Abstract
Korn’s inequalities, involving quadratic functionals subject to various side conditions, have played an important role in elasticity theory. Here a formulation of Korn’s inequality is given where the admissible vector fields are subject to the constraint of incompressibility. The work is mainly concerned with investigation of the Korn’s constant for different regions. A variational approach is used to establish an associated eigenvalue problem, whose spectral properties are examined. For simply-connected two-dimensional regions, this leads to consideration of uniqueness and nonuniqueness questions for plane isotropic linear elastostatics. New optimal lower bounds for the Korn constants are also established. Finally, some applications to incompressible elasticity and incompressible viscous flow are outlined.
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