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Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications

Zhaolong Han, Xiaochuan Tian

Abstract

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection-diffusion, nonlocal correspondence model of linear elasticity, and nonlocal Helmholtz decomposition on bounded domains.

Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications

Abstract

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection-diffusion, nonlocal correspondence model of linear elasticity, and nonlocal Helmholtz decomposition on bounded domains.
Paper Structure (16 sections, 38 theorems, 220 equations)

This paper contains 16 sections, 38 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Nonlocal half-ball vector operators
  3. Definitions and integration by parts
  4. Distributional nonlocal operators
  5. Fourier symbols of nonlocal operators
  6. Nonlocal vector identities for smooth functions
  7. Nonlocal Sobolev-type spaces
  8. Definitions and properties of nonlocal Sobolev-type spaces
  9. Nonlocal vector inequalities and identities
  10. Nonlocal Poincaré inequality for integrable kernels with compact support
  11. Nonlocal Poincaré inequality for general kernel functions
  12. Applications
  13. Nonlocal convection-diffusion equation
  14. Nonlocal correspondence models of isotropic linear elasticity
  15. Nonlocal Helmholtz decomposition
  16. ...and 1 more sections

Key Result

Lemma 2.1

Suppose that $u\in C^\infty_c(\mathbb{R}^d)$ and $\bm v\in C^\infty_c(\mathbb{R}^d;\mathbb{R}^d)$. Then $\mathcal{G}^{\bm\nu}_w u$, $\mathcal{D}_w^{\bm\nu} \bm v$ and $\mathcal{C}_w^{\bm\nu} \bm v$ ($d=3$) are $C^\infty$ functions with and if $d=3$, For $p\in [1,\infty]$ and multi-index $\alpha\in \mathbb{N}^d$, there is a constant $C$ depending on $p$ such that the following estimates hold: an

Theorems & Definitions (81)

  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.1: Nonlocal "half-ball" integration by parts
  • Remark 2.3
  • Definition 2.2
  • Remark 2.4
  • Corollary 2.1
  • ...and 71 more