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Nonlocal operators in divergence form and existence theory for integrable data

David Arcoya, Serena Dipierro, Edoardo Proietti Lippi, Caterina Sportelli, Enrico Valdinoci

TL;DR

The article develops a robust existence/uniqueness theory for weak solutions of nonlocal elliptic problems in divergence form with data in $L^1(Ω)$, using a kernel $K$ that satisfies sharp upper/lower bounds and symmetry. It introduces a variational framework and a minimization approach to obtain solutions for the nonlocal problem with nonlinear term $a h(u)$, establishing uniform a priori bounds that persist in the limiting process. A central contribution is the careful analysis of the asymptotic limit as the fractional parameter $s\to1$, showing convergence to a classical local problem $-\mathrm{div}(A\nabla u) + a h(u) = f$, where the local matrix $A$ is recovered from a suitable nonlocal construction involving an auxiliary matrix $M$. The paper also provides a reverse procedure: from a given local matrix $A$ to a nonlocal operator via an explicit formula for $M$, enabling a unified view that encompasses both local and nonlocal problems and justifies the local problem as a limit of nonlocal ones. These results yield a unified variational or approximation-based route to classical PDE theory from nonlocal models and furnish explicit mechanisms to translate between local and nonlocal operators in divergence form.

Abstract

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(Ω)$ and to be suitably dominated. We also prove that the solution that we find converges, as $s\nearrow 1$, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in $L^1(Ω)$ and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as $s\nearrow 1$, every classical operator in divergence form.

Nonlocal operators in divergence form and existence theory for integrable data

TL;DR

The article develops a robust existence/uniqueness theory for weak solutions of nonlocal elliptic problems in divergence form with data in , using a kernel that satisfies sharp upper/lower bounds and symmetry. It introduces a variational framework and a minimization approach to obtain solutions for the nonlocal problem with nonlinear term , establishing uniform a priori bounds that persist in the limiting process. A central contribution is the careful analysis of the asymptotic limit as the fractional parameter , showing convergence to a classical local problem , where the local matrix is recovered from a suitable nonlocal construction involving an auxiliary matrix . The paper also provides a reverse procedure: from a given local matrix to a nonlocal operator via an explicit formula for , enabling a unified view that encompasses both local and nonlocal problems and justifies the local problem as a limit of nonlocal ones. These results yield a unified variational or approximation-based route to classical PDE theory from nonlocal models and furnish explicit mechanisms to translate between local and nonlocal operators in divergence form.

Abstract

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to and to be suitably dominated. We also prove that the solution that we find converges, as , to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as , every classical operator in divergence form.

Paper Structure

This paper contains 18 sections, 30 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Nonlocal existence theory for integrable data
  3. Back to the classical case (from $M$ to $A$)
  4. Back to the classical case (from $A$ to $M$)
  5. Existence theory and proof of Theorems \ref{['main2']} and \ref{['thmminimo']}
  6. Functional analysis preliminaries
  7. Proof of Theorem \ref{['thmminimo']}
  8. Proof of Theorem \ref{['main2']}
  9. Asymptotic behavior of nonlocal solutions and proof of Theorem \ref{['main3']}
  10. Tailoring a matrix $M$ satisfying \ref{['aijdefn']} and proof of Theorems \ref{['THMFROMATOM']} and \ref{['fuweioifhqweio7564']}
  11. The algebraic roadmap leading to Theorem \ref{['THMFROMATOM']} and proof of Lemma \ref{['lemma3']}
  12. Proof of Theorem \ref{['THMFROMATOM']}
  13. Reconstructing the classical solution through the limit $s\nearrow 1$ and proof of Theorem \ref{['fuweioifhqweio7564']}
  14. Calculating ${\mathcal{L}}_K$ in the pointwise sense
  15. Bounds on nondegenerate matrices
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $h:\mathbb{R}\to\mathbb{R}$. Suppose that and let Let $a$, $f\in L^1(\Omega)$. Assume that Then, there exists a unique weak solution $u\in H^s_0(\Omega)\cap L^\infty(\Omega)$ of In addition, we have that

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 51 more