<p>In this article, we study the necessary and sufficient conditions for the existence of solutions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$W_0^{1,\infty }(\Omega ;\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mn>0</mn> <mrow> <mn>1</mn> <mo>,</mo> <mi>∞</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the minimal dimension of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\operatorname {span}E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>span</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> for the following problem: <Equation ID="Equ10"> <EquationSource Format="TEX">$\begin{aligned} P(D)u\in E \text { a.e. in }\Omega , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>∈</mo> <mi>E</mi> <mspace width="0.333333em" /> <mtext>a.e. in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$P(D)= D$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>D</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$D+D^{\top }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>+</mo> <msup> <mi>D</mi> <mi>⊤</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$E\subseteq \mathbb {R}^{n\times n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is a given set. We conclude this article with some properties of real symmetric matrices.</p>

Differential inclusions for gradient and symmetrized gradient operators

期刊论文
发表日期：

2026年4月9日

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Nurun Nesha

摘要

In this article, we study the necessary and sufficient conditions for the existence of solutions in $W_0^{1,\infty }(\Omega ;\mathbb {R}^n)$ $W_{0}^{1, \infty} (Ω; R^{n})$ in the minimal dimension of $\operatorname {span}E$ $span E$ for the following problem: $\begin{aligned} P(D)u\in E \text { a.e. in }\Omega , \end{aligned}$ $\begin{matrix} P (D) u \in E a.e. in Ω, \end{matrix}$ where $P(D)= D$ $P (D) = D$ or $D+D^{\top }$ $D + D^{⊤}$ , and $E\subseteq \mathbb {R}^{n\times n}$ $E \subseteq R^{n \times n}$ is a given set. We conclude this article with some properties of real symmetric matrices.