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.[1][2][3]

## Contents

* 1 Definition
* 2 Approximation
* 3 Homotopy
* 4 Extension of traces
* 5 Lifting
* 6 References
* 7 Further reading

## Definition

Given Riemannian manifolds M and N, which is assumed by Nash's smooth embedding theorem without loss of generality to be isometrically embedded into ℝν as [4][5] Ws,p(M,N):={u∈Ws,p(M,ℝν)|u(x)∈N for almost every x∈M}. First-order (s=1) Sobolev mappings can also be defined in the context of metric spaces.[6][7]

## Approximation

The strong approximation problem consists in determining whether smooth mappings from M to N are dense in Ws,p(M,N) with respect to the norm topology. When sp>dim⁡M, Morrey's inequality implies that Sobolev mappings are continuous and can thus be strongly approximated by smooth maps. When sp=dim⁡M, Sobolev mappings have vanishing mean oscillation[8] and can thus be approximated by smooth maps.[9]

When sp<dim⁡M, the question of density is related to obstruction theory: C∞(M,N) is dense in W1,p(M,N) if and only if every continuous mapping on a from a ⌊p⌋–dimensional triangulation of M into N is the restriction of a continuous map from M to N.[10][5]

The problem of finding a sequence of weak approximation of maps in W1,p(M,N) is equivalent to the strong approximation when p is not an integer.[10] When p is an integer, a necessary condition is that the restriction to a ⌊p−1⌋-dimensional triangulation of every continuous mapping from a ⌊p⌋–dimensional triangulation of M into N coincides with the restriction a continuous map from M to N.[5] When p=2, this condition is sufficient.[11] For W1,3(M,𝕊2) with dim⁡M≥4, this condition is not sufficient.[12]

## Homotopy

The homotopy problem consists in describing and classifying the path-connected components of the space Ws,p(M,N) endowed with the norm topology. When 0<s≤1 and dim⁡M≤sp, then the path-connected components of Ws,p(M,N) are essentially the same as the path-connected components of C(M,N): two maps in Ws,p(M,N)∩C(M,N) are connected by a path in Ws,p(M,N) if and only if they are connected by a path in C(M,N), any path-connected component of Ws,p(M,N) and any path-connected component of C(M,N) intersects Ws,p(M,N)∩C(M,N) non trivially.[13][14][15] When dim⁡M>p, two maps in W1,p(M,N) are connected by a continuous path in W1,p(M,N) if and only if their restrictions to a generic ⌊p−1⌋-dimensional triangulation are homotopic.[5]: th. 1.1

## Extension of traces

The classical trace theory states that any Sobolev map u∈W1,p(M,N) has a trace Tu∈W1−1/p,p(∂M,N) and that when N=ℝ, 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 π1(N)≃⋯π⌊p−1⌋(N)≃{0}[16] or when p≥3, π1(N) is finite and π2(N)≃⋯π⌊p−1⌋(N)≃{0}.[17] The surjectivity of the trace operator fails if π⌊p−1⌋(N)≄{0} [16][18] or if πℓ(N) is infinite for some ℓ∈{1,…,⌊p−1⌋}.[17][19]

## Lifting

Given a covering map π:N~->N, the lifting problem asks whether any map u∈Ws,p(M,N) can be written as u=π∘u~ for some u~∈Ws,p(M,N~), as it is the case for continuous or smooth u and u~ when M is simply-connected in the classical lifting theory. If the domain M is simply connected, any map u∈Ws,p(M,N) can be written as u=π∘u~ for some u~∈Ws,p(M,N) when sp≥dim⁡M,[20][21] when s≥1 and 2≤sp<dim⁡M[22][21] and when N is compact, 0<s<1 and 2≤sp<dim⁡M.[23] There is a topological obstruction to the lifting when sp<2 and an analytical obstruction when 1≤sp<dim⁡M.[20][21]

## References

1. ↑ Hélein, Frédéric; Wood, John C. (2008). "Harmonic maps". Handbook of Global Analysis: 417–491. doi:10.1016/B978-044452833-9.50009-7. ISBN 978-0-444-52833-9.
2. ↑ Eells, J.; Lemaire, L. (March 1978). "A Report on Harmonic Maps". Bulletin of the London Mathematical Society 10 (1): 1–68. doi:10.1112/blms/10.1.1.
3. ↑ Eells, J.; Lemaire, L. (September 1988). "Another Report on Harmonic Maps". Bulletin of the London Mathematical Society 20 (5): 385–524. doi:10.1112/blms/20.5.385.
4. ↑ 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.
5. ↑ 5.0 5.1 5.2 5.3 Hang, Fengbo; Lin, Fanghua (2003). "Topology of sobolev mappings, II". Acta Mathematica 191 (1): 55–107. doi:10.1007/BF02392696.
6. ↑ 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. https://hal.science/hal-00809291.
7. ↑ Hajłasz, Piotr (2009). "Sobolev Mappings between Manifolds and Metric Spaces". Sobolev Spaces in Mathematics I. International Mathematical Series. 8. pp. 185–222. doi:10.1007/978-0-387-85648-3_7. ISBN 978-0-387-85647-6.
8. ↑ 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.
9. ↑ Schoen, Richard; Uhlenbeck, Karen (1 January 1982). "A regularity theory for harmonic maps". Journal of Differential Geometry 17 (2). doi:10.4310/jdg/1214436923.
10. ↑ 10.0 10.1 Bethuel, Fabrice (1991). "The approximation problem for Sobolev maps between two manifolds". Acta Mathematica 167: 153–206. doi:10.1007/BF02392449.
11. ↑ 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.
12. ↑ 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.
13. ↑ 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.
14. ↑ Brezis, Haim; Li, Yanyan (July 2001). "Topology and Sobolev Spaces". Journal of Functional Analysis 183 (2): 321–369. doi:10.1006/jfan.2000.3736.
15. ↑ 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.
16. ↑ 16.0 16.1 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.
17. ↑ 17.0 17.1 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.
18. ↑ 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.
19. ↑ 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.
20. ↑ 20.0 20.1 Bourgain, Jean; Brezis, Haim; Mironescu, Petru (December 2000). "Lifting in Sobolev spaces". Journal d'Analyse Mathématique 80 (1): 37–86. doi:10.1007/BF02791533.
21. ↑ 21.0 21.1 21.2 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.
22. ↑ 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.
23. ↑ 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

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

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