  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.
