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The Talenti comparison result in a quantitative form

Vincenzo Amato, Rosa Barbato, Alba Lia Masiello, Gloria Paoli

Abstract

In this paper, we obtain a quantitative version of the classical comparison result of Talenti for elliptic problems with Dirichlet boundary conditions. The key role is played by quantitative versions of the Pólya-Szego inequality and of the Hardy-Littlewood inequality.

The Talenti comparison result in a quantitative form

Abstract

In this paper, we obtain a quantitative version of the classical comparison result of Talenti for elliptic problems with Dirichlet boundary conditions. The key role is played by quantitative versions of the Pólya-Szego inequality and of the Hardy-Littlewood inequality.
Paper Structure (10 sections, 22 theorems, 191 equations)

This paper contains 10 sections, 22 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Rearrangement of functions
  4. Some quantitative inequalities
  5. Rescaling properties
  6. Proof of Theorem \ref{['asimmetryindex']}
  7. Proof of the Theorem \ref{['confrontol1']}
  8. Proof of Theorem \ref{['confrontof']}
  9. Conclusions and Open Problems
  10. Appendix: $L^2-$ bound of the asymmetry

Key Result

Theorem 1.1

Let $\Omega$ be a bounded open set of $\mathop{\mathrm{\mathbb{R}}}\nolimits^n$ and let $f$ be a non-negative function in $L^2(\Omega)$. Suppose that $u$ and $v$ are the solutions to mainproblem and symmetricproblem, respectively. Then, there exist some positive constants $\theta_1=\theta_1(n), \the Moreover, the dependence of $C_1,\, C_2$ and $C_3$ from ${\left|\Omega\right|}$ and $f^\sharp$ is

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Coarea formula
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Lemma 2.2
  • ...and 35 more