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

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