On functions with given boundary data and convex constraints on the gradient

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be an open set. Given a boundary datum $g$ on $\mathrm{\partial }\mathrm{\Omega }$ and a function $K:\bar{\mathrm{\Omega }}\to \mathcal{K}$, the family of all compact convex subsets of ${\mathbb{R}}^d$, we prove the existence of functions $u:\mathrm{\Omega }\to \mathbb{R}$ such that $u=g$ on $\mathrm{\partial }\mathrm{\Omega }$ and $\mathrm{\nabla }u(x)\in K(x)$ a.e. and we investigate the regularity of such solutions on the set $\mathcal{U}\subset \bar{\mathrm{\Omega }}$ of points at which they all coincide.