<p>In this paper, we focus on the famous Talenti’s symmetrization inequality, more precisely, its <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> corollary which asserts that the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-norm of the solution to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$-\Delta v=f^\sharp $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>♯</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> is higher than the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-norm of the solution to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$-\Delta u=f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> (we are considering Dirichlet boundary conditions, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f^\sharp $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mo>♯</mo> </msup> </math></EquationSource> </InlineEquation> denotes the Schwarz symmetrization of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f:\Omega \rightarrow \mathbb {R}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation>). We focus on the particular case where functions <i>f</i> are defined on the unit ball, and are characteristic functions of a subset of this unit ball. We show in this case that stability occurs for the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-Talenti inequality with the sharp exponent 2.</p>

Sharp Quantitative Talenti Inequality in Particular Cases

期刊论文
发表日期：

2026年4月3日

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P. Acampora,
J. Lamboley

摘要

In this paper, we focus on the famous Talenti’s symmetrization inequality, more precisely, its $L^p$ $L^{p}$ corollary which asserts that the $L^p$ $L^{p}$ -norm of the solution to $-\Delta v=f^\sharp $ $- Δ v = f^{♯}$ is higher than the $L^p$ $L^{p}$ -norm of the solution to $-\Delta u=f$ $- Δ u = f$ (we are considering Dirichlet boundary conditions, and $f^\sharp $ $f^{♯}$ denotes the Schwarz symmetrization of $f:\Omega \rightarrow \mathbb {R}_+$ $f : Ω \to R_{+}$ ). We focus on the particular case where functions f are defined on the unit ball, and are characteristic functions of a subset of this unit ball. We show in this case that stability occurs for the $L^p$ $L^{p}$ -Talenti inequality with the sharp exponent 2.