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SUMMARY:From Ornstein's non-inequalities to rank-one convexity - Jan Krist
 ensen (University of Oxford\, OxPDE)
DTSTART:20121203T150000Z
DTEND:20121203T160000Z
UID:TALK41641@talks.cam.ac.uk
CONTACT:Filip Rindler
DESCRIPTION:Questions about sharp integral estimates for partial derivativ
 es of mappings can often be recast as questions about quasiconvexity of as
 sociated integrands. Quasiconvexity was introduced by Morrey in his work o
 n weak lower semicontinuity in the Calculus of Variations. It is by now re
 cognized as a central notion within the Calculus of Variations\, but it re
 mains somewhat mysterious. A closely related notion is that of rank-one co
 nvexity. Rank-one convexity is a necessary condition for quasiconvexity\, 
 and it is easy to check whether or not a given integrand is rank-one conve
 x. Unfortunately\, rank-one convexity is not equivalent to quasiconvexity.
  An example of Sverak shows that\, in high dimensions\, there exists a qua
 rtic polynomial that is rank-one convex but not quasiconvex. However\, it 
 is still plausible that rank-one convexity could be equivalent to quasicon
 vexity within more restricted classes of integrands. An interesting class 
 being the positively one homogeneous integrands. Their quasiconvexity prop
 erties correspond to L1-estimates.\n\nIn this talk I briefly review the co
 nvexity notions from the Calculus of Variations. I then show how the above
  viewpoint can be used to give a proof\, and a generalization\, of Ornstei
 n's non-L1-inequalities.
LOCATION:CMS\, MR11
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