Inspired by Morrey's Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals Bp, p > 2, are quasiconcave, when tested on deformations of identity f in Id+Coinifinty (omega) with Bp (Df(x)) > 0 pointwise, or equivalently, deformations such that abs[Df]2 < (p/(p-2))Jf. In particular, this holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible Lp-estimates for the gradient of a principal solution to the Beltrami equation fz = mu(z)fz , for any p in the critical interval 2 < 1+1/ abs[mu f]infinity. Examples of local maxima lacking symmetry manifest the intricate nature of the problem.
Astala, Kari; Iwaniec, Tadeusz; Prause, Istvan; and Saksman, Eero, "Burkholder Integrals, Morrey's Problem and Quasiconformal Mappings" (2010). Mathematics Faculty Scholarship. 81.
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