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Perron solutions and boundary regularity for nonlocal nonlinear Dirichlet problems

This paper establishes the equivalence between Sobolev and Perron regularity for nonlinear fractional $p$ -Laplace operators by introducing a generalized definition of Perron solutions for arbitrary exterior data, proving their resolutivity and coincidence with Sobolev solutions, and demonstrating the invariance of these solutions under perturbations on sets of zero fractional capacity.

Original authors: Anders Björn, Jana Björn, Minhyun Kim

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anders Björn, Jana Björn, Minhyun Kim

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 you are trying to predict the weather inside a specific room (let's call it $\Omega$ ). You know the temperature outside the room perfectly, but because the air in this room behaves in a very strange, "nonlocal" way, the temperature at any single point inside isn't just determined by its immediate neighbors. Instead, every point inside the room "feels" the temperature of every point outside, though points closer to the window have a stronger influence than those far away.

This paper is about solving the puzzle of: "If we know the temperature outside, what is the temperature inside, and does the temperature inside smoothly match the temperature right at the window?"

Here is the breakdown of their work using simple analogies:

1. The Two Ways to Guess the Answer

The authors look at this problem using two different "guessing strategies" (mathematical methods) that usually give different answers in complex situations:

The "Sobolev" Strategy (The Smooth Averager): This method looks at the problem like a statistical average. It asks, "What is the smoothest, most energy-efficient temperature distribution that fits the outside data?" It's like smoothing out a crumpled piece of paper until it lies flat, but it only cares about the average behavior, not necessarily what happens at every single tiny speck on the edge.
The "Perron" Strategy (The Boundary Watcher): This method is more like a strict inspector. It builds a "ceiling" and a "floor" of possible temperature distributions based on the outside data. It asks, "What is the highest possible temperature we can have without breaking the rules, and what is the lowest?" The answer is the space right in the middle. This method is very sensitive to the exact conditions right at the window (the boundary).

The Big Discovery: For a long time, mathematicians wondered if these two strategies would agree on whether the temperature at the window is "regular" (smooth and predictable) or "irregular" (jumpy and chaotic). The authors prove that they always agree. If the window is smooth for the "Smooth Averager," it is also smooth for the "Boundary Watcher," and vice versa.

2. The "Nonlocal" Twist

In normal physics (like heat in a metal rod), what happens at the window only depends on the metal right next to it. But in this paper's world (fractional $p$ -Laplacian), the window "talks" to the whole outside world.

The Analogy: Imagine the room is a party. In a normal room, you only hear the people standing next to you. In this "nonlocal" room, you can hear the whispers of people standing across the street, though they sound quieter.
The Challenge: Because the room listens to the entire outside world, you can't just set the temperature at the window; you have to set the temperature for the entire "outside" (the complement of the room).

3. The "Barrier" (The Guard at the Door)

To determine if a specific point on the window is "regular" (meaning the inside temperature will match the outside temperature exactly as you approach it), the authors use a concept called a Barrier.

The Metaphor: Imagine a guard standing at a specific spot on the window. If this guard can stand there and say, "I can see the outside temperature clearly, and I can prove that the inside temperature must meet me there," then that spot is regular.
The paper proves that if you can build this "guard" (a mathematical function that acts as a barrier), the temperature will be smooth. If you can't, the temperature might jump or behave erratically at that spot.

4. Handling "Bad" Data (The Messy Outside)

Sometimes, the data outside the room is messy, incomplete, or even "non-measurable" (mathematically speaking, it's like trying to measure a cloud that keeps changing shape).

The Innovation: The authors created a new, more flexible definition for the "Perron" strategy. Their new definition works even if the outside data is messy or undefined in some spots.
The Result: They showed that even with messy data, the "Perron" solution behaves nicely inside the room. Furthermore, if you change the outside data only on a tiny, invisible set of points (a set with "zero capacity," like a single speck of dust that doesn't affect the overall view), the solution inside the room does not change at all. This is called invariance.

