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Difference between revisions of "Convex integration"

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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gromov,  "Partial differential relations" , ''Ergebn. Math. Grenzgeb. (3)'' , '''9''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Spring,  "Convex integration theory" , ''Monogr. Math.'' , '''92''' , Birkhäuser  (1998)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gromov,  "Partial differential relations" , ''Ergebn. Math. Grenzgeb. (3)'' , '''9''' , Springer  (1986) {{MR|0864505}} {{ZBL|0651.53001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Spring,  "Convex integration theory" , ''Monogr. Math.'' , '''92''' , Birkhäuser  (1998) {{MR|1488424}} {{ZBL|0997.57500}} </TD></TR></table>

Revision as of 11:58, 27 September 2012

One of the methods developed by M. Gromov to prove the -principle. The essence of this method is contained in the following statement: If the convex hull of some path-connected subset contains a small neighbourhood of the origin, then there exists a mapping whose derivative sends into . This is equivalent to saying that the differential relation for mappings given by requiring for all satisfies the -principle. More generally, the method of convex integration allows one to prove the -principle for so-called ample relations . In the simplest case of a -jet bundle over a -dimensional manifold , this means that the convex hull of is all of for any fibre of (notice that this fibre is an affine space). The extension to arbitrary dimension and higher-order jet bundles is achieved by studying codimension-one hyperplane fields in and intermediate affine bundles defined in terms of .

One particular application of convex integration is to the construction of divergence-free vector fields and related geometric problems.

References

[a1] M. Gromov, "Partial differential relations" , Ergebn. Math. Grenzgeb. (3) , 9 , Springer (1986) MR0864505 Zbl 0651.53001
[a2] D. Spring, "Convex integration theory" , Monogr. Math. , 92 , Birkhäuser (1998) MR1488424 Zbl 0997.57500
How to Cite This Entry:
Convex integration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_integration&oldid=28164
This article was adapted from an original article by H. Geiges (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article