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Second-order estimates for the $p$-Laplacian in RCD spaces

Luca Benatti, Ivan Yuri Violo

Abstract

We establish quantitative second-order Sobolev regularity for functions having a $2$-integrable $p$-Laplacian in bounded RCD spaces, with $p$ in a suitable range. In the finite-dimensional case, we also obtain Lipschitz regularity under the assumption that $p$-Laplacian is sufficiently integrable. Our results cover both $p$-Laplacian eigenfunctions and $p$-harmonic functions having relatively compact level sets.

Second-order estimates for the $p$-Laplacian in RCD spaces

Abstract

We establish quantitative second-order Sobolev regularity for functions having a -integrable -Laplacian in bounded RCD spaces, with in a suitable range. In the finite-dimensional case, we also obtain Lipschitz regularity under the assumption that -Laplacian is sufficiently integrable. Our results cover both -Laplacian eigenfunctions and -harmonic functions having relatively compact level sets.
Paper Structure (20 sections, 42 theorems, 267 equations)

This paper contains 20 sections, 42 theorems, 267 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. First order calculus on metric measure spaces
  4. p-Poisson equation in metric measure spaces
  5. Second-order calculus on RCD spaces
  6. Sobolev-Poincaré inequalities in RCD spaces
  7. Approximation results for the p-Laplacian
  8. Approximation via regularized p-Laplacian operators
  9. Approximation via regularized source term
  10. Uniform a priori estimates for _(p,)
  11. Preliminary properties and estimates for the developed (p,)-Laplacian
  12. Second-order regularity estimates
  13. Gradient estimates
  14. Existence of regular solutions of _(p,)=f
  15. Existence for an auxiliary operator
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $({\rm X},{\sf d},\mathfrak{m})$ be an $\mathrm{RCD}(K,\infty)$ space with ${\rm diam}({\rm X}) \le D<\infty$. Fix $p \in \mathcal{RI}_{{\rm X}}$ and suppose that $u \in {\sf D}(\Delta_p)$ with $\Delta_p u \in L^2(\mathfrak{m})$. Then $|\nabla u|^{p-2}\nabla u\in H^{1,2}_{C}(T{\rm X})$ and in where $C>0$ is a constant depending only on $p,K$ and $D.$

Theorems & Definitions (96)

  • Definition 1.1: Regularity interval
  • Theorem 1.2: Regularity of $p$-Laplacian, $N=\infty$
  • Theorem 1.3: Regularity of $p$-Laplacian, $N<\infty$
  • Corollary 1.4: Regularity of $p$-eigenfunctions
  • Corollary 1.5: Regularity of $p$-harmonic function with relatively compact level sets
  • Remark 2.1
  • Definition 2.2: PI space
  • Theorem 2.3: Sobolev and Poincaré inequalities
  • Theorem 2.4: $(p,p)$-Poincaré inequality
  • proof
  • ...and 86 more