Poincare meets Korn via Maxwell: Extending Korn's First Inequality to Incompatible Tensor Fields

Details Poincare meets Korn via Maxwell: Extending Korn's First Inequality to Incompatible Tensor Fields Poincare meets Korn via Maxwell: Extending Korn's First Inequality to Incompatible Tensor Fields

2012-03-13

arxiv.org/abs/1203.2744v1

Mathematics

Analysis of PDEs

Key Words: Korn's inequality, incompatible tensors, Maxwell's equations, Helmholtz decomposition, Poincare type inequalities,

Scientific paper

For a bounded three-dimensional domain with Lipschitz boundary we extend Korn's first inequality to incompatible tensor fields. For compatible tensor fields our estimate reduces to a non-standard variant of the well known Korn's first inequality. On the other hand, for skew-symmetric tensor fields our new estimate turns to Poincare's inequality. Therefore, our result may be viewed as a natural common generalization of Korn's first and Poincare's inequality. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditions and the Maxwell estimate.

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