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A Talenti-type comparison theorem for the $p$-Laplacian on $\mathrm{RCD}(K,N)$ spaces and some applications

Wenjing Wu

Abstract

In this paper, we prove a Talenti-type comparison theorem for the $p$-Laplacian with Dirichlet boundary conditions on open subsets of a $\mathrm{RCD}(K,N)$ space with $K>0$ and $N\in (1,\infty)$. The obtained Talenti-type comparison theorem is sharp, rigid and stable with respect to measured Gromov-Hausdorff topology. As an application of such Talenti-type comparison, we establish a sharp and rigid reverse Hölder inequality for first eigenfunctions of the $p$-Laplacian and a related quantitative stability result.

A Talenti-type comparison theorem for the $p$-Laplacian on $\mathrm{RCD}(K,N)$ spaces and some applications

Abstract

In this paper, we prove a Talenti-type comparison theorem for the -Laplacian with Dirichlet boundary conditions on open subsets of a space with and . The obtained Talenti-type comparison theorem is sharp, rigid and stable with respect to measured Gromov-Hausdorff topology. As an application of such Talenti-type comparison, we establish a sharp and rigid reverse Hölder inequality for first eigenfunctions of the -Laplacian and a related quantitative stability result.
Paper Structure (17 sections, 38 theorems, 226 equations)

This paper contains 17 sections, 38 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemannian curvature-dimension condition
  4. Local Sobolev spaces and BV functions
  5. Measured Gromov-Hausdorff convergence and stability of $\operatorname{RCD}(K, N)$
  6. 1-dimensional model spaces
  7. Rearrangements and symmetrizations
  8. Poisson problem on the model space
  9. The Talenti-type comparison theorem
  10. Proof of the Talenti-type comparison theorem
  11. Rigidity in the Talenti-type theorem
  12. Almost rigidity in the Talenti-type theorem
  13. The continuity of the local Poisson problem
  14. Applications
  15. Improved Sobolev embeddings
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $p,q\in (1,\infty)$ be conjugate exponents. Let $(\mathrm{X}, \mathrm{d}, \mathfrak{m})$ be a $\operatorname{RCD}(K, N)$ space with $K>0$, $N \in(1, \infty)$, and $\Omega \subset \mathrm{X}$ an open set with $\mathfrak{m}(\Omega)=v \in(0,1)$. Let $f \in L^q(\Omega)$. Assume that $u \in W_0^{1,p

Theorems & Definitions (79)

  • Theorem 1.1: A Talenti-type comparison theorem
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.1: Basic properties of $\mathrm{CD}(K, N)$ spaces
  • Definition 2.2: $q$-test plan
  • Definition 2.3: Sobolev classes
  • Definition 2.4
  • Remark 2.2
  • Definition 2.5
  • Lemma 2.3
  • ...and 69 more