is not an integer.[7] When [math]\displaystyle{ p }[/math] is an integer, a necessary condition is that the restriction to a [math]\displaystyle{ \lfloor p - 1\rfloor }[/math]-dimensional triangulation of every continuous mapping from a [math]\displaystyle{ \lfloor p\rfloor }[/math]–dimensional triangulation of [math]\displaystyle{ M }[/math] into [math]\displaystyle{ N }[/math] coincides with the restriction a continuous map from [math]\displaystyle{ M }[/math] to [math]\displaystyle{ N }[/math].[2] When [math]\displaystyle{ p = 2 }[/math], this condition is sufficient[8] For [math]\displaystyle{ W^{1, 3} (M, \mathbb{S}^2) }[/math] with [math]\displaystyle{ \dim M \ge 4 }[/math], this condition is not sufficient.[9]

Homotopy

The homotopy problem consists in describing and classifying the path-connected components of the space [math]\displaystyle{ W^{s, p}(M, N) }[/math] endowed with the norm topology. When [math]\displaystyle{ 0 \lt s \le 1 }[/math] and [math]\displaystyle{ \dim M \le sp }[/math], then the path-connected components of [math]\displaystyle{ W^{s, p} (M, N) }[/math] are essentially the same as the path-connected components of [math]\displaystyle{ C(M, N) }[/math]: two maps in [math]\displaystyle{ W^{s, p} (M, N) \cap C (M, N) }[/math] are connected by a path in [math]\displaystyle{ W^{s, p} (M, N) }[/math] if and only if they are connected by a path in [math]\displaystyle{ C(M, N) }[/math], any path-connected component of [math]\displaystyle{ W^{s, p} (M, N) }[/math] and any path-connected component of [math]\displaystyle{ C (M, N) }[/math] intersects [math]\displaystyle{ W^{s, p} (M, N) \cap C (M, N) }[/math] non trivially.[10][11][12] When [math]\displaystyle{ \dim M \gt p }[/math], two maps in [math]\displaystyle{ W^{1, p} (M, N) }[/math] are connected by a continuous path in [math]\displaystyle{ W^{1, p} (M, N) }[/math] if and only if their restrictions to a generic [math]\displaystyle{ \lfloor p - 1\rfloor }[/math]-dimensional triangulation are homotopic.[2]:th. 1.1

Extension of traces

The classical trace theory states that any Sobolev map [math]\displaystyle{ u \in W^{1, p} (M, N) }[/math] has a trace [math]\displaystyle{ Tu \in W^{1 - 1/p, p} (\partial M, N) }[/math] and that when [math]\displaystyle{ N = \mathbb{R} }[/math], 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 [math]\displaystyle{ \pi_{1} (N) \simeq \dotsb \pi_{\lfloor p - 1\rfloor}(N) \simeq \{0\} }[/math][13] or when [math]\displaystyle{ p\ge 3 }[/math], [math]\displaystyle{ \pi_{1} (N) }[/math] is finite and [math]\displaystyle{ \pi_{2} (N) \simeq \dotsb \pi_{\lfloor p - 1\rfloor}(N) \simeq \{0\} }[/math].[14] The surjectivity of the trace operator fails if [math]\displaystyle{ \pi_{\lfloor p - 1\rfloor} (N)\not \simeq \{0\} }[/math] [13][15] or if [math]\displaystyle{ \pi_{\ell} (N) }[/math] is infinite for some [math]\displaystyle{ \ell \in \{1, \dotsc, \lfloor p - 1\rfloor\} }[/math].[14][16]

Lifting

Given a covering map [math]\displaystyle{ \pi : \tilde{N} \to N }[/math], the lifting problem asks whether any map [math]\displaystyle{ u \in W^{s, p} (M, N) }[/math] can be written as [math]\displaystyle{ u = \pi \circ \tilde{u} }[/math] for some [math]\displaystyle{ \tilde{u} \in W^{s, p} (M, \tilde{N}) }[/math], as it is the case for continuous or smooth [math]\displaystyle{ u }[/math] and [math]\displaystyle{ \tilde{u} }[/math] when [math]\displaystyle{ M }[/math] is simply-connected in the classical lifting theory. If the domain [math]\displaystyle{ M }[/math] is simply connected, any map [math]\displaystyle{ u \in W^{s, p} (M, N) }[/math] can be written as [math]\displaystyle{ u = \pi \circ \tilde{u} }[/math] for some [math]\displaystyle{ \tilde{u} \in W^{s, p} (M, N) }[/math] when [math]\displaystyle{ sp \ge \dim M }[/math],[17][18] when [math]\displaystyle{ s\ge 1 }[/math] and [math]\displaystyle{ 2 \le sp \lt \dim M }[/math][19][18] and when [math]\displaystyle{ N }[/math] is compact, [math]\displaystyle{ 0 \lt s \lt 1 }[/math] and [math]\displaystyle{ 2 \le sp \lt \dim M }[/math].[20] There is a topological obstruction to the lifting when [math]\displaystyle{ sp \lt 2 }[/math] and an analytical obstruction when [math]\displaystyle{ 1 \le sp \lt \dim M }[/math].[17][18]

