Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Burkholder integrals, Morrey's problem and quasiconformal mappings


Authors: Kari Astala, Tadeusz Iwaniec, István Prause and Eero Saksman
Journal: J. Amer. Math. Soc. 25 (2012), 507-531
MSC (2010): Primary 30C62, 30C70, 49K10, 49K30
Published electronically: October 6, 2011
MathSciNet review: 2869025
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Inspired by Morrey's Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals $ \text {B}_p$, $ p \geqslant 2$, are quasiconcave, when tested on deformations of the identity $ f\in \textup {Id} + {\mathscr C}^\infty _\circ (\Omega )$ with $ \text {B}_p\,(Df(x)) \geqslant 0$ pointwise, or equivalently, deformations such that $ \vert D f \vert^2 \leqslant \frac {p}{p-2}J_f $. In particular, quasiconcavity holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible $ \mathscr L^p$-estimates for the gradient of a principal solution to the Beltrami equation $ f_{\bar {z}}=\mu (z) f_z$, for any $ p$ in the critical interval $ 2\leqslant p \leqslant 1+1/\Vert\mu \Vert _\infty $.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 30C62, 30C70, 49K10, 49K30

Retrieve articles in all journals with MSC (2010): 30C62, 30C70, 49K10, 49K30


Additional Information

Kari Astala
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland
Email: kari.astala@helsinki.fi

Tadeusz Iwaniec
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244, USA, and Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland
Email: tiwaniec@syr.edu

István Prause
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland
Email: istvan.prause@helsinki.fi

Eero Saksman
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland
Email: eero.saksman@helsinki.fi

DOI: http://dx.doi.org/10.1090/S0894-0347-2011-00718-2
PII: S 0894-0347(2011)00718-2
Keywords: Rank-one convex and quasiconvex variational integrals, critical Sobolev exponents, extremal quasiconformal mappings, Jacobian inequalities
Received by editor(s): December 5, 2010
Received by editor(s) in revised form: August 19, 2011
Published electronically: October 6, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.