A new approach to counterexamples to L1 estimates:: Korn's inequality, geometric rigidity, and regularity for gradients of separately convex functions

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 --> R which is separately convex but whose gradient is not in BVloc, in the sense that the mixed derivative (2) f/partial derivativex(1) partial derivativex(2) is not a bounded measure.