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An overview of regularity results for the Laplacian and $p$-Laplacian in metric spaces

Ivan Yuri Violo

TL;DR

This survey analyzes regularity theory for the Laplacian and $p$-Laplacian on metric measure spaces, emphasizing interior Hölder and Lipschitz estimates and second-order regularity under local doubling, Poincaré, and lower Ricci curvature bounds via the CD/RCD framework. It juxtaposes the variational, boundary-value, and interior-regularity aspects in general spaces with the enhanced analytic structure provided by $ ext{RCD}(K,N)$ spaces, including Bochner-type inequalities and Hessian calculus. A central highlight is that in bounded $ ext{RCD}(K,\infty)$ spaces, if $\Delta u\in L^2$, then $u\in W^{2,2}$, and for the $p$-Laplacian, $\Delta_p u\in L^2$ implies $|D u|^{p-2}\nabla u\in H^{1,2}_C(T\mathbf{X})$ under suitable $p$, with a detailed four-step regularization argument. These results underpin Lipschitz regularity for broad classes of solutions and eigenfunctions, advancing nonlinear potential theory in non-smooth settings and suggesting directions for unique continuation and boundary regularity in metric spaces.

Abstract

We review some regularity results for the Laplacian and $p$-Laplacian in metric measure spaces. The focus is mainly on interior Hölder, Lipschitz and second-regularity estimates and on spaces supporting a Poincaré inequality or having Ricci curvature bounded below.

An overview of regularity results for the Laplacian and $p$-Laplacian in metric spaces

TL;DR

This survey analyzes regularity theory for the Laplacian and -Laplacian on metric measure spaces, emphasizing interior Hölder and Lipschitz estimates and second-order regularity under local doubling, Poincaré, and lower Ricci curvature bounds via the CD/RCD framework. It juxtaposes the variational, boundary-value, and interior-regularity aspects in general spaces with the enhanced analytic structure provided by spaces, including Bochner-type inequalities and Hessian calculus. A central highlight is that in bounded spaces, if , then , and for the -Laplacian, implies under suitable , with a detailed four-step regularization argument. These results underpin Lipschitz regularity for broad classes of solutions and eigenfunctions, advancing nonlinear potential theory in non-smooth settings and suggesting directions for unique continuation and boundary regularity in metric spaces.

Abstract

We review some regularity results for the Laplacian and -Laplacian in metric measure spaces. The focus is mainly on interior Hölder, Lipschitz and second-regularity estimates and on spaces supporting a Poincaré inequality or having Ricci curvature bounded below.

Paper Structure

This paper contains 17 sections, 36 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Standing assumptions and preliminaries
  3. First order theory for the Laplacian and p-Laplacian in metric setting
  4. Variational approach and integration by parts
  5. Boundary value problems
  6. Interior and boundary regularity of solutions
  7. Laplacian under Ricci lower bounds
  8. Second order regularity and the definition of RCD space
  9. Lipschitz regularity
  10. Eigenfunctions
  11. Unique continuation property
  12. Second order regularity for the -Laplacian
  13. Proof of second order regularity estimates for the p-Laplacian
  14. Step 1: regularization of the problem
  15. Step 2: Uniform a-priori estimates for regularized solutions
  16. ...and 2 more sections

Key Result

Proposition 3.3

A function $u\in W^{1,p}_\mathsf{loc}(\Omega)$ is $p$-subharmonic (resp. $p$-superharmonic) in $\Omega$ if and only if $\Delta_p \ge 0$ (resp. $\Delta_p\le 0$) in $\Omega$. In particular $u$ is $p$-harmonic if and only if $\Delta_p u=0.$

Theorems & Definitions (58)

  • Definition 3.1: $p$-harmonic functions
  • Definition 3.2: $p$-Laplacian
  • Proposition 3.3
  • Definition 3.4: Local Poincaré inequality
  • Proposition 3.5: Poincaré inequality with zero-Dirichlet boundary conditions
  • Proposition 3.6: Poisson equation
  • proof : Sketch of the proof
  • Proposition 3.7: Eigenvalue problem
  • Proposition 3.8: $p$-Electrostatic potential
  • Remark 3.9
  • ...and 48 more