<p>This paper is concerned with weak solutions of elliptic equations and of systems of the form <Equation ID="Equ52"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\sum _{i=1}^{n}D_i \left( A^\alpha _i\left( x,Du(x)\right) \right) = 0, \ \ & \textrm{in }\ \Omega , \ \ \forall \ \alpha \in \left\{ 1, \cdots , m\right\} ,\\ u(x)=u_(x), & \textrm{on }\ \partial \Omega . \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>-</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>D</mi> <mi>i</mi> </msub> <mfenced close=")" open="("> <msubsup> <mi>A</mi> <mi>i</mi> <mi>α</mi> </msubsup> <mfenced close=")" open="("> <mi>x</mi> <mo>,</mo> <mi>D</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mfenced> </mfenced> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> </mrow> </mstyle> </mtd> <mtd columnalign="left"> <mrow> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="4pt" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <mo>∀</mo> <mspace width="4pt" /> <mi>α</mi> <mo>∈</mo> <mfenced close="}" open="{"> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>m</mi> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mmultiscripts> <mi>u</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>on</mtext> <mspace width="0.333333em" /> <mspace width="4pt" /> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>We show that, assuming some high degree of integrability of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Du_$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mmultiscripts> <mi>u</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation> the gradient of the boundary datum, we can obtain bounds of the difference <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ |u^\gamma -u^\gamma _* |_\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi>u</mi> <mi>γ</mi> </msup> <mo>-</mo> <mmultiscripts> <mi>u</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mi>γ</mi> </mmultiscripts> <msub> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>∞</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma \in \left\{ 1, \cdots , m\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mfenced close="}" open="{"> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>m</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. In particular, when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, things are easier and we can get a better result.</p>

On the difference between weak solution and boundary datum in some elliptic equations and systems

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2025年12月28日

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Aiping Zhang,
Renato Colucci,
Francesco Leonetti,
Hongya Gao

摘要

This paper is concerned with weak solutions of elliptic equations and of systems of the form $\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\sum _{i=1}^{n}D_i \left( A^\alpha _i\left( x,Du(x)\right) \right) = 0, \ \ & \textrm{in }\ \Omega , \ \ \forall \ \alpha \in \left\{ 1, \cdots , m\right\} ,\\ u(x)=u_*(x), & \textrm{on }\ \partial \Omega . \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} - \sum_{i = 1}^{n} D_{i} (A_{i}^{α}, (x, D u (x))) = 0, & in Ω, \forall α \in \{1, \dots, m\}, \\ u (x) = u_{*} (x), & on \partial Ω . \end{matrix}) \end{matrix}$ We show that, assuming some high degree of integrability of $Du_*$ $D u_{*}$ the gradient of the boundary datum, we can obtain bounds of the difference $ |u^\gamma -u^\gamma _* |_\infty $ $| u^{γ} - u_{*}^{γ} |_{\infty}$ , $\gamma \in \left\{ 1, \cdots , m\right\} $ $γ \in \{1, \dots, m\}$ . In particular, when $m=1$ $m = 1$ , things are easier and we can get a better result.