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MiS Preprint
69/2004

Rigidity and Gamma convergence for solid-solid phase transitions with SO(2)-invariance

Sergio Conti and Ben Schweizer

Abstract

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

Received:
Oct 12, 2004
Published:
Oct 12, 2004

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inJournal
2006 Repository Open Access
Sergio Conti and Ben Schweizer

Rigidity and Gamma convergence for solid-solid phase transitions with SO(2)-invariance

In: Communications on pure and applied mathematics, 59 (2006) 6, pp. 830-868