  Korn's inequality has been at the heart of much exciting research since its
first appearance in the beginning of the 20th century. Many are the
applications of this inequality to the analysis and construction of
discretizations of a large variety of problems in continuum mechanics. In this
paper, we prove that the classical Korn inequality holds true in {\it
anisotropic Sobolev spaces}. We also prove that an extension of Korn's
inequality, involving non-linear continuous maps, is valid in such spaces.
Finally, we point out that another classical inequality, Poincare's inequality,
also holds in anisotropic Sobolev spaces.
