<p>This note is devoted to the failure of the Calderon-Zygmund theory for linear differential operators with discontinuous coefficients. It is known that the theory holds if the data belong to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^{m}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>m</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1 < m \le \frac{2N}{N+2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>m</mi> <mo>≤</mo> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> (see Boccardo and Gallouët in Commun Partial Diff Equ 17:641–655, 1992). In this paper we prove that the theory fails if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{2N}{N+2}< m < N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <mo><</mo> <mi>m</mi> <mo><</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, thus extending to the case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frac{2N}{N+2}< m < \frac{N}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <mo><</mo> <mi>m</mi> <mo><</mo> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> the results of Boccardo (Atti Accad Naz Lincei Rend Lincei Mat Appl 26:215–221, 2015).</p>

Failure of the Calderon–Zygmund theory for linear elliptic equations with discontinuous coefficients

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2026年4月17日

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Lucio Boccardo,
Luigi Orsina

摘要

This note is devoted to the failure of the Calderon-Zygmund theory for linear differential operators with discontinuous coefficients. It is known that the theory holds if the data belong to $L^{m}(\Omega )$ $L^{m} (Ω)$ , with $1 < m \le \frac{2N}{N+2}$ $1 < m \leq \frac{2 N}{N + 2}$ (see Boccardo and Gallouët in Commun Partial Diff Equ 17:641–655, 1992). In this paper we prove that the theory fails if $\frac{2N}{N+2}< m < N$ $\frac{2 N}{N + 2} < m < N$ , thus extending to the case $\frac{2N}{N+2}< m < \frac{N}{2}$ $\frac{2 N}{N + 2} < m < \frac{N}{2}$ the results of Boccardo (Atti Accad Naz Lincei Rend Lincei Mat Appl 26:215–221, 2015).