Two-well rigidity and multidimensional sharp-interface limits for solid-solid phase transitions

TL;DR

This paper develops a quantitative rigidity estimate for two-well nonlinear energies with a single rank-one connection, and analyzes the sharp-interface limit of solid-solid phase transitions, showing convergence to a laminate-structured energy proportional to interface length.

Contribution

It introduces a novel rigidity estimate for two-well energies with a single rank-one connection and analyzes the sharp-interface limit of phase transitions in arbitrary dimensions.

Findings

Rigidity estimate for two-well energies established.

Solid-solid phase transitions converge to a laminate-structured sharp-interface model.

Limiting energy proportional to total interface length.

Abstract

We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid-solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of $\Gamma$-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases.