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> | + | <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 |
Convex integration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_integration&oldid=14622