A Compactness and Structure Result for a Discrete Multi-Well Problem with $SO(n)$ Symmetry in Arbitrary Dimension

TL;DR

This paper introduces a new, robust proof for the compactness of discrete multi-well energies in phase transition models, applicable to various physical phenomena including shape-memory and magnetic transformations.

Contribution

It combines existing techniques to reduce the multi-well problem to a one-well problem, providing a novel and versatile approach for analyzing phase transitions.

Findings

Proves compactness of discrete multi-well energies.

Applicable to models of shape-memory and magnetic materials.

Introduces a reduction to incompatible one-well problems.

Abstract

In this note we combine the "spin-argument" from [KLR15] and the $n$-dimensional incompatible, one-well rigidity result from [LL16], in order to infer a new proof for the compactness of discrete multi-well energies associated with the modelling of surface energies in certain phase transitions. Mathematically, a main novelty here is the reduction of the problem to an incompatible one-well problem. The presented argument is very robust and applies to a number of different physically interesting models, including for instance phase transformations in shape-memory materials but also anti-ferromagnetic transformations or related transitions with an "internal" microstructure on smaller scales.