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MiS Preprint

A sharp-interface limit for the geometrically linear two-well problem in the gradient theory of phase transitions in two dimensions

Sergio Conti and Ben Schweizer


We obtain a Gamma-convergence result for the gradient theory of solid-solid phase transitions, in the case of two geometrically linear wells in two dimensions. We consider the functionals \[ I_\epsilon[u] = \int_\Omega \frac{1}{\epsilon} W(\nabla u) + \epsilon |\nabla^2u|^2\] where $u:\Omega\subset R^2\to R^2$, W depends only on the symmetric part of $\nabla u$, and W(F)=0 for two distinct values of F, say A and B. We show that, under suitable growth assumptions on W and for star-shaped domains $\Omega$, as $\epsilon\to0$ $I_\epsilon$ converges, in the sense of Gamma convergence, to a functional $I_0$. The limit $I_0$ is finite only on functions u such that the symmetric part of $\nabla u$ is a function of bounded variation which takes only values A and B. On those functions, the energy concentrates on the jump set J of $\nabla u$, with a surface energy depending on the normal $\nu$ to J, and is given by \[I_0[u]= \int_{J} k(\nu) dH^1\,. \] The interfaces can have, in general, two orientations.


Related publications

2006 Repository Open Access
Sergio Conti and Ben Schweizer

A sharp-interface limit for a two-well problem in geometrically linear elasticity

In: Archive for rational mechanics and analysis, 179 (2006) 3, pp. 413-452