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

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