  Geometric rigidity states that a gradient field which is $L^p$-close to the
set of proper rotations is necessarily $L^p$-close to a fixed rotation, and is
one key estimate in nonlinear elasticity. In several applications, as for
example in the theory of plasticity, energy densities with mixed growth appear.
We show here that geometric rigidity holds also in $L^p+L^q$ and in $L^{p,q}$
interpolation spaces. As a first step we prove the corresponding linear
inequality, which generalizes Korn's inequality to these spaces.
