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 $f\in Id + C^\infty_0(\Omega)$ with $B_p(Df(x)) \ge 0$ pointwise, or equivalently, deformations such that $|Df|^2 \leq \frac{p}{p-2} J_f$. 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 $\f_{\bar{z}} = \mu(z) f_z$, for any $p$ in the critical interval $2 \leq p \leq 1+1/\|\mu_f\|_\infty$. Examples of local maxima lacking symmetry manifest the intricate nature of the problem.