Burkholder integrals, Morrey's problem and quasiconformal mappings

TL;DR

This paper demonstrates that Burkholder functionals are quasiconcave under certain deformations, leading to optimal $L^p$ estimates for solutions to the Beltrami equation, connecting Morrey's problem and quasiconformal mappings.

Contribution

It establishes quasiconcavity of Burkholder functionals near the identity, providing new $L^p$ bounds for Beltrami equation solutions and linking Morrey's problem with quasiconformal mapping theory.

Findings

Burkholder functionals are quasiconcave near the identity.

Strongest $L^p$ estimates for Beltrami solutions are obtained.

Examples show complex local maxima without symmetry.

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.