Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p \ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. For any small enough <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$r> \max \{p-1,1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Lambda > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> there exists a Lipschitz function u and a bounded vectorfield f such that<Equation ID="Equ15"> <EquationSource Format="TEX">$ {\left\{ \begin{array}{ll} \textrm{div}(|\nabla u|^{p-2} \nabla u) = \textrm{div} (f) \quad & \text {in }\mathbb {B}^2\\ u=0 & \text {on }\partial \mathbb {B}^2 \end{array}\right. } $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mtext>div</mtext> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>in</mtext> <mspace width="0.333333em" /> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>on</mtext> <mspace width="0.333333em" /> <mi>∂</mi> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>but <Equation ID="Equ16"> <EquationSource Format="TEX">$ \int _{\mathbb {B}^2} |\nabla u|^r \not \le \Lambda \int _{\mathbb {B}^2} |f|^{\frac{r}{p-1}}. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mn>2</mn> </msup> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> </msup> <mo>≰</mo> <mi mathvariant="normal">Λ</mi> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mn>2</mn> </msup> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi>f</mi> <mo stretchy="false">|</mo> </mrow> <mfrac> <mi>r</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </msup> <mo>.</mo> </mrow> </math></EquationSource> </Equation>This disproves a conjecture by Iwaniec from 1983.

Failure of Calderón-Zygmund Estimates for the p-Laplace Equation

期刊论文
发表日期：

2026年4月2日

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Armin Schikorra

摘要

Let $p \ne 2$ $p \neq 2$ . For any small enough $r> \max \{p-1,1\}$ $r > max {p - 1, 1}$ and for any $\Lambda > 1$ $Λ > 1$ there exists a Lipschitz function u and a bounded vectorfield f such that $ {\left\{ \begin{array}{ll} \textrm{div}(|\nabla u|^{p-2} \nabla u) = \textrm{div} (f) \quad & \text {in }\mathbb {B}^2\\ u=0 & \text {on }\partial \mathbb {B}^2 \end{array}\right. } $ $\{\begin{matrix} div (| \nabla u |^{p - 2} \nabla u) = div (f) & in B^{2} \\ u = 0 & on \partial B^{2} \end{matrix})$ but $ \int _{\mathbb {B}^2} |\nabla u|^r \not \le \Lambda \int _{\mathbb {B}^2} |f|^{\frac{r}{p-1}}. $ $\int_{B^{2}} {| \nabla u |}^{r} ≰ Λ \int_{B^{2}} {| f |}^{\frac{r}{p - 1}} .$ This disproves a conjecture by Iwaniec from 1983.