Burkholder integrals, Morrey's problem and quasiconformal mappings

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 $f\in Id+C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ with $B_p(Df(x))\ge 0$ pointwise, or equivalently, deformations such that $\mathrm{\mid }Df{\mathrm{\mid }}^2\le \frac{p}{p-2}{J}_f$. 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 ${\text{f}}_{\bar{z}}=\mu (z)f_z$, 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.