Sobolev mapping explained

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

and

N

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

R\nu

as [1] [2] W^ (M, N) :=\.First-order (

s=1

) 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

M

to

N

are dense in

Ws,(M,N)

with respect to the norm topology.When

sp>\dimM

, Morrey's inequality implies that Sobolev mappings are continuous and can thus be strongly approximated by smooth maps.When

sp=\dimM

, Sobolev mappings have vanishing mean oscillation[5] and can thus be approximated by smooth maps.[6]

When

sp<\dimM

, the question of density is related to obstruction theory:

Cinfty(M,N)

is dense in

W1,(M,N)

if and only if every continuous mapping on a from a

\lfloorp\rfloor

–dimensional triangulation of

M

into

N

is the restriction of a continuous map from

M

to

N

.[7] [2]

The problem of finding a sequence of weak approximation of maps in

W1,(M,N)

is equivalent to the strong approximation when

p

is not an integer.When

p

is an integer, a necessary condition is that the restriction to a

\lfloorp-1\rfloor

-dimensional triangulation of every continuous mapping from a

\lfloorp\rfloor

–dimensional triangulation of

M

into

N

coincides with the restriction a continuous map from

M

to

N

.[2] When

p=2

, this condition is sufficient.[8] For

W1,(M,S2)

with

\dimM\ge4

, this condition is not sufficient.[9]

Homotopy

The homotopy problem consists in describing and classifying the path-connected components of the space

Ws,(M,N)

endowed with the norm topology.When

0<s\le1

and

\dimM\lesp

, then the path-connected components of

Ws,(M,N)

are essentially the same as the path-connected components of

C(M,N)

: two maps in

Ws,(M,N)\capC(M,N)

are connected by a path in

Ws,(M,N)

if and only if they are connected by a path in

C(M,N)

, any path-connected component of

Ws,(M,N)

and any path-connected component of

C(M,N)

intersects

Ws,(M,N)\capC(M,N)

non trivially.[10] [11] [12] When

\dimM>p

, two maps in

W1,(M,N)

are connected by a continuous path in

W1,(M,N)

if and only if their restrictions to a generic

\lfloorp-1\rfloor

-dimensional triangulation are homotopic.[2]

Extension of traces

The classical trace theory states that any Sobolev map

u\inW1,(M,N)

has a trace

Tu\inW1(\partialM,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

\pi1(N)\simeq...b\pi\lfloor(N)\simeq\{0\}

[13] or when

p\ge3

,

\pi1(N)

is finite and

\pi2(N)\simeq...b\pi\lfloor(N)\simeq\{0\}

.[14] The surjectivity of the trace operator fails if

\pi\lfloor(N)\not\simeq\{0\}

[15] or if

\pi\ell(N)

is infinite for some

\ell\in\{1,...c,\lfloorp-1\rfloor\}

.[14] [16]

Lifting

\pi:\tilde{N}\toN

, the lifting problem asks whether any map

u\inWs,(M,N)

can be written as

u=\pi\circ\tilde{u}

for some

\tilde{u}\inWs,(M,\tilde{N})

, as it is the case for continuous or smooth

u

and

\tilde{u}

when

M

is simply-connected in the classical lifting theory.If the domain

M

is simply connected, any map

u\inWs,(M,N)

can be written as

u=\pi\circ\tilde{u}

for some

\tilde{u}\inWs,(M,N)

when

sp\ge\dimM

,[17] [18] when

s\ge1

and

2\lesp<\dimM

[19] and when

N

is compact,

0<s<1

and

2\lesp<\dimM

.[20] There is a topological obstruction to the lifting when

sp<2

and an analytical obstruction when

1\lesp<\dimM

.[17] [18]

