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

Geometric rigidity on Sobolev spaces with variable exponent and applications