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Continuity of solutions to equations with weakly singular nonlocal operators

Sven Jarohs, Moritz Kassmann, Tobias Weth

TL;DR

This work develops a robust regularity theory for weak solutions to nonlocal equations driven by integro-differential operators with highly anisotropic and weakly singular kernels. By formulating minimal conditions on the jump density $j$ and the kernel $K$, and introducing a nonlocal growth lemma, the authors prove that bounded weak solutions are continuous and admit a uniform modulus of continuity. The results extend the classical De Giorgi–Nash–Moser-type regularity to broad kernel classes, including anisotropic densities that may vanish near the origin, and account for non-symmetric kernels via a suitable domination by the symmetric part. The findings have notable implications for the qualitative behavior of nonlocal equations in diverse applications, providing quantitative continuity and stability bounds under weak differentiability assumptions on the operator.

Abstract

We prove boundedness and regularity estimates for weak solutions to a class of linear nonlocal equations involving integro-differential operators with almost no order of differentiability. In particular, we show that bounded weak solutions are continuous, and we provide a uniform a-priori estimates for the modulus of continuity. In contrast to earlier works, we allow the nonlocal operators to be highly anisotropic and weakly singular, and we allow the associated kernel functions to vanish close to the singularity.

Continuity of solutions to equations with weakly singular nonlocal operators

TL;DR

This work develops a robust regularity theory for weak solutions to nonlocal equations driven by integro-differential operators with highly anisotropic and weakly singular kernels. By formulating minimal conditions on the jump density and the kernel , and introducing a nonlocal growth lemma, the authors prove that bounded weak solutions are continuous and admit a uniform modulus of continuity. The results extend the classical De Giorgi–Nash–Moser-type regularity to broad kernel classes, including anisotropic densities that may vanish near the origin, and account for non-symmetric kernels via a suitable domination by the symmetric part. The findings have notable implications for the qualitative behavior of nonlocal equations in diverse applications, providing quantitative continuity and stability bounds under weak differentiability assumptions on the operator.

Abstract

We prove boundedness and regularity estimates for weak solutions to a class of linear nonlocal equations involving integro-differential operators with almost no order of differentiability. In particular, we show that bounded weak solutions are continuous, and we provide a uniform a-priori estimates for the modulus of continuity. In contrast to earlier works, we allow the nonlocal operators to be highly anisotropic and weakly singular, and we allow the associated kernel functions to vanish close to the singularity.
Paper Structure (8 sections, 10 theorems, 181 equations)

This paper contains 8 sections, 10 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment.
  3. Preliminaries and Examples
  4. Examples
  5. Function spaces
  6. Boundedness of solutions
  7. The growth lemma
  8. Proof of Theorem 1.1

Key Result

Theorem 1.3

Assume that eq:cond-K holds with some measurable function $j: \mathbb{R}^N \setminus \{0\} \to [0,\infty)$ satisfying (eq:cond-A1) for some $\gamma \le 2$, and suppose also that (eq:cond-K_as) holds (if $j \not \in L^1(\mathbb{R}^N)$). Moreover, let $\Omega \subset \mathbb{R}^N$ be a bounded open se then every weak solution $u \in V(\Omega|\mathbb{R}^N)$ of (general-op-eq) with $u \in L^\infty(\Om

Theorems & Definitions (28)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Example 1
  • Example 2
  • Remark 2.1
  • Definition 2.2
  • ...and 18 more