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

TL;DR

This paper proves the existence of functions with prescribed boundary data and convex gradient constraints, and studies their regularity where solutions coincide, advancing understanding of constrained boundary value problems.

Contribution

It establishes existence results for functions with boundary data and convex gradient constraints, and analyzes their regularity on the coincidence set.

Findings

Existence of solutions with prescribed boundary and gradient constraints.

Regularity properties of solutions on the set where all solutions coincide.

Characterization of the coincidence set within the domain.

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.