Inspired by Morrey's Problem (on rank-one convex functionals) and theBurkholder integrals (of his martingale theory) we find that the Burkholderfunctionals $B_p$, $p \ge 2$, are quasiconcave, when tested on deformations ofidentity $f\in Id + C^\infty_0(\Omega)$ with $B_p(Df(x)) \ge 0$ pointwise, orequivalently, deformations such that $|Df|^2 \leq \frac{p}{p-2} J_f$. Inparticular, this holds in explicit neighbourhoods of the identity map. Amongthe 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 manifestthe intricate nature of the problem.
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机译：受莫里问题（关于一阶凸函数）和Burkholder积分（他的mar理论的启发）的启发，我们发现Burkholder函数$ B_p $，$ p \ ge 2 $，在对恒等式$ f \ in Id进行测试时是拟凹的+ C ^ \ infty_0（\ Omega）$和$ B_p（Df（x））\ ge 0 $逐点等效地变形，使得$ | Df | ^ 2 \ leq \ frac {p} {p-2} J_f $ 。特别是，这适用于身份地图的显式邻域。在许多直接后果中，这给出了Beltrami方程的主解的梯度$ \ f _ {\ bar {z}} = \ mu（z）f_z $的最大可能的$ L ^ p $估计。临界间隔$ 2中的p $ \ leq p \ leq 1 + 1 / \ || \ mu_f \ | _ \ infty $。缺乏对称性的局部最大值的例子表明了问题的复杂性。
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