  We present a quantitative geometric rigidity estimate in dimensions $d=2,3$
generalizing the celebrated result by Friesecke, James, and M\"uller to the
setting of variable domains. Loosely speaking, we show that for each $y \in
H^1(U;\mathbb{R}^d)$ and for each connected component of a smooth open, bounded
set $U \subset \mathbb{R}^d$, the $L^2$-distance of $\nabla y$ from a single
rotation can be controlled up to a constant by its $L^2$-distance from the
group $SO(d)$, with the constant not depending on the precise shape of $U$, but
only on an integral curvature functional related to $\partial U$. We further
show that for linear strains the estimate can be refined, leading to a uniform
control independent of the set $U$. The estimate can be used to establish
compactness in the space of generalized special functions of bounded
deformation ($GSBD^2$) for sequences of displacements related to deformations
with uniformly bounded elastic energy. As an application, we rigorously derive
linearized models for nonlinearly elastic materials with free surfaces by means
of $\Gamma$-convergence. In particular, we study energies related to
epitaxially strained crystalline films and to the formation of material voids
inside elastically stressed solids.

Analysis of PDEs