Calculus of Variations and Geometric Measure Theory

S. Conti - D. Faraco - F. Maggi

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

created on 26 Nov 2003
modified by maggi on 19 Dec 2005


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

Preprint Nr. 932003, Max Planck Institute for Mathematics in the Sciences, DOWNLOAD AT http:/www.mis.mpg.depreprints2003index.html


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