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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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