The Burkholder functional B_p is perhaps the most important rank-one convex functional in the plane due to its various connections to singular integrals, martingales, and the vector valued calculus of variations. B_p is therefore a prime candidate to consider the validity of the notorious Morrey conjecture for. The Iwaniec conjecture concerning the exact norms of the Beurling transform is also equivalent to the quasiconvexity of the Burkholder functional.
In this talk we discuss recent joint work together with K. Astala, D. Faraco, A. Guerra, and J. Kristensen. We study the Burkholder functional from the context of nonlinear elasticity, where the axiom of non-interpenetration of matter gives natural motivation to consider a restricted class of mappings which satisfy B_p(Df) ≤ 0 pointwise. In particular, we show the quasiconvexity of B_p in this class.