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Nonlocal Boundary Value Problems Governed by Symmetric Nonlocal Operators

Leonhard Frerick, Julia Huschens, Michael Vu

TL;DR

The paper develops a comprehensive variational theory for nonlocal boundary value problems driven by symmetric transition kernels, extending beyond the standard diffusion kernel to a general operator $\mathcal{L}u = \mathrm{PV} \int_{\mathbb{R}^d} (u(x)-u(y)) \, K(x,\mathrm{d}y)$. It introduces a notion of $\lambda$-symmetry that enables a nonlocal integration-by-parts identity, and constructs nonlocal function spaces $V$ and $V_0$ with a bilinear form $\mathcal{B}$ to formulate weak solutions for Dirichlet and Neumann problems. Central to the analysis are nonlocal Friedrichs and Poincaré-type inequalities, which ensure well-posedness via the Lax–Milgram theorem and provide stability and uniqueness up to the natural kernel. The theory is illustrated by connecting to the discrete Poisson problem through stencil operators, yielding matrix formulations and a discrete maximum principle that align with the continuous nonlocal framework. This work broadens the scope of nonlocal models and offers a rigorous foundation for both analysis and discretization of nonlocal boundary value problems.

Abstract

Nonlocal boundary value problems with Dirichlet or Neumann boundary are well-studied for nonlocal operators of the type $\mathcal{L}_γu = \operatorname{PV} \int_{\mathbb{R}^d} \big(u(\cdot)-u(y)\big) γ(\cdot,y) \, \mathrm{d}y$ where the underlying kernel function $γ: \mathbb{R}^d \times \mathbb{R}^d \rightarrow [0,\infty)$ is assumed to be measurable and symmetric. In this paper, a theory is introduced for problems whose governing operator is of the more general type \[\mathcal{L}u:= \operatorname{PV} \int_{\mathbb{R}^d}\big(u(\cdot)-u(y)\big) \, K(\cdot, \mathrm{d}y)\] where ${K: \mathbb{R}^d \times \mathcal{B}(\mathbb{R}^d) \rightarrow [0,\infty]}$ is a symmetric transition kernel. Our main focus is on nonlocal Dirichlet and Neumann problems and a classical Hilbert space approach is developed for solving designated weak formulations. As an example, the discrete Poisson problem on $Ω=(0,1)^d$ is discussed.

Nonlocal Boundary Value Problems Governed by Symmetric Nonlocal Operators

TL;DR

The paper develops a comprehensive variational theory for nonlocal boundary value problems driven by symmetric transition kernels, extending beyond the standard diffusion kernel to a general operator . It introduces a notion of -symmetry that enables a nonlocal integration-by-parts identity, and constructs nonlocal function spaces and with a bilinear form to formulate weak solutions for Dirichlet and Neumann problems. Central to the analysis are nonlocal Friedrichs and Poincaré-type inequalities, which ensure well-posedness via the Lax–Milgram theorem and provide stability and uniqueness up to the natural kernel. The theory is illustrated by connecting to the discrete Poisson problem through stencil operators, yielding matrix formulations and a discrete maximum principle that align with the continuous nonlocal framework. This work broadens the scope of nonlocal models and offers a rigorous foundation for both analysis and discretization of nonlocal boundary value problems.

Abstract

Nonlocal boundary value problems with Dirichlet or Neumann boundary are well-studied for nonlocal operators of the type where the underlying kernel function is assumed to be measurable and symmetric. In this paper, a theory is introduced for problems whose governing operator is of the more general type where is a symmetric transition kernel. Our main focus is on nonlocal Dirichlet and Neumann problems and a classical Hilbert space approach is developed for solving designated weak formulations. As an example, the discrete Poisson problem on is discussed.
Paper Structure (16 sections, 29 theorems, 213 equations, 1 figure)

This paper contains 16 sections, 29 theorems, 213 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Notion Of Symmetry
  3. Variational Analysis
  4. Nonlocal Function Spaces
  5. A Variational Formulation
  6. Nonlocal Poincaré-Type Inequalities
  7. The Nonlocal Friedrichs Inequality
  8. The Nonlocal Poincaré Inequality
  9. The Nonlocal Dirichlet Problem
  10. The Nonlocal Neumann Problem
  11. A Weak Maximum Principle
  12. Example – The Discrete Poisson Problem
  13. The Dirichlet Problem
  14. The Neumann Problem
  15. The Discrete Maximum Principle
  16. ...and 1 more sections

Key Result

Lemma 2.5

Let $\lambda$ be a Borel measure on $\mathbb{R}^d$ and let $K \in \mathcal{K}$. Then, $K$ is $\lambda$-symmetric if and only if for all $f \in \mathcal{M}$ quasi-integrable with respect to $\lambda \otimes K$, it is valid that

Figures (1)

  • Figure 1: Illustration of the discretization of the unit cube $\overline{\Omega}= [0,1]^2$ in $d=2$. Note that the grid nodes in $(\partial \Omega)_h$ are contained in the nonlocal boundary $\Gamma$.

Theorems & Definitions (73)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Lemma 3.1
  • proof
  • ...and 63 more