  In this article we continue our study of higher Sobolev regularity of
flexible convex integration solutions to differential inclusions arising from
applications in materials sciences. We present a general framework yielding
higher Sobolev regularity for Dirichlet problems with affine data in
$\text{int}(K^{lc})$. This allows us to simultaneously deal with linear and
nonlinear differential inclusion problems. We show that the derived higher
integrability and differentiability exponent has a lower bound, which is
independent of the position of the Dirichlet boundary data in
$\text{int}(K^{lc})$. As applications we discuss the regularity of weak
isometric immersions in two and three dimensions as well as the differential
inclusion problem for the geometrically linear hexagonal-to-rhombic and the
cubic-to-orthorhombic phase transformations occurring in shape memory alloys.
