Burkholder integrals, Morrey's problem and quasiconformal mappings

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 $B_p$, $p\ge 2$, are quasiconcave, when tested on deformations of identity $f\in Id+C_0^{\infty }(\Omega )$ with $B_p(Df(x))\ge 0$ pointwise, or equivalently, deformations such that $|Df{|}^2\le \frac{p}{p-2}{J}_f$. In particular, this holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible $L^p$- estimates for the gradient of a principal solution to the Beltrami equation $\f_{\bar{z}} = μ(z) f_z$, for any $p$ in the critical interval $2\le p\le 1+1/\Vert {\mu }_f{\Vert }_{\infty }$. Examples of local maxima lacking symmetry manifest the intricate nature of the problem.

33 pages, 1 figure

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