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Stability of the Gaussian Faber-Krahn inequality

Alessandro Carbotti, Simone Cito, Domenico Angelo La Manna, Diego Pallara

Abstract

We prove a quantitative version of the Gaussian Faber-Krahn type inequality proved by Betta, Chiacchio and Ferone for the first Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator, estimating the deficit in terms of the Gaussian Fraenkel asymmetry. As expected, the multiplicative constant only depends on the prescribed Gaussian measure.

Stability of the Gaussian Faber-Krahn inequality

Abstract

We prove a quantitative version of the Gaussian Faber-Krahn type inequality proved by Betta, Chiacchio and Ferone for the first Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator, estimating the deficit in terms of the Gaussian Fraenkel asymmetry. As expected, the multiplicative constant only depends on the prescribed Gaussian measure.
Paper Structure (5 sections, 5 theorems, 88 equations)

This paper contains 5 sections, 5 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Properties of eigenvalues and eigenfunctions of $-\Delta_\gamma$
  4. Local bilipschitz continuity of the Faber-Krahn profile
  5. Proof of the Main Theorem

Key Result

Lemma 2.1

Let $\Omega\subset\mathbb{R}^N$ be an open connected set with $\gamma(\Omega)<1$. Then, we have that

Theorems & Definitions (9)

  • Lemma 2.1
  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof : Proof of the Main Theorem