MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

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.

Oct 14, 2003
Oct 14, 2003

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