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Lipschitz regularity for fully nonlinear elliptic equations with $(p,q)$ -growth

This paper establishes interior and global Lipschitz regularity for solutions to fully nonlinear elliptic equations with $(p,q)$ -growth, demonstrating that solutions are Lipschitz continuous when the gap between $p$ and $q$ is sufficiently small or under improved bounds for Hölder continuous solutions.

Original authors: Sun-Sig Byun, Hongsoo Kim

Published 2026-05-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sun-Sig Byun, Hongsoo 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 shape of a stretched rubber sheet or the flow of traffic in a city. In mathematics, these shapes and flows are described by complex equations called elliptic equations. Usually, mathematicians like it when these equations are "well-behaved" (uniformly elliptic), meaning the rules governing the shape don't change drastically no matter how fast or slow things are moving.

However, this paper tackles a much trickier scenario: Fully Nonlinear Elliptic Equations with (p, q)-growth.

The Problem: A Rulebook That Changes with Speed

Think of the equation as a set of rules for how a surface bends.

In a normal, "well-behaved" world, the rules are consistent.
In this paper's world, the rules change depending on how steep the slope is (the gradient, or $Du$).

Specifically, the equation has a (p, q)-growth condition. Imagine a car driving on a road.

At low speeds, the road is smooth and predictable (governed by power $p$ ).
At high speeds, the road suddenly becomes bumpy and unpredictable (governed by power $q$ ).

The "gap" between $p$ and $q$ represents how much the rules change as the speed increases. If the gap is too wide, the math breaks down, and the solution (the shape of the sheet) might become jagged, jagged, or even infinite—like a cliff appearing out of nowhere. This is called a lack of Lipschitz regularity. In simple terms, "Lipschitz regularity" just means the solution is smooth enough that it doesn't have any sudden, infinite spikes; its slope is always under control.

The Goal: Proving the Surface Stays Smooth

The authors, Sun-Sig Byun and Hongsoo Kim, want to prove that even with these changing rules, the solution remains smooth (Lipschitz continuous), provided the gap between $p$ and $q$ isn't too big.

They look at two scenarios:

Interior Regularity: Looking at the middle of the domain (away from the edges).
Global Regularity: Looking at the entire domain, including the edges (boundaries).

The Key Discovery: How Big Can the Gap Be?

The paper finds a "safe zone" for the gap between $p$ and $q$ .

The Basic Rule: If the gap ( $q - p$ ) is smaller than a specific number related to how "smooth" the environment is (denoted by $\alpha$ ), the solution stays smooth.
The "Pre-Knowledge" Boost: If we already know the solution is somewhat smooth (Hölder continuous, like a slightly bumpy but connected surface), the authors prove that the gap can be slightly larger before things break.

They compare their findings to similar problems in physics (called "double phase problems") and show that their results match the best-known limits for those problems.

The Tools: How They Solved It

To prove this, the authors used a powerful mathematical technique called the Ishii-Lions method.

The Analogy of the "Double-Blind" Test:
Imagine you are trying to find the highest point on a hilly landscape, but you can't see the whole map. You pick two points, $x$ and $y$ , and you want to see how far apart they can be before the hill becomes too steep to climb.

You create a "penalty" function. If you pick two points that are too far apart, the penalty gets huge.
You assume the worst-case scenario: that the hill is too steep (the solution is not smooth).
You use the Ishii-Lions method to look at the "curvature" (how much the hill bends) at these two points simultaneously.
By comparing the rules at point $x$ and point $y$ , they show that if the gap ( $q-p$ ) is small enough, the math forces a contradiction. The "hill" cannot be that steep; it must be smooth.

The Boundary Trick:
For the edges of the domain (the boundary), they used a "barrier" method. Imagine building a temporary fence around the edge of the rubber sheet. They constructed a specific shape (a barrier) that is guaranteed to be higher than the solution at the edge. By proving the solution can't poke through this fence, they ensured the solution stays smooth right up to the boundary.

Why This Matters (According to the Paper)

The paper highlights that while mathematicians have studied these "changing rule" problems for decades in the context of energy minimization (variational theory), this is one of the first times such rigorous smoothness results have been proven for fully nonlinear equations (where the rules depend on the shape itself, not just the energy).

They also note a crucial difference from a similar model (the "product-type" model): In their model, the rules for the "steepness" and the "curvature" are mixed together in a way that makes the gap between $p$ and $q$ critical. If the gap is too wide, the smoothness is lost. In