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Learning how to figure out sharp $L^p$-estimates of nonlinear differentialexpressions, to prove and use them, is a fundamental part of the development ofPDEs and Geometric Function Theory (GFT). Our survey presents, among what isknown to date, some notable recent efforts and novelties made in thisdirection. We focus attention here on the historic Morrey's Conjecture andBurkholder's martingale inequalities for stochastic integrals. Some of thesetopics have already been discussed by the present authors [5] and by RodrigoBa\~{n}uelos [10]. Nevertheless, there is always something new to add.

A hunt for sharp $L ^p$-estimates and rank-one convex variational integrals