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Capacities, Wiener criteria and fine continuity for nonlocal nonlinear equations

This paper investigates the boundary regularity of solutions to nonlocal nonlinear equations of the $s$ -fractional $p$ -Laplacian type by establishing the equivalence between condenser and Sobolev capacities via precise comparison estimates, thereby characterizing the conditions under which regularity transfers between different parameters and proving the fine continuity of superharmonic functions and the coincidence of polar sets with zero capacity sets.

Original authors: Anders Björn, Jana Björn, Minhyun Kim

Published 2026-08-26

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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 a world where the rules of smoothness are not absolute but depend on the texture of the space itself. In mathematics, this space is often a region of our familiar three-dimensional reality, or perhaps a higher-dimensional version of it. Scientists who study these regions are interested in how things change from one point to another, specifically how solutions to certain equations behave when they reach the very edge of a shape. These equations describe physical phenomena like heat flow or the distribution of electric charge, but in a more complex, non-local form where a point is influenced by distant neighbors, not just its immediate surroundings. The central question for these researchers is simple yet profound: when does a solution behave nicely right up to the boundary, and when does it break down? The answer depends on two hidden parameters that define the equation's character: one describing the strength of the long-range influence, and another describing the non-linear nature of the interaction.

In a recent study, Anders Björn, Jana Björn, and Minhyun Kim tackled this question for a broad class of these non-local equations. They sought to map out exactly how the behavior of a solution at a boundary point changes as these two parameters shift. Their work reveals a precise map of cause and effect. They discovered that if a boundary point is "regular"—meaning the solution arrives there smoothly and matches the expected value—for one specific pair of parameters, it will automatically be regular for a different pair of parameters, but only if that new pair falls into a very specific relationship with the first. The researchers proved that this relationship is not a simple matter of one parameter being larger or smaller than the other. Instead, it is a delicate balance involving the dimension of the space and the specific values of the parameters. For instance, if the influence is strong enough relative to the dimension of the space, regularity is guaranteed. But in more subtle cases, the researchers found that regularity for a weaker influence might imply regularity for a stronger one, or vice versa, depending on a precise inequality that links the two sets of parameters.

To reach these conclusions, the team had to bridge two different ways of measuring the "size" of a set of points near a boundary. One method measures how much energy is required to create a specific disturbance, while the other measures the capacity of a set to hold a charge. The authors showed that these two measurements are essentially equivalent for the purpose of determining regularity, provided the parameters stay within a certain range. This equivalence allowed them to translate a complex condition involving one type of measurement into a more manageable form involving the other. They then used this translation to prove that the condition for a point to be regular is the same regardless of which measurement tool is used, as long as the parameters are below a critical threshold. When the parameters exceed this threshold, the rules change entirely, and the point is always regular, regardless of the shape of the boundary.

The paper also explored the concept of "fine continuity," a specialized form of smoothness that ignores tiny, negligible sets of points. The researchers demonstrated that solutions to these equations are always finely continuous, meaning they behave predictably even if they encounter these tiny, invisible obstacles. They further proved that the sets of points where a solution might blow up to infinity are exactly the same as the sets that have zero capacity in this mathematical sense. This result connects the behavior of the solution to the fundamental geometry of the space in a rigorous way. By establishing these connections, the authors provided a complete picture of when and why solutions to these complex equations remain well-behaved at the edges of their domains. Their findings do not just apply to one specific equation but cover a vast family of them, including the classical local equations that have been studied for decades, showing how the old rules fit into this new, broader framework.

The implications of this work are that the boundary behavior of these systems is not a mystery but a predictable outcome of the underlying parameters. The researchers did not find that any change in parameters leads to chaos; rather, they found a structured landscape where regularity is preserved across specific transitions. If a boundary point is regular for a certain set of conditions, it remains regular for a whole range of other conditions, provided those conditions satisfy the specific inequalities they derived. Conversely, if a point is irregular for one set of conditions, it will remain irregular for others that fall outside the safe zones they identified. This clarity allows mathematicians to predict the behavior of solutions without having to solve the equations explicitly for every new scenario. The study confirms that the interplay between the strength of the non-local influence and the non-linearity of the system creates a stable, understandable structure, even in the complex, non-local world these equations describe.

Technical Summary: Capacities, Wiener criteria and fine continuity for nonlocal nonlinear equations

Problem Statement
This paper investigates the boundary regularity of solutions to nonlocal nonlinear equations of the $s$ -fractional $p$ -Laplacian type in open subsets of $\mathbb{R}^n$ . The operator $L$ is defined via a symmetric measurable kernel $k(x,y)$ satisfying ellipticity conditions comparable to $|x-y|^{-n-sp}$ . The central problem is to determine precisely how the regularity of a boundary point $x_0$ depends on the parameters $s$ (fractional order) and $p$ (nonlinearity exponent). Specifically, the authors seek to establish the exact conditions under which regularity for a pair $(s_1, p_1)$ implies regularity for a different pair $(s_2, p_2)$ , covering both the fractional case ( $0 < s < 1$ ) and the local limit case ( $s=1$ , corresponding to the classical $p$ -Laplacian).

