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Archive for Rational Mechanics and Analysis

, Volume 175, Issue 2, pp 287–300 | Cite as

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
  • Francesco Maggi
Article

Abstract.

The derivation of counterexamples to L1 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 L1. Secondly, we construct a function f:ℝ2→ℝ which is separately convex but whose gradient is not in BVloc, in the sense that the mixed derivative ∂2f/∂x1x2 is not a bounded measure.

Keywords

Neural Network Complex System Nonlinear Dynamics Convex Function Electromagnetism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Dipartimento di Matematica “U. Dini”Università di Firenze Viale Morgagni 67/AFirenzeItaly

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