# Convex Integration Solutions for the Geometrically Non-linear Two-Well Problem with Higher Sobolev Regularity

## TL;DR

This paper constructs higher Sobolev regularity solutions for a non-linear two-well problem in shape-memory alloys using convex integration, advancing understanding of microstructure regularity and phase transformations.

## Contribution

It provides the first convex integration solutions with higher Sobolev regularity for the non-linear two-well problem, relevant to shape-memory alloy modeling.

## Key findings

- Constructed solutions with higher Sobolev regularity for the two-well problem.
- Demonstrated the applicability of convex integration in non-linear matrix spaces.
- Linked regularity of solutions to microstructure selection mechanisms.

## Abstract

In this article we discuss higher Sobolev regularity of convex integration solutions for the geometrically non-linear two-well problem. More precisely, we construct solutions to the differential inclusion $\nabla u\in K$ subject to suitable affine boundary conditions for $ u$ with $$ K:= SO(2)\left[\begin{array}{ ccc } 1 & \delta \\ 0 & 1 \end{array}\right] \cup SO(2)\left[\begin{array}{ ccc } 1 & -\delta \\ 0 & 1 \end{array}\right] $$ such that the associated deformation gradients $\nabla u$ enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape-memory alloys where $K^{qc} \neq K^{c}$, and where the energy minimisers constructed by convex integration satisfy higher Sobolev regularity. We show that in spite of additional difficulties arising from the treatment of the non-linear matrix space geometry, it is possible to deal with the geometrically non-linear two-well problem within the framework outlined in \cite{RZZ18}. Physically, our investigation of convex integration solutions at higher Sobolev regularity is motivated by viewing regularity as a possible selection mechanism of microstructures.