Convex hull-like property and supported images of open sets

Abstract

In this note, as a particular case of a more general result, we obtain the following theorem: Let $\Omega \subseteq {\mathbf{R}}^n$ be a non-empty bounded open set and let $f:\overline{\Omega }\to {\mathbf{R}}^n$ be a continuous function which is $C^1$ in $\Omega$. Then, at least one of the following assertions holds: $(a)$ $f(\Omega )\subseteq conv(f(\partial \Omega )).$ $(b)$ There exists a non-empty open set $X\subseteq \Omega$, with $\overline{X}\subseteq \Omega$, satisfying the following property: for every continuous function $g:\Omega \to {\mathbf{R}}^n$ which is $C^1$ in $X$, there exists $\tilde{\lambda }>0$ such that, for each $\lambda >\tilde{\lambda }$, the Jacobian determinant of the function $g+\lambda f$ vanishes at some point of $X$. As a consequence, if $n=2$ and $h:\Omega \to \mathbf{R}$ is a non-negative function, for each $u\in C^2(\Omega )\cap C^1(\overline{\Omega })$ satisfying in $\Omega$ the Monge-Ampère equation $$u_{xx}{u}_{yy}-u_{xy}^2=h,$$ one has $$\nabla u(\Omega )\subseteq conv(\nabla u(\partial \Omega )).$$

References in corpus (1)

• Convex Hull Property and Maximum Principle for Finite Element Minimisers of General Convex Functionals