References

  1. ↑ Mironescu, Petru (2007). "Sobolev maps on manifolds: degree, approximation, lifting". Contemporary Mathematics 446: 413–436. doi:10.1090/conm/446/08642. ISBN 9780821841907. https://hal.archives-ouvertes.fr/hal-00747679/file/sobolev_spaces_survey_20070120.pdf. 
  2. ↑ 2.0 2.1 2.2 2.3 Hang, Fengbo; Lin, Fanghua (2003). "Topology of sobolev mappings, II". Acta Mathematica 191 (1): 55–107. doi:10.1007/BF02392696. 
  3. ↑ Chiron, David (August 2007). "On the definitions of Sobolev and BV spaces into singular spaces and the trace problem". Communications in Contemporary Mathematics 09 (4): 473–513. doi:10.1142/S0219199707002502. 
  4. ↑ Hajłasz, Piotr (2009). "Sobolev Mappings between Manifolds and Metric Spaces". Sobolev Spaces in Mathematics I. International Mathematical Series 8: 185–222. doi:10.1007/978-0-387-85648-3_7. ISBN 978-0-387-85647-6. 
  5. ↑ Brezis, H.; Nirenberg, L. (September 1995). "Degree theory and BMO; part I: Compact manifolds without boundaries". Selecta Mathematica 1 (2): 197–263. doi:10.1007/BF01671566. 
  6. ↑ Schoen, Richard; Uhlenbeck, Karen (1 January 1982). "A regularity theory for harmonic maps". Journal of Differential Geometry 17 (2). doi:10.4310/jdg/1214436923. 
  7. ↑ 7.0 7.1 Bethuel, Fabrice (1991). "The approximation problem for Sobolev maps between two manifolds". Acta Mathematica 167: 153–206. doi:10.1007/BF02392449. 
  8. ↑ Pakzad, M.R.; Rivière, T. (February 2003). "Weak density of smooth maps for the Dirichlet energy between manifolds". Geometric and Functional Analysis 13 (1): 223–257. doi:10.1007/s000390300006. 
  9. ↑ Bethuel, Fabrice (February 2020). "A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces". Inventiones Mathematicae 219 (2): 507–651. doi:10.1007/s00222-019-00911-3. Bibcode: 2020InMat.219..507B. 
  10. ↑ Brezis, Haı̈m; Li, YanYan (September 2000). "Topology and Sobolev spaces". Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 331 (5): 365–370. doi:10.1016/S0764-4442(00)01656-6. Bibcode: 2000CRASM.331..365B. 
  11. ↑ Brezis, Haim; Li, Yanyan (July 2001). "Topology and Sobolev Spaces". Journal of Functional Analysis 183 (2): 321–369. doi:10.1006/jfan.2000.3736. 
  12. ↑ Bousquet, Pierre (February 2008). "Fractional Sobolev spaces and topology". Nonlinear Analysis: Theory, Methods & Applications 68 (4): 804–827. doi:10.1016/j.na.2006.11.038. 
  13. ↑ 13.0 13.1 Hardt, Robert; Lin, Fang-Hua (September 1987). "Mappings minimizing the Lp norm of the gradient". Communications on Pure and Applied Mathematics 40 (5): 555–588. doi:10.1002/cpa.3160400503. 
  14. ↑ 14.0 14.1 Mironescu, Petru; Van Schaftingen, Jean (9 July 2021). "Trace theory for Sobolev mappings into a manifold". Annales de la Faculté des sciences de Toulouse: Mathématiques 30 (2): 281–299. doi:10.5802/afst.1675. 
  15. ↑ Bethuel, Fabrice; Demengel, Françoise (October 1995). "Extensions for Sobolev mappings between manifolds". Calculus of Variations and Partial Differential Equations 3 (4): 475–491. doi:10.1007/BF01187897. 
  16. ↑ Bethuel, Fabrice (March 2014). "A new obstruction to the extension problem for Sobolev maps between manifolds". Journal of Fixed Point Theory and Applications 15 (1): 155–183. doi:10.1007/s11784-014-0185-0. 
  17. ↑ 17.0 17.1 Bourgain, Jean; Brezis, Haim; Mironescu, Petru (December 2000). "Lifting in Sobolev spaces". Journal d'Analyse Mathématique 80 (1): 37–86. doi:10.1007/BF02791533. 
  18. ↑ 18.0 18.1 18.2 Bethuel, Fabrice; Chiron, David (2007). "Some questions related to the lifting problem in Sobolev spaces". Contemporary Mathematics 446: 125–152. doi:10.1090/conm/446/08628. ISBN 9780821841907. 
  19. ↑ Bethuel, Fabrice; Zheng, Xiaomin (September 1988). "Density of smooth functions between two manifolds in Sobolev spaces". Journal of Functional Analysis 80 (1): 60–75. doi:10.1016/0022-1236(88)90065-1. 
  20. ↑ Mironescu, Petru; Van Schaftingen, Jean (7 September 2021). "Lifting in compact covering spaces for fractional Sobolev mappings". Analysis & PDE 14 (6): 1851–1871. doi:10.2140/apde.2021.14.1851. 

Further reading



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