  We present extensions of rigidity estimates and of Korn's inequality to the
setting of (mixed) variable exponents growth. The proof techniques, based on a
classical covering argument, rely on the log-H\"older continuity of the
exponent to get uniform regularity estimates on each cell of the cover, and on
an extension result \`a la Nitsche in Sobolev spaces with variable exponents.
As an application, by means of $\Gamma$-convergence we perform a passage from
nonlinear to linearized elasticity under variable subquadratic energy growth
far from the energy well.
