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

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