Differential Inclusions Involving the Curl Operator

TL;DR

This paper investigates the existence of vector fields with bounded derivatives whose curl lies within a specified set, advancing the mathematical understanding of differential inclusions involving the curl operator.

Contribution

It establishes conditions for the existence of solutions to curl-based differential inclusions in bounded domains, extending prior theoretical frameworks.

Findings

Existence results for curl-inclusion problems in bounded domains

Conditions under which vector fields with prescribed curl properties exist

Mathematical characterization of curl differential inclusions

Abstract

In this article, we study the existence of $\eta\in W_0^{1,\infty}(\Omega;\mathbb R^n)$ satisfying $$\textrm{curl} \ \eta\in E \textrm{ a.e. in }\Omega,$$ where $n\in \mathbb N, \Omega\subseteq \mathbb R^n$ is open, bounded and $E\subseteq \Lambda^2.$