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Sharp quantitative Talenti's inequality in particular cases

Paolo Acampora, Jimmy Lamboley

Abstract

In this paper, we focus on the famous Talenti's symmetrization inequality, more precisely its $L^p$ corollary asserting that the $L^p$-norm of the solution to $-Δv=f^\sharp$ is higher than the $L^p$-norm of the solution to $-Δu=f$ (we are considering Dirichlet boundary conditions, and $f^\sharp$ denotes the Schwarz symmetrization of $f:Ω\to\mathbb{R}_+$). 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 $L^p$-Talenti inequality with the sharp exponent 2.

Sharp quantitative Talenti's inequality in particular cases

Abstract

In this paper, we focus on the famous Talenti's symmetrization inequality, more precisely its corollary asserting that the -norm of the solution to is higher than the -norm of the solution to (we are considering Dirichlet boundary conditions, and denotes the Schwarz symmetrization of ). We focus on the particular case where functions 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 -Talenti inequality with the sharp exponent 2.

Paper Structure

This paper contains 21 sections, 36 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Tools and first results
  3. Symmetrization and Talenti's inequality
  4. Proof of Theorem \ref{['thm: pnorm']} for $p=1$
  5. Optimization among densities
  6. Sharpness of the exponent
  7. Proof of Theorem \ref{['thm: maintheorem']}
  8. Step 1: local stability implies global stability
  9. Step 2: \ref{['thm: fuglede']} implies \ref{['thm: maintheorem']}
  10. Proof of \ref{['thm: fuglede']}
  11. Computation of shape derivatives
  12. Step 2: coercivity of the second order shape derivative in $B_*$
  13. Step 3: improved continuity of the second order shape derivative
  14. Conclusion to the proof of \ref{['thm: fuglede']}
  15. The case $\mathcal{J}(E)=\lVert u_E\rVert_\infty$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $n\ge 1$, $p\in [1,+\infty]$, and $m\in(0,|B_1|)$. Then there exists $c=c(p,m,n)>0$ such that for every measurable set $E\subset B_1$ with $\lvert E\rvert=m$ we have where $B_*$ is the centered ball of volume $m$.

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • proof : Proof of \ref{['thm: pnorm']} from \ref{['thm: maintheorem']} when $p\in(1,\infty)$:
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 2.1: Talenti's comparison
  • Theorem 2.2
  • Definition 4
  • ...and 69 more