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Regularity for nonlocal problems with non-standard growth

This paper establishes local boundedness and Hölder continuity for minimizers of nonlocal functionals with $(p,q)$ -type non-standard growth and their corresponding weak solutions by analyzing associated De Giorgi classes.

Original authors: Jamil Chaker, Minhyun Kim, Marvin Weidner

Published 2026-08-28

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CC BY 4.0

Original authors: Jamil Chaker, Minhyun Kim, Marvin Weidner

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 governing how things change are not written in the smooth, predictable lines of classical physics, but in a language of jumps and long-range connections. In the standard view of the physical world, if you want to know how a temperature shifts or how a fluid flows, you look at the immediate neighbors of a point. The change at one spot depends entirely on what is happening right next to it. But many natural phenomena, from the movement of particles in a turbulent fluid to the way information spreads through a complex network, do not follow these local rules. Instead, they are influenced by events happening far away, across vast distances, in a way that defies simple, step-by-step calculation. Mathematicians describe these systems using "nonlocal" equations, where the state of a system at any single point is a weighted average of its state everywhere else.

For decades, scientists have understood how to handle these long-range interactions when the changes are uniform and predictable, much like a gentle, steady slope. However, the real world is often messier. Sometimes the forces at play change their nature depending on how strong they are; a system might behave one way under a gentle push and a completely different way under a violent shove. This is known as "non-standard growth," where the rules of the game shift based on the intensity of the activity. Until now, it has been difficult to prove that solutions to these complex, shifting rules are well-behaved. Specifically, mathematicians have struggled to show that these solutions do not suddenly spike to infinity or become jagged and unpredictable. Without proof of this stability, the models remain theoretical curiosities rather than reliable tools for describing reality.

A team of researchers has now taken a significant step forward in understanding these chaotic systems. They focused on a specific class of energy functionals, which are mathematical expressions that measure the total "cost" or "energy" of a system's configuration. In their study, they examined systems where this energy cost grows in a non-standard way, meaning the rules for how energy accumulates change depending on the size of the differences between points. The researchers wanted to know if the systems that minimize this energy—essentially, the most stable states a system can settle into—remain smooth and bounded, or if they could explode into chaos.

To answer this, the authors developed a new framework to analyze the behavior of these systems. They proved that even when the growth rules are complex and variable, the solutions to these equations possess a high degree of regularity. In plain terms, this means the solutions do not have sudden, infinite spikes, nor do they jump erratically. Instead, they are locally bounded, meaning they stay within a finite range of values in any given area, and they are Hölder continuous. This last property is a precise mathematical way of saying the solutions are smooth; if you zoom in on any part of the system, the changes happen gradually and predictably, without jagged breaks.

The researchers achieved this by studying a special category of functions known as De Giorgi classes. These are sets of functions that satisfy a specific type of inequality, which acts as a guardrail against wild behavior. The team demonstrated that the minimizers of their complex energy functionals, as well as the weak solutions to the related equations, naturally fall into these classes. By showing that these solutions obey the guardrails, they could then apply established mathematical techniques to prove that the solutions must be smooth and bounded. A key part of their success was handling the "tail" of the system. In nonlocal problems, the behavior at a distant point can influence the center. The researchers carefully quantified this distant influence, showing that even though the system reaches far out, the effect of these distant points is controlled and does not disrupt the local smoothness.

One of the most striking aspects of their findings is the robustness of the results. The mathematical constants that guarantee the smoothness of the solutions do not depend on a specific parameter that controls the "fractional" nature of the distance. This means their proof holds true even as the system approaches the limit of standard, local physics. In other words, the smoothness they proved is a fundamental property of these nonlocal systems, persisting whether the interactions are short-range or long-range. This provides a solid foundation for using these complex models in real-world applications, assuring scientists that the solutions they compute will not be mathematical artifacts but will represent stable, physical realities.

The work also clarifies the relationship between different types of growth conditions. The researchers showed that their results hold even when the gap between the lower and upper bounds of the growth is large. In previous studies, such large gaps often led to a breakdown in regularity, but this paper demonstrates that for this specific class of nonlocal problems, the solutions remain well-behaved regardless of how wide that gap is. This is a crucial distinction, as it expands the range of physical phenomena that can be reliably modeled.

Ultimately, this paper bridges a gap between abstract mathematical theory and the messy reality of nonlocal interactions. By proving that solutions to these complex, non-standard growth problems are smooth and bounded, the researchers have provided a rigorous guarantee that these systems are stable. This allows for greater confidence in using nonlocal models to describe everything from anomalous diffusion in porous media to the behavior of materials with micro-structures. The study confirms that even in a world governed by long-range jumps and shifting rules, there is an underlying order that keeps things from falling apart.

Technical Summary: Regularity for Nonlocal Problems with Non-Standard Growth

Problem Statement
This paper investigates the regularity properties of local minimizers and weak solutions to a class of nonlocal functionals and equations characterized by non-standard growth of $(p, q)$ -type. The primary object of study is the energy functional:
$I_f(u) = (1-s) \iint_{(\Omega^c \times \Omega^c)^c} f\left(\frac{|u(x) - u(y)|}{|x-y|^s}\right) \frac{k(x, y)}{|x-y|^d} \, dy \, dx,$
where $s \in (0, 1)$ , $\Omega \subset \mathbb{R}^d$ , and $f: [0, \infty) \to [0, \infty)$ is a convex, increasing function. The growth of $f$ is controlled by the condition $(f^q_p)$ , which implies bounds of the form $c_1 t^p \leq f(t) \leq c_2 t^q$ for large and small $t$ , with $1 < p \leq q$ . The kernel $k(x, y)$ is symmetric and bounded.

The authors also consider a related class of nonlocal equations $L_h u = 0$ in $\Omega$ , where the operator $L_h$ involves a measurable function $h$ satisfying structural conditions comparable to $f'$ . The specific challenge addressed is establishing local boundedness and Hölder continuity for these solutions without assuming the standard $p$ -growth ( $f(t) \sim t^p$ ) or restricting the gap between $p$ and $q$ for Hölder estimates.

Methodology
The core methodology relies on the theory of De Giorgi classes adapted to the nonlocal setting with non-standard growth. The approach proceeds through the following steps:

Auxiliary Functions and Growth Conditions: The authors introduce an auxiliary convex function $F$ and a related function $g(t) = F(t^p)$ to handle the non-standard growth. They establish comparability between $f$ , $g$ , and their derivatives, which is crucial for transferring regularity estimates.
Fractional De Giorgi Classes: The paper defines fractional De Giorgi classes, denoted $G(\Omega; q, c, s, f)$ , which consist of functions satisfying a specific improved fractional Caccioppoli inequality. This inequality includes a "tail" term accounting for the nonlocal behavior of the function outside the domain of interest.
Improved Caccioppoli Inequalities: By testing the minimization property (for minimizers) or the weak formulation (for solutions) with carefully constructed cutoff functions, the authors derive these improved inequalities. These inequalities serve as the bridge between the variational/equation structure and the De Giorgi class definitions.
Iterative Arguments:
- Hölder Continuity: Using the De Giorgi class properties, the authors employ an iterative scheme (similar to the local case in [MN91] and the fractional $p$ -Laplacian case in [Coz17]) to prove that functions in these classes are locally Hölder continuous. This involves a growth lemma that controls the measure of sets where the function deviates significantly from its bounds.
- Local Boundedness: Under the assumption $q < p^* = \frac{dp}{d-sp}$ , the authors prove local boundedness. This requires a Moser-type iteration combined with fractional Sobolev embeddings and isoperimetric inequalities.

Key Contributions and Results
The paper establishes two main theorems regarding the regularity of solutions:

Theorem 1.1 (Local Minimizers): For a local minimizer $u$ of the functional $I_f$ :
- If $f$ satisfies the upper growth condition $(f^q)$ , $u$ is locally Hölder continuous. The Hölder exponent $\alpha$ and the constant $C$ depend on $d, s_0, p, q, \Lambda$ but remain uniform as $s \to 1^-$ .
- If $sp < d$ and $q < p^*$ , and $f$ satisfies a lower growth condition $(f \gtrsim t^p)$ , then $u$ is locally bounded ( $u \in L^\infty_{loc}(\Omega)$ ). Explicit estimates for the supremum are provided in terms of the $L^q$ norm and the nonlocal tail.
Theorem 1.2 (Weak Solutions): For a weak solution $u$ to the equation $L_h u = 0$ :
- Under the same structural assumptions on $h$ and $f$ , $u$ satisfies the same Hölder continuity and local boundedness results as the minimizers.

A critical technical contribution is the proof that both local minimizers and weak solutions belong to the appropriate fractional De Giorgi classes (Theorems 6.2 and 7.3). This allows the authors to decouple the specific variational/equation structure from the regularity proofs, relying instead on the general properties of the De Giorgi classes.

Significance and Claims
The authors position their work as the first to study regularity properties for local minimizers of nonlocal functionals of the form (1.1) with general non-standard growth, as well as the local boundedness of weak solutions to the corresponding equations (1.6).

Robustness: The results are claimed to be "robust" in the sense that the regularity constants do not degenerate as the fractional order $s$ approaches 1, provided $s$ stays within a compact subset of $(0, 1)$ .
Generality: The approach generalizes previous results for the fractional $p$ -Laplacian (where $f(t) = t^p$ ) and extends the local theory of non-standard growth (e.g., [MN91], [Lie91]) to the nonlocal setting.
No Gap Restriction for Hölder: Unlike some local results where the gap between $p$ and $q$ must be small for Hölder continuity, this paper establishes Hölder estimates without such a restriction on the gap between $p$ and $q$ .
Methodological Shift: The paper distinguishes itself from recent works on fractional Orlicz-Sobolev spaces (e.g., [BSV20]) by focusing on local minimizers (not just weak solutions), allowing for $p > 1$ (not restricted to $p \geq 2$ ), and utilizing a De Giorgi class approach rather than direct Orlicz-space techniques.

The paper concludes by noting that while it establishes Hölder continuity and local boundedness, it does not address $C^{1,\beta}$ regularity, which was achieved in the local case by Lieberman [Lie91] under more restrictive assumptions on $f$ .

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