In the study of solid-solid phase transitions and in particular of shape-memory alloys one is interested in studying variational models of the form

$\displaystyle \int_\Omega W(\nabla u) dx, u:\Omega\subset \mathbb{R}^n\to\mathbb{R}^n,$

where the energy density $ W$ is invariant under rotations and it is minimized by several copies of the set $ SO(n)$ of the proper rotations, i.e., by sets of the form

$\displaystyle K:=SO(n)U_1\cup \dots \cup SO(n)U_{m}, \det U_i>0, i=1,\dots,{m}.$

While the set $ SO(n)$ (case $ m=1$) is rigid, in the sense that there are no nontrivial gradient fields taking values in $ SO(n)$, the set $ K$ is in general not rigid.

After a brief review of known rigidity results, we present a quantitative rigidity estimate for a multiwell problem ($ m\geq2$) in dimension $ n\geq 2$. Precisely, we show that if a gradient field is $ L^1$-close to the set $ K$, a set of the form $ SO(n)U_1\cup\dots\cup SO(n)U_l$, and and an appropriate bound on the length of the interfaces holds, then the gradient field is actually close to only one of the wells $ SO(n)U_i$. The estimate holds for any connected subdomain, and has the optimal scaling.

Results of these kind have several applications, e.g., in studying the scaling of singularly perturbed problem under Dirichlet boundary conditions or in proving compactness and $ \Gamma$-convergence for a sequence of singularly perturbed functionals of the kind

$\displaystyle I_\varepsilon[u]:=\int \frac{1}{\varepsilon} W(\nabla u) + \varepsilon \vert\nabla^2 u\vert^2 dx.$