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The Wiener criterion for nonlocal Dirichlet problems

This paper establishes a nonlocal counterpart of the Wiener criterion that characterizes regular boundary points for solutions to Dirichlet problems involving integro-differential operators with differentiability order $s \in (0, 1)$ and summability $p>1$ within the framework of nonlocal nonlinear potential theory.

Original authors: Minhyun Kim, Ki-Ahm Lee, Se-Chan Lee

Published 2026-08-28

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CC BY 4.0

Original authors: Minhyun Kim, Ki-Ahm Lee, Se-Chan Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to predict how heat spreads through a metal rod, or how a drop of ink disperses in water. In the classical world of physics, these processes are governed by local rules: the temperature at a specific point depends only on the temperatures of its immediate neighbors, and the ink spreads step-by-step to the molecules touching it. For centuries, mathematicians have used a set of tools to describe these smooth, continuous flows, including a famous test developed by Norbert Wiener in the 1920s. This test, known as the Wiener criterion, acts like a geometric litmus test for the edges of a container. It determines whether a boundary point is "regular," meaning that if you set a specific temperature or concentration at the edge, the solution inside will smoothly and predictably match that value right up to the very edge. If the point is "irregular," the solution might jump or behave erratically, failing to connect with the boundary condition.

However, the real world is not always so local. In many modern systems, from the movement of particles in a turbulent fluid to the pricing of complex financial assets, changes at one point can instantly influence points far away. These are called nonlocal interactions. For these systems, the old rules of smooth, neighbor-to-neighbor flow break down. Instead of a simple diffusion, the process involves long-range jumps, where a particle might skip over a large gap to land somewhere else. For decades, mathematicians struggled to find a version of Wiener's test that worked for these long-range, nonlocal systems. They knew that certain simple geometric conditions, like a domain having a sharp corner or a smooth curve, were enough to guarantee regularity, but they lacked a precise, necessary-and-sufficient condition that could handle the messy, complex shapes found in nature. Without this test, it was impossible to know for sure if a solution would behave well at the boundary of a strange, irregular shape.

In a recent study, a team of researchers has finally constructed this missing piece of the puzzle. They established a nonlocal version of the Wiener criterion, a mathematical rule that characterizes exactly when a boundary point is regular for a broad class of these long-range interaction problems. The researchers focused on a specific type of mathematical operator that models these nonlocal effects, which can be thought of as a generalized version of the fractional Laplacian. This operator describes systems where the influence between two points decays with distance but never truly vanishes, allowing for those characteristic long-range jumps. The team proved that the regularity of a boundary point in these systems is determined entirely by the local geometry of the boundary itself, specifically by how much of the surrounding space is "blocked" by the domain's complement at different scales.

The core of their discovery is a new integral test, a formula that sums up the capacity of the space outside the domain as one zooms in closer and closer to a boundary point. In plain terms, this test measures how "thick" the obstacle is around the point at every level of magnification. If this accumulated measure of thickness becomes infinitely large as one approaches the point, the boundary is regular, and the solution will behave smoothly. If the sum remains finite, the point is irregular, and the solution may fail to connect properly. This result is significant because it mirrors the classical Wiener criterion for local heat flow, showing that despite the complex, long-range nature of the interactions, the ultimate behavior at the edge is still dictated by the immediate shape of the boundary.

The researchers did not just propose this test; they rigorously proved that it is both necessary and sufficient. This means the test works in both directions: if the integral diverges, the point is regular, and if the point is regular, the integral must diverge. They achieved this by developing a new framework for nonlocal potential theory, which allowed them to analyze the behavior of solutions near the boundary with unprecedented precision. A key challenge they overcame was the "tail" effect, a mathematical term describing the lingering influence of distant parts of the domain on the local behavior. In nonlocal equations, what happens far away can theoretically affect what happens right at the edge. The authors demonstrated that while these long-range effects are real and must be accounted for during the intermediate steps of the analysis, they ultimately cancel out in the final criterion. The information contained in the local geometry is so strong that it absorbs the nonlocal effects, leaving a clean, geometric condition that depends only on the shape of the boundary.

This work resolves a long-standing open problem in the field, specifically addressing a question that had been posed by other mathematicians regarding the behavior of these operators. The findings apply to a wide range of scenarios where the order of differentiability and the summability of the solutions vary, covering cases that were previously out of reach. By proving that the regularity of a boundary point depends only on the dimension of the space and the specific parameters of the equation, and not on the specific details of the operator itself, the authors have provided a universal tool. This allows scientists and engineers to determine the stability and predictability of solutions in complex, nonlocal systems simply by looking at the geometry of the domain, bringing the clarity of classical potential theory to the chaotic world of long-range interactions.

