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Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels

Helmut Abels, Christoph Hurm, Patrik Knopf

Abstract

We study nonlocal convolution-type operators with singular, possibly anisotropic kernels. Our main objective is to establish and quantify their nonlocal-to-local convergence to a local differential operator with natural boundary conditions, as the kernels concentrate at the origin in a suitable way. Such convergence results provide a useful tool for the physical justification of mathematical models, particularly in situations where the desired local differential operator cannot be directly derived from microscopic laws. The present work substantially extends previous results by allowing kernels with stronger singularities (comparable to those of fractional Laplacians), anisotropic and non-localized kernels, and by proving strong convergence in general $L^p$ spaces together with explicit convergence rates.

Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels

Abstract

We study nonlocal convolution-type operators with singular, possibly anisotropic kernels. Our main objective is to establish and quantify their nonlocal-to-local convergence to a local differential operator with natural boundary conditions, as the kernels concentrate at the origin in a suitable way. Such convergence results provide a useful tool for the physical justification of mathematical models, particularly in situations where the desired local differential operator cannot be directly derived from microscopic laws. The present work substantially extends previous results by allowing kernels with stronger singularities (comparable to those of fractional Laplacians), anisotropic and non-localized kernels, and by proving strong convergence in general spaces together with explicit convergence rates.
Paper Structure (11 sections, 7 theorems, 41 equations)

This paper contains 11 sections, 7 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Main objective of the paper.
  3. State of the art and related results.
  4. Novelties and main results of our paper.
  5. Structure of the paper.
  6. Assumptions and preliminaries
  7. Nonlocal-to-local convergence of the nonlocal operator
  8. Well-definedness of the nonlocal operator
  9. Convergence on the full space
  10. Convergence on a curved half space
  11. Convergence on a domain with compact boundary

Key Result

Lemma 2.4

Suppose that Assumption ASS:RHO holds.

Theorems & Definitions (10)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Corollary 3.5
  • Theorem 3.6