Differential Inclusions for Gradient and Symmetrized Gradient Operators

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