Technical Summary: The Wiener Criterion for Nonlocal Dirichlet Problems

Problem Statement
This paper addresses the boundary regularity of solutions to Dirichlet problems associated with a class of nonlinear nonlocal elliptic operators. Specifically, the authors consider operators $L$ of the form:
$Lu(x) = 2 \, \text{p.v.} \int_{\mathbb{R}^n} |u(x) - u(y)|^{p-2}(u(x) - u(y))k(x, y) \, dy$
where $s \in (0, 1)$ , $p > 1$ , and the kernel $k(x, y)$ satisfies the ellipticity condition $\Lambda^{-1}|x-y|^{-n-sp} \le k(x, y) \le \Lambda|x-y|^{-n-sp}$ . This class includes the fractional $p$ -Laplacian as a model case.

The central question is to characterize the regularity of a boundary point $x_0 \in \partial \Omega$ . A point is defined as regular if, for any continuous boundary data $g$ , the unique $L$ -harmonic solution $u$ with boundary values $g$ satisfies $\lim_{\Omega \ni x \to x_0} u(x) = g(x_0)$ . While the Wiener criterion provides a necessary and sufficient condition for regularity in the local case (Laplacian and quasilinear second-order operators), a general counterpart for nonlinear nonlocal operators with measurable coefficients had not been established. Previous results for nonlocal operators were limited to sufficient conditions (e.g., exterior sphere or cone conditions) or specific cases like the fractional Laplacian with compactly supported data.

Methodology
The authors employ nonlocal nonlinear potential theory to establish the Wiener criterion. The proof is divided into two main parts: sufficiency and necessity.

Local Estimates up to the Boundary:
The authors first establish boundary variants of interior estimates for weak solutions. They prove local boundedness for weak subsolutions and a weak Harnack inequality for weak supersolutions up to the boundary. A significant technical challenge in the nonlocal setting is the "nonlocal tail" term, which encodes long-range interactions. The authors adapt localization techniques (Caccioppoli-type estimates and Moser's iteration) to handle these tail terms, showing that while the tail affects intermediate estimates, its influence can be controlled in the context of the Wiener integral.
Sufficiency (The "If" Direction):
To prove that the divergence of the Wiener integral implies regularity, the authors assume the contrary: that a boundary point is irregular despite the integral diverging. They construct a specific weak supersolution related to the boundary data and derive an estimate linking the $(s, p)$ -capacity of the complement of the domain (near the boundary point) to the solution's behavior. By decomposing the domain into annuli and analyzing the tail term iteratively, they show that the divergence of the Wiener integral forces the solution to converge to the boundary data, leading to a contradiction.
Necessity (The "Only If" Direction):
For the converse, the authors utilize the relationship between the operator $L$ and the $(s, p)$ -capacity via $L$ -superharmonic functions and $L$ -potentials. They establish a pointwise estimate for nonnegative $L$ -superharmonic functions in terms of the Wolff potential of the associated measure (the $L$ -distribution).
- Technical Innovation: Existing pointwise estimates for solutions obtained as limits of approximations (SOLA) required the restriction $p > 2 - s/n$ . To cover the full range $p > 1$ , the authors introduce a new proof for the Wolff potential estimate tailored to $L$ -superharmonic functions, following an iteration scheme similar to that used for local operators but adapted for nonlocal tails.
- By relating the Wolff potential of the $L$ -distribution of the complement set to the Wiener integral, they demonstrate that if the Wiener integral converges, the solution fails to attain the boundary value, implying irregularity.

Key Contributions and Results
The main result is Theorem 1.1, which states that a boundary point $x_0 \in \partial \Omega$ is regular with respect to $L$ if and only if the following Wiener integral diverges:
$\int_0 \left( \frac{\text{cap}_{s,p}(B_\rho(x_0) \setminus \Omega, B_{2\rho}(x_0))}{\rho^{n-sp}} \right)^{\frac{1}{p-1}} \frac{d\rho}{\rho} = +\infty$
where $\text{cap}_{s,p}$ denotes the $(s, p)$ -capacity.

Key aspects of this result include:

Generality: The criterion applies to a broad class of nonlinear nonlocal operators with measurable coefficients, not just the fractional Laplacian.
Sharpness: It provides a necessary and sufficient condition, unlike previous geometric conditions (exterior cone, corkscrew) which are only sufficient.
Independence from Operator: As a direct corollary (Corollary 1.2), the regularity of a boundary point depends only on the dimension $n$ , the order $s$ , and the exponent $p$ , but not on the specific kernel $k$ (provided it satisfies the ellipticity bounds).
Range of $p$ : The results hold for all $p > 1$ , overcoming previous limitations where existence of solutions for certain measures required $p > 2 - s/n$ .

Significance
The paper claims to resolve Open Problem 7 from reference [46] by establishing the full Wiener criterion for nonlocal Dirichlet problems. The authors highlight a surprising phenomenon: despite the nonlocal nature of the equations (where solutions depend on values far outside the domain), the condition for boundary regularity is determined entirely by the local geometry of the boundary, specifically the capacity of the complement near the point. The long-range interactions (tail terms) present in the intermediate estimates are shown to be "absorbed" by the strength of the Wiener integral condition.

This work extends the classical theory of boundary regularity from local second-order operators to the nonlocal nonlinear setting, unifying the understanding of regularity across local and nonlocal frameworks through the lens of potential theory.

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