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Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea, J. L. Varona

TL;DR

The paper studies nonlocal discrete diffusion on the one-dimensional lattice $\mathbb{Z}_h$ driven by the fractional power $(-\Delta_h)^s$ with $0<s<1$, deriving a pointwise nonlocal kernel, a discrete mean value property, and an extension problem interpretation. It develops a complete regularity framework in discrete Hölder spaces, proves existence/uniqueness and discrete Sobolev/Poincaré inequalities for the nonlocal Dirichlet problem, and constructs the negative power $(-\Delta_h)^{-s}$ with a discrete Hardy-Littlewood-Sobolev bound. It then proves that $(-\Delta_h)^s$ converges to the continuous operator $(-\Delta)^s$ as $h\to0$ with explicit Hölder-based error rates, and that continuous fractional Poisson problems can be approximated by discrete Dirichlet problems with uniform control. The work also establishes a discrete maximum principle and a robust semigroup-based framework tying together kernel representations, Bessel-function heat kernels, and extension problems for both positive and negative fractional powers.

Abstract

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h>0$ \[ (-Δ_h)^su=f, \] for $u,f:\mathbb{Z}_h\to\mathbb{R}$, $0<s<1$, is performed. The pointwise nonlocal formula for $(-Δ_h)^su$ and the nonlocal discrete mean value property for discrete $s$-harmonic functions are obtained. We observe that a characterization of $(-Δ_h)^s$ as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution) \[ u=(-Δ_h)^{-s}f, \] which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy--Littlewood--Sobolev inequality for $(-Δ_h)^{-s}$. As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as $h\to0$ in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of $h$ under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian \[ (-Δ)^sU=F, \] in $\mathbb{R}$, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of~$h$.

Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

TL;DR

The paper studies nonlocal discrete diffusion on the one-dimensional lattice driven by the fractional power with , deriving a pointwise nonlocal kernel, a discrete mean value property, and an extension problem interpretation. It develops a complete regularity framework in discrete Hölder spaces, proves existence/uniqueness and discrete Sobolev/Poincaré inequalities for the nonlocal Dirichlet problem, and constructs the negative power with a discrete Hardy-Littlewood-Sobolev bound. It then proves that converges to the continuous operator as with explicit Hölder-based error rates, and that continuous fractional Poisson problems can be approximated by discrete Dirichlet problems with uniform control. The work also establishes a discrete maximum principle and a robust semigroup-based framework tying together kernel representations, Bessel-function heat kernels, and extension problems for both positive and negative fractional powers.

Abstract

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size for , , is performed. The pointwise nonlocal formula for and the nonlocal discrete mean value property for discrete -harmonic functions are obtained. We observe that a characterization of as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution) which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy--Littlewood--Sobolev inequality for . As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian in , can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of~.

Paper Structure

This paper contains 15 sections, 24 theorems, 212 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['thm:basicProperties']}
  3. Proof of Theorem \ref{['lem:kernelFractionalIntegral']}
  4. Proof of Theorems \ref{['thm:Holder']} and \ref{['thm:consistencia1d']}
  5. Proof of Theorem \ref{['thm:Holder']}
  6. Proof of Theorem \ref{['thm:consistencia1d']}
  7. Proof of Theorem \ref{['RegularityfractionalIntegral']}
  8. The Dirichlet problem for the fractional discrete Laplacian
  9. The discrete maximum principle
  10. Proof of Theorem \ref{['thm:Convergence-result']}
  11. Technical lemmas, Bessel functions, the continuous Poisson problem and the fractional discrete Sobolev and Poincaré inequalities
  12. Some technical lemmas
  13. Properties of Bessel functions $I_k$
  14. The fractional discrete Sobolev and Poincaré inequalities
  15. The continuous Poisson problem

Key Result

Theorem 1.1

For $0\leq s\leq1$, we let

Theorems & Definitions (50)

  • Theorem 1.1: Pointwise nonlocal formula and limits
  • Remark 1.2: Mean value formula and probabilistic interpretation
  • Theorem 1.3: Fundamental solution and Hardy--Littlewood--Sobolev inequality
  • Remark 1.4: Extension problem for $(-\Delta_h)^s$ and $(-\Delta_h)^{-s}$
  • Theorem 1.5: Fractional discrete Laplacian in discrete Hölder spaces
  • Theorem 1.6: Discrete Schauder estimates
  • Theorem 1.7: Uniform comparison with fractional Laplacian
  • Theorem 1.8: Convergence of discrete solutions to continuous ones
  • Remark 2.1
  • Definition 4.1: Continuous Hölder spaces
  • ...and 40 more