Sufficient Conditions for an Existence of a Solution to a Differential Inclusion

We formulate geometric conditions induced by the compact set $K\subset\mathbb{R}^{m\times n}$, which imply existence of a Lipschitz solution $u$ to the differential inclusion $Du\in K$. The solutions are obtained using the convex integration method. We illustrate our result for the known example $K=SO(2)\cup SO(2)B$, where $B$ is a $2\times2$ diagonal matrix with $\det B=1$