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MiS Preprint
93/2003

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

Sergio Conti, Daniel Faraco and Francesco Maggi

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.

Received:
Nov 21, 2003
Published:
Nov 21, 2003

Related publications

inJournal
2005 Repository Open Access
Sergio Conti, Daniel Faraco and Francesco Maggi

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

In: Archive for rational mechanics and analysis, 175 (2005) 2, pp. 287-300