5. The "Wiener Criterion" (The Rulebook)

The paper connects their findings to a famous rule called the Wiener Criterion.

The Analogy: Think of the window as a gate. The Wiener Criterion is a formula that calculates how "crowded" the outside is near the gate. If the outside is "crowded" enough (in a specific mathematical sense), the gate is regular, and the temperature flows smoothly. If the outside is too "empty" or sparse near the gate, the gate is irregular, and the temperature might misbehave.
The authors proved that this rule works perfectly for their complex, nonlocal, nonlinear equations, just as it does for simpler, classical physics.

Summary of the Main Takeaways

Two Methods, One Truth: The two different ways mathematicians usually solve these problems (Sobolev and Perron) actually lead to the exact same conclusion about whether the boundary is smooth or not.
Robustness: The solution is very stable. If you tweak the outside data on a tiny, negligible set of points, the solution inside the room remains exactly the same.
The Barrier Test: You can tell if a point on the boundary is "regular" simply by checking if a specific type of "guard" (barrier function) can be built there.
Generalization: They extended these rules to work with very general, messy data, not just the clean, perfect data usually assumed in textbooks.

In short, this paper provides a unified, robust framework for understanding how "nonlocal" forces (where everything affects everything else) behave at the edges of a domain, proving that the rules of smoothness are consistent regardless of which mathematical lens you use to look at them.

Technical Summary: Perron Solutions and Boundary Regularity for Nonlocal Nonlinear Dirichlet Problems

Problem Statement
The paper addresses the Dirichlet problem for a class of nonlocal nonlinear operators $L$ of fractional $p$ -Laplace type, defined on a bounded open set $\Omega \subset \mathbb{R}^n$ . The operator is given by:
$Lu(x) = 2 \text{ p.v.} \int_{\mathbb{R}^n} |u(x) - u(y)|^{p-2}(u(x) - u(y))k(x, y) \, dy,$
where $1 < p < \infty$ , $0 < s < 1$ , and $k$ is a symmetric measurable kernel satisfying ellipticity conditions comparable to $|x-y|^{-n-sp}$ . Due to the nonlocal nature of $L$ , the Dirichlet data $g$ must be prescribed on the entire complement $\Omega^c$ , not just the boundary $\partial \Omega$ .

The central challenge is that, even for bounded and continuous exterior data, the prescribed boundary values are not always attained as limits of solutions. Consequently, the problem must be understood in a relaxed sense. The authors investigate two distinct approaches to defining solutions:

Sobolev solutions ( $H_g$ ): Weak solutions based on fractional Sobolev spaces $V^{s,p}(\Omega)$ .
Perron solutions ( $P_g$ ): Solutions constructed via the Perron method using $L$ -superharmonic functions.

These approaches lead to two distinct concepts of boundary regularity: Sobolev regularity and Perron regularity. The primary objective is to establish the relationship between these two notions and to characterize the regularity of boundary points.

Methodology
The authors develop a comprehensive theory of Perron solutions for nonlocal nonlinear equations, extending previous work on fractional linear and nonlinear operators. The methodology involves:

Redefinition of Perron Solutions: The authors propose a new definition for the upper class of Perron solutions. Unlike previous definitions (e.g., Korvenpää–Kuusi–Palatucci [36]) which required equality $u=g$ almost everywhere in $\Omega^c$ , the new definition requires only the inequality $u \ge g$ in $\Omega^c$ . This modification allows for the definition of Perron solutions for arbitrary exterior data $g: \Omega^c \to [-\infty, \infty]$ , including nonmeasurable functions, and ensures the monotonicity property $P_{g_1} \le P_{g_2}$ whenever $g_1 \le g_2$ .
Function Spaces and Capacities: The study utilizes fractional Sobolev spaces $V^{s,p}(\Omega)$ and tail spaces $L^{p-1}_{sp}(\Omega^c)$ . A crucial technical component is the analysis of two capacities: the Sobolev capacity $C_{s,p}$ and the condenser capacity $c_{s,p}$ . The authors establish their equivalence regarding sets of zero capacity.
Kellogg Property: A significant methodological step is the proof of the Kellogg property, which asserts that the set of Sobolev irregular boundary points has zero Sobolev capacity. This is achieved by relating the nonlocal problem to a weighted $p$ -Laplace equation in $\mathbb{R}^{n+1}$ via the Caffarelli–Silvestre extension framework (implicitly used through capacity equivalences).
Obstacle Problems: The authors utilize the obstacle problem as a tool to study convergence and regularity. They prove convergence results for sequences of solutions to obstacle problems, which are essential for establishing the equality between Sobolev and Perron solutions.
Comparison Principles: The proof relies heavily on comparison principles for both the Dirichlet and obstacle problems, as well as for $L$ -superharmonic functions.