Methodology
The analysis relies on the equivalence between two distinct formulations of the Wiener criterion: one utilizing condenser capacity ( $\text{cap}_{s,p}$ ) and the other utilizing Sobolev capacity ( $C_{s,p}$ ).

Capacity Comparison: The authors derive precise comparison estimates between condenser and Sobolev capacities. A key technical contribution is proving that these two capacities are comparable for sets of small diameter when $sp \leq n$ , and establishing their equivalence in the context of Wiener integrals. This involves proving that both capacities are Choquet capacities and utilizing properties of fractional Sobolev spaces ( $W^{s,p}$ ) and tail spaces.
Fine Topology and Thinness: The paper defines the fine topology and thinness via the divergence of a specific Wiener integral involving condenser capacity. The authors demonstrate that this definition coincides with the fine topology defined via Sobolev capacity (as established in the local case by Adams–Hedberg).
Superharmonic Functions: The study extends to $L$ -superharmonic functions (essentially regularized supersolutions). The authors utilize lower semicontinuity, Perron solutions, and obstacle problems to analyze the pointwise behavior of these functions.
Counterexamples: To prove the necessity of their conditions, the authors construct specific bounded open sets $\Omega$ where regularity fails for one parameter set but holds for another, often adapting constructions from Adams–Hedberg and Adams–Meyers to the nonlocal setting.

Key Contributions and Results

Equivalence of Wiener Criteria: The paper establishes that for $sp \leq n$ , the Wiener criterion formulated with condenser capacity diverges if and only if the corresponding integral with Sobolev capacity diverges (Theorem 1.4). This equivalence is crucial for transferring known results from the local and Bessel potential settings to the general nonlocal nonlinear framework.
Complete Classification of Regularity Implications: The main result (Theorem 1.2 / Theorem 8.1) provides a necessary and sufficient condition for the implication: "Regularity for $(s_1, p_1) \implies$ $(s_{1}, p_{1}) ⟹$ Regularity for $(s_2, p_2)$ $(s_{2}, p_{2})$ ." The conditions are:
- (a) $s_2 p_2 > n$ (Points are always regular).
- (b) $s_1 p_1 = s_2 p_2 = n$ and $p_1 > p_2$ .
- (c) $s_1 p_1 < s_2 p_2 = n$ .
- (d) $s_1 p_1 < s_2 p_2 < n$ and a specific inequality involving $s$ and $p$ holds: $s_1(p_2 - 1) + \frac{n}{p_2} \leq s_2(p_1 - 1) + \frac{n}{p_1}$ .
  The paper notes that if $s_2 \geq s_1$ and $p_2 \geq p_1$ , regularity is preserved, but the full picture is more complex when parameters vary independently.
Fine Continuity of Superharmonic Functions: The authors prove that every $L$ -superharmonic function is finely continuous (Theorem 1.5). In the case where $sp > n$, this implies standard continuity. This result confirms that superharmonic functions serve as "good" representatives of supersolutions.
Characterization of Polar Sets: The paper proves that a set is $L$ -polar (the set of infinities of an $L$ -superharmonic function) if and only if it has zero Sobolev capacity (Proposition 1.6). This generalizes classical results by Cartan and Choquet to the fractional nonlinear setting.
Local vs. Fractional Comparison: The analysis clarifies the relationship between the local case ( $s=1$ ) and fractional cases. For instance, regularity for $(1, p_1)$ with $p_1 \geq n$ does not imply regularity for any fractional $(s, p)$ with $sp \leq n$ (except trivial cases), whereas regularity for $(1, p_1)$ with $p_1 < n$ implies regularity for all $(s, p)$ where $sp > p_1$ .

Significance and Claims
The authors claim that this work provides the first complete characterization of how boundary regularity transitions between different fractional and nonlinear parameters. By rigorously establishing the equivalence between condenser and Sobolev capacities in the nonlocal setting, the paper bridges a gap between existing literature on local equations (e.g., Malý–Ziemer) and fractional equations.

The paper modestly notes that while experts may have suspected the equivalence of the two Wiener integrals (provided $sp \leq n$ ), there was no published proof. The derivation of precise comparison estimates (Lemma 1.3) and the proof of the equivalence (Theorem 1.4) are presented as the primary technical innovations enabling the classification of regularity. Furthermore, the extension of the theory of polar sets and fine continuity to nonlocal nonlinear operators generalizes foundational potential theory results to a broader class of equations. The paper concludes by visualizing these implications in the $(1/p, s)$ plane, delineating regions where regularity is preserved, lost, or indeterminate.

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