*Published Paper*

**Inserted:** 26 nov 2003

**Last Updated:** 19 dec 2005

**Journal:** Arch. Ration. Mech. Anal.

**Volume:** 175

**Number:** 2

**Pages:** 287-300

**Year:** 2005

**Notes:**

Preprint Nr. 93*2003, Max Planck Institute for Mathematics in the Sciences, DOWNLOAD AT http:/www.mis.mpg.de*preprints*2003*index.html

**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 $f:R^2\rightarrow R$ 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.

**Keywords:**
bounded variation, convex integration, Korn's inequality, separate convexity, rank-one convexity, laminates