MiS Preprint Repository

The singularly perturbed two-well problem in the theory of solid-solid phase transitions takes the form $$I_{ϵ}[u]={\int }_{\mathrm{\Omega }}\frac{1}{ϵ}W(\mathrm{\nabla }u)+ϵ|{\mathrm{\nabla }}^{2}u{|}^2,$$ where $u:\mathrm{\Omega }\subset R^n\to R^n$ is the deformation, and $W$ vanishes for all matrices in $K=SO(n)A\cup SO(n)B$. We focus on the case $n=2$ and derive, by means of Gamma convergence, a sharp-interface limit for $I_{ϵ}$. The proof is based on a rigidity estimate for low-energy functions. Our rigidity argument also gives an optimal two-well Liouville estimate: if $\mathrm{\nabla }u$ has a small $BV$ norm (compared to the diameter of the domain), then, in the $L^1$ sense, either the distance of $\mathrm{\nabla }u$ from $SO(2)A$ or the one from $SO(2)B$ is controlled by the distance of $\mathrm{\nabla }u$ from $K$. This implies that the oscillation of $\mathrm{\nabla }u$ in weak-$L^1$ is controlled by the $L^1$ norm of the distance of $\mathrm{\nabla }u$ to $K$.