A new approach to counterexamples to L^1 estimates: Korn's inequality, geometric rigidity, and regularity for gradients of separately convex functions

Abstract: The derivation of counterexamples to L^1 estimates can be reduced to a geometric decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korn’s inequality, and of the corresponding geometrically nonlinear rigidity result, in L^1. Secondly, we construct a function f : R^2 \to R which is separately convex but whose gradient is not in BV_loc, in the sense that the mixed derivative f_{12} is not a bounded measure.