Differential Inclusions for Gradient and Symmetrized Gradient Operators

TL;DR

This paper investigates the conditions for the existence of solutions to differential inclusion problems involving gradient and symmetrized gradient operators, focusing on minimal dimensionality and properties of symmetric matrices.

Contribution

It provides necessary and sufficient conditions for solutions in specific Sobolev spaces for problems involving gradient and symmetrized gradient operators, extending understanding of differential inclusions.

Findings

Characterization of solution existence conditions in minimal dimension

Analysis of properties of real symmetric matrices

Extension of differential inclusion theory to gradient operators

Abstract

In this article, we study the necessary and sufficient conditions for the existence of solutions in $W_0^{1,\infty}(\Omega;\mathbb R^n)$ in the minimal dimension of $\textrm{span }E$ for the following problem: \begin{equation*} P(D)u\in E \textrm{ a.e. in }\Omega, \end{equation*} where $P(D)= D$ or $D+D^{\top}$, and $E\subseteq \mathbb R^{n\times n}$ is a given set. We conclude this paper with some properties of real symmetric matrices.