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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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

Related publications

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