Variational analysis of the Burkholder functional

The quasiconvexity of the Burkholder function implies the Iwaniec's conjecture on Beurling-Ahlfors transform and Iwaniec-Martin conjecture for quasiregular mappings. The quasiconvexity of the Burkholder function is equivalent to a global minimization problem of its integral functional over the space $W_0^{1,p}$. In this talk, we introduce the technique of non-smooth analysis and non-smooth optimization in the variational study of the Burkholder functional. We will demonstrate how to derive the first and second order derivatives of only locally Lipschitz integral functionals and how these derivative tests imply a strict local minimum as the classical $C^2$ case. This method does not completely solve the quasiconvexity problem since it only gives a local minimum over every compact subsets of the space $W_0^{1,p}$.