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Boundary regularity for non-local operators with symmetric kernels and vanishing horizon

Philipp Svinger

TL;DR

The paper analyzes boundary regularity for a non-local, symmetric operator with horizon parameter, proving optimal $C^{2s-1}$ regularity up to the boundary and, under the same assumptions, a higher-order boundary expansion $u \approx q_0 \delta_\Omega^{2s-1}$ with a Hölder remainder. The approach blends barrier methods, explicit half-space calculations, and a blow-up/ Liouville framework to transfer local boundary information to the global domain, supplemented by a one-dimensional boundary Harnack principle. Key contributions include establishing sharp boundary Hölder regularity for all $f\in L^{\infty}$, deriving a 1D boundary Harnack principle for the symmetric kernel, and obtaining a Liouville-type classification in the half-space that underpins the higher-order boundary expansion. These results clarifies how symmetry and horizon vanishing impact boundary behavior, contrasting with non-symmetric or translationally invariant non-local operators and contributing to a more complete regularity theory for censored/regional-type operators.

Abstract

We prove optimal Hölder boundary regularity for a non-local operator with a singular, symmetric kernel that depends on the distance to the boundary of the underlying domain. Additionally, we prove higher boundary regularity of solutions.

Boundary regularity for non-local operators with symmetric kernels and vanishing horizon

TL;DR

The paper analyzes boundary regularity for a non-local, symmetric operator with horizon parameter, proving optimal regularity up to the boundary and, under the same assumptions, a higher-order boundary expansion with a Hölder remainder. The approach blends barrier methods, explicit half-space calculations, and a blow-up/ Liouville framework to transfer local boundary information to the global domain, supplemented by a one-dimensional boundary Harnack principle. Key contributions include establishing sharp boundary Hölder regularity for all , deriving a 1D boundary Harnack principle for the symmetric kernel, and obtaining a Liouville-type classification in the half-space that underpins the higher-order boundary expansion. These results clarifies how symmetry and horizon vanishing impact boundary behavior, contrasting with non-symmetric or translationally invariant non-local operators and contributing to a more complete regularity theory for censored/regional-type operators.

Abstract

We prove optimal Hölder boundary regularity for a non-local operator with a singular, symmetric kernel that depends on the distance to the boundary of the underlying domain. Additionally, we prove higher boundary regularity of solutions.

Paper Structure

This paper contains 19 sections, 26 theorems, 294 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Related literature
  4. Strategy of the proof
  5. Outline
  6. Acknowledgments
  7. Preliminaries
  8. Weak solution concept
  9. Interior regularity
  10. Calculations in half-space
  11. Barriers
  12. Hölder regularity up to the boundary
  13. One dimensional boundary Harnack
  14. Barriers for $\sigma =1$
  15. Oscillation decay
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\Omega \subset \mathds{R}^d$ be a bounded $C^{1,1}$ domain, $s\in (1/2,1)$, and $\sigma \in (0,1]$. Assume that $u\in H^s_0(\Omega )$ is a weak solution to the Dirichlet problem eq:DirichletProblem for some $f\in L^\infty (\Omega )$. Then $u\in C^{2s-1 }( \overline{\Omega })$ with for some constant $c=c(\Omega ,d,s,\sigma ) >0$.

Figures (2)

  • Figure 1: Shows the support of $\mathcal{B}_{\mathds{R}^2_+,\sigma }(e_2,\cdot )$ for $\sigma =2/3$.
  • Figure 2: Shows the support of $\mathcal{B}_{\mathds{R}^2_+,\sigma }(e_2,\cdot )$ for $\sigma =1$.

Theorems & Definitions (57)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Corollary 4
  • Definition 5: Weak solution concept
  • Remark 6
  • Definition 7
  • Lemma 8
  • proof
  • Corollary 9
  • ...and 47 more