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

Sergio Conti, Daniel Faraco, and Francesco Maggi

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Submission date: 21. Nov. 2003
Pages: 17
published in: Archive for rational mechanics and analysis, 175 (2005) 2, p. 287-300 
DOI number (of the published article): 10.1007/s00205-004-0350-5
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Abstract:
The derivation of counterexamples to 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 . Secondly, we construct a function which is separately convex but whose gradient is not in , in the sense that the mixed derivative is not a bounded measure.