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The convergence rate of $p$-harmonic to infinity-harmonic functions

Leon Bungert

Abstract

The purpose of this paper is to prove a uniform convergence rate of the solutions of the $p$-Laplace equation $Δ_p u = 0$ with Dirichlet boundary conditions to the solution of the infinity-Laplace equation $Δ_\infty u = 0$ as $p\to\infty$. The rate scales like $p^{-1/4}$ for general solutions of the Dirichlet problem and like $p^{-1/2}$ for solutions with positive gradient. An explicit example shows that it cannot be better than $p^{-1}$. The proof of this result solely relies on the comparison principle with the fundamental solutions of the $p$-Laplace and the infinity-Laplace equation, respectively. Our argument does not use viscosity solutions, is purely metric, and is therefore generalizable to more general settings where a comparison principle with Hölder cones and Hölder regularity is available.

The convergence rate of $p$-harmonic to infinity-harmonic functions

Abstract

The purpose of this paper is to prove a uniform convergence rate of the solutions of the -Laplace equation with Dirichlet boundary conditions to the solution of the infinity-Laplace equation as . The rate scales like for general solutions of the Dirichlet problem and like for solutions with positive gradient. An explicit example shows that it cannot be better than . The proof of this result solely relies on the comparison principle with the fundamental solutions of the -Laplace and the infinity-Laplace equation, respectively. Our argument does not use viscosity solutions, is purely metric, and is therefore generalizable to more general settings where a comparison principle with Hölder cones and Hölder regularity is available.
Paper Structure (10 sections, 8 theorems, 75 equations)

This paper contains 10 sections, 8 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Notation
  4. Solutions of the p-Laplace and infinity-Laplace equations
  5. Main result
  6. Extensions of our main result
  7. Proof of the main result
  8. Perturbations of infinity-harmonic functions
  9. Approximate consistency of p-harmonic functions
  10. Convergence rates

Key Result

Proposition 2.1

Let $u_p$ solve eq:p-Laplace_equation for $d<p<\infty$ or eq:inf-Laplace_equation for $p=\infty$. Furthermore, define the function Then for all domains $D\Subset\Omega$, compactly contained in $\Omega$, for all $a\geq 0$, and for all $x_0\in\mathbb{R}^d\setminus D$ it holds

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1: Existence and uniqueness
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 1: Explicit convergence rate
  • Remark 2.3: Boundary term
  • Remark 2.4: Asymptotic rate
  • Remark 2.5: The assumption $\mathsf H < \infty$
  • ...and 20 more