Further reading

Notes and References

  1. Mironescu . Petru . Sobolev maps on manifolds: degree, approximation, lifting . Contemporary Mathematics . 2007 . 446 . 413–436 . 10.1090/conm/446/08642. 9780821841907 .
  2. Hang . Fengbo . Lin . Fanghua . Topology of sobolev mappings, II . Acta Mathematica . 2003 . 191 . 1 . 55–107 . 10.1007/BF02392696. 121520479 . free .
  3. Chiron . David . On the definitions of Sobolev and BV spaces into singular spaces and the trace problem. Communications in Contemporary Mathematics . August 2007 . 09 . 4 . 473–513 . 10.1142/S0219199707002502.
  4. Hajłasz . Piotr . Sobolev Mappings between Manifolds and Metric Spaces . Sobolev Spaces in Mathematics I . International Mathematical Series . 2009 . 8 . 185–222 . 10.1007/978-0-387-85648-3_7. 978-0-387-85647-6 .
  5. Brezis . H. . Nirenberg . L. . Degree theory and BMO; part I: Compact manifolds without boundaries . Selecta Mathematica . September 1995 . 1 . 2 . 197–263 . 10.1007/BF01671566. 195270732 .
  6. Schoen . Richard . Uhlenbeck . Karen . A regularity theory for harmonic maps . Journal of Differential Geometry . 1 January 1982 . 17 . 2 . 10.4310/jdg/1214436923. free .
  7. Bethuel . Fabrice . The approximation problem for Sobolev maps between two manifolds . Acta Mathematica . 1991 . 167 . 153–206 . 10.1007/BF02392449. 122996551 . free .
  8. Pakzad . M.R. . Rivière . T. . Weak density of smooth maps for the Dirichlet energy between manifolds . Geometric and Functional Analysis . February 2003 . 13 . 1 . 223–257 . 10.1007/s000390300006. 121794503 .
  9. Bethuel . Fabrice . A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces . Inventiones Mathematicae . February 2020 . 219 . 2 . 507–651 . 10.1007/s00222-019-00911-3. 1401.1649 . 2020InMat.219..507B . 119627475 .
  10. Brezis . Haı̈m . Li . YanYan . Topology and Sobolev spaces . Comptes Rendus de l'Académie des Sciences - Series I - Mathematics . September 2000 . 331 . 5 . 365–370 . 10.1016/S0764-4442(00)01656-6. 2000CRASM.331..365B .
  11. Brezis . Haim . Li . Yanyan . Topology and Sobolev Spaces . Journal of Functional Analysis . July 2001 . 183 . 2 . 321–369 . 10.1006/jfan.2000.3736. free .
  12. Bousquet . Pierre . Fractional Sobolev spaces and topology . Nonlinear Analysis: Theory, Methods & Applications . February 2008 . 68 . 4 . 804–827 . 10.1016/j.na.2006.11.038.
  13. Hardt . Robert . Lin . Fang-Hua . Mappings minimizing the Lp norm of the gradient . Communications on Pure and Applied Mathematics . September 1987 . 40 . 5 . 555–588 . 10.1002/cpa.3160400503.
  14. Mironescu . Petru . Van Schaftingen . Jean . Trace theory for Sobolev mappings into a manifold . Annales de la Faculté des sciences de Toulouse: Mathématiques . 9 July 2021 . 30 . 2 . 281–299. 2001.02226. 10.5802/afst.1675. 210023485 .
  15. Bethuel . Fabrice . Demengel . Françoise . Extensions for Sobolev mappings between manifolds . Calculus of Variations and Partial Differential Equations . October 1995 . 3 . 4 . 475–491 . 10.1007/BF01187897. 121749565 .
  16. Bethuel . Fabrice . A new obstruction to the extension problem for Sobolev maps between manifolds . Journal of Fixed Point Theory and Applications . March 2014 . 15 . 1 . 155–183 . 10.1007/s11784-014-0185-0. 1402.4614 . 119614310 .
  17. Bourgain . Jean . Jean Bourgain . Brezis . Haim . Haïm Brezis . Mironescu . Petru . Lifting in Sobolev spaces . . December 2000 . 80 . 1 . 37–86 . 10.1007/BF02791533 . free.
  18. Bethuel . Fabrice . Chiron . David . Some questions related to the lifting problem in Sobolev spaces . Contemporary Mathematics . 2007 . 446 . 125–152 . 10.1090/conm/446/08628. 9780821841907 .
  19. Bethuel . Fabrice . Zheng . Xiaomin . Density of smooth functions between two manifolds in Sobolev spaces . Journal of Functional Analysis . September 1988 . 80 . 1 . 60–75 . 10.1016/0022-1236(88)90065-1. free .
  20. Mironescu . Petru . Van Schaftingen . Jean . Lifting in compact covering spaces for fractional Sobolev mappings . Analysis & PDE . 7 September 2021 . 14 . 6 . 1851–1871 . 1907.01373. 10.2140/apde.2021.14.1851. 195776361 .