Geometric rigidity on Sobolev spaces with variable exponent and applications
Stefano Almi, Maicol Caponi, Manuel Friedrich, Francesco Solombrino
TL;DR
The paper develops a rigorous framework to extend geometric rigidity and Korn-type inequalities to Sobolev spaces with variable exponent $p(\cdot)$ under global log-Hölder continuity. It adapts the classical Friesecke–James–Müller approach using a Whitney-type covering, Lusin truncations, and a Nitsche-type extension to obtain a global rigidity estimate $\|\nabla u - R\|_{L^{p(\cdot)}(\Omega)} \le C \|d(\nabla u,SO(n))\|_{L^{p(\cdot)}(\Omega)}$ and a mixed-growth version with a decomposition $\nabla u - S = F+G$ in $L^{p(\cdot)}$ and $L^{q(\cdot)}$. It then proves $\Gamma$-convergence of nonlinear elasticity energies with variable-exponent growth to a linearized quadratic form, ensuring compactness and strong convergence of minimizers under the log-Hölder regularity of $p(\cdot)$. The results enable rigorous nonlinear-to-linear elasticity transitions in heterogeneous media where growth rates vary spatially, providing a solid foundation for modeling composites and materials with spatially varying mechanical response.
Abstract
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-Hölder continuity of the exponent to get uniform regularity estimates on each cell of the cover, and on an extension result à la Nitsche in Sobolev spaces with variable exponents. As an application, by means of $Γ$-convergence we perform a passage from nonlinear to linearized elasticity under variable subquadratic energy growth far from the energy well.