Key Contributions and Results

Equivalence of Regularity: The paper's main result (Theorem 1.7) establishes that Sobolev regularity and Perron regularity are equivalent. A boundary point $x_0$ is Sobolev regular if and only if it is Perron regular. This equivalence implies that the distinction between the two types of regularity is unnecessary for this class of operators.
Characterization of Regular Points: The authors provide multiple equivalent characterizations for regular boundary points, including:
- The Wiener criterion involving the condenser capacity (Theorem 4.2, Theorem 1.7(e)).
- The existence of a barrier at the point (Theorem 1.7(f)).
- The limit behavior of the solution for specific test data (e.g., $d_{x_0}(x) = \min\{1, |x-x_0|\}$ ).
- The behavior of the solution to the obstacle problem with obstacle $d_{x_0}$ .
Resolutivity and Invariance: The authors prove resolutivity results for a broad class of exterior data. Specifically, if $g$ is bounded and continuous at the boundary (or continuous outside a set of zero capacity), the Perron solution $P_g$ coincides with the Sobolev solution $H_g$ . Furthermore, they show that Perron solutions are invariant under perturbations of the exterior data on sets of zero Sobolev capacity (Theorem 1.3, Theorem 1.4).
Uniqueness: A uniqueness result is obtained for bounded solutions taking prescribed continuous exterior data quasieverywhere on the boundary (Theorem 1.6).
Properties of Perron Solutions: The paper establishes that for arbitrary data $g$ , the upper Perron solution $P_g$ is either continuous in $\Omega$ or identically $\pm \infty$ . If $g$ belongs to the tail space $L^{p-1}_{sp}(\Omega^c)$ , $P_g$ is either $L$ -harmonic in $\Omega$ or identically $\pm \infty$ (Theorem 1.2).
Extension to Obstacle Problems: The equivalence of regularity notions is extended to obstacle problems, showing that boundary regularity for obstacle problems is equivalent to that for the Dirichlet problem (Theorem 10.3).

Significance
The paper claims to substantially extend the nonlocal nonlinear theory initiated in previous works (specifically [36] and [39]). Its significance lies in:

Unification: By proving the equivalence of Sobolev and Perron regularity, the paper unifies two previously distinct approaches to boundary regularity in the nonlocal nonlinear setting.
Generality: The new definition of Perron solutions allows for arbitrary exterior data, including nonmeasurable functions, which was not possible under previous restrictive definitions.
Robustness: The results on resolutivity and invariance under zero-capacity perturbations provide a robust framework for solving Dirichlet problems where data may be irregular or defined only almost everywhere.
Foundational Tools: The establishment of the Kellogg property and the detailed analysis of capacities for nonlocal operators provide foundational tools for further research in nonlinear potential theory for fractional operators.

The authors note that while the Wiener criterion for Sobolev regularity was recently obtained by Kim–Lee–Lee [31] and Björn [8], the equivalence to Perron regularity and the comprehensive characterization of regular points via barriers and obstacle problems are novel contributions of this work. The results hold for general kernels satisfying the ellipticity condition, not just the standard fractional $p$ -Laplacian.