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

Abstract

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.