The aim of this paper is to derive some Calderón-Zygmund estimates for local solutions u of nonlinear elliptic systems of the type <Equation ID="Equ102"> <EquationSource Format="TEX">$\begin{aligned} \textrm{div}\textbf{A}(x, Du) = \textrm{div}|\textbf{G}|^{p - 2}\textbf{G} \qquad \textrm{in} \quad \Omega \subset \mathbb {R}^n, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mtext>div</mtext> <mi mathvariant="bold">A</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>D</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mtext>div</mtext> <mo stretchy="false">|</mo> <mi mathvariant="bold">G</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi mathvariant="bold">G</mi> <mspace width="2em" /> <mtext>in</mtext> <mspace width="1em" /> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a bounded domain. We assume that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u: \Omega \rightarrow \mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> belongs to a weighted Sobolev space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$W_{loc}^{1, p}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>W</mi> <mrow> <mi mathvariant="italic">loc</mi> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p \in \left( \frac{2n}{n + 2}, 2\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <mn>2</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textbf{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">G</mi> </math></EquationSource> </InlineEquation> belongs to a weighted <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L_{loc}^{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mrow> <mi mathvariant="italic">loc</mi> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> space and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x \rightarrow \textbf{A}(x, \xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="bold">A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has growth coefficients in the John and Nirenberg space BMO. As an application of similar techniques, we also consider weak solutions u of linear elliptic systems of the type <Equation ID="Equ103"> <EquationSource Format="TEX">$\begin{aligned} \textrm{div}(A(x)Du) = \textrm{div}\, G, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mtext>div</mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>D</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext>div</mtext> <mspace width="0.166667em" /> <mi>G</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where the matrix A(x) lies in the space BMO. In this case, an improvement of the admissible exponent range in the Calderón-Zygmund estimate is obtained.

CZ estimates for some linear and nonlinear elliptic systems with discontinuous coefficients

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2026年7月1日

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Giulio Pascale

摘要

The aim of this paper is to derive some Calderón-Zygmund estimates for local solutions u of nonlinear elliptic systems of the type $\begin{aligned} \textrm{div}\textbf{A}(x, Du) = \textrm{div}|\textbf{G}|^{p - 2}\textbf{G} \qquad \textrm{in} \quad \Omega \subset \mathbb {R}^n, \end{aligned}$ $\begin{matrix} div A (x, D u) = {div | G |}^{p - 2} G in Ω \subset R^{n}, \end{matrix}$ where $\Omega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ is a bounded domain. We assume that $u: \Omega \rightarrow \mathbb {R}^N$ $u : Ω \to R^{N}$ belongs to a weighted Sobolev space $W_{loc}^{1, p}$ $W_{loc}^{1, p}$ , with $p \in \left( \frac{2n}{n + 2}, 2\right) $ $p \in (\frac{2 n}{n + 2}, 2)$ , $\textbf{G}$ $G$ belongs to a weighted $L_{loc}^{p}$ $L_{loc}^{p}$ space and $x \rightarrow \textbf{A}(x, \xi )$ $x \to A (x, ξ)$ has growth coefficients in the John and Nirenberg space BMO. As an application of similar techniques, we also consider weak solutions u of linear elliptic systems of the type $\begin{aligned} \textrm{div}(A(x)Du) = \textrm{div}\, G, \end{aligned}$ $\begin{matrix} div (A (x) D u) = div G, \end{matrix}$ where the matrix A(x) lies in the space BMO. In this case, an improvement of the admissible exponent range in the Calderón-Zygmund estimate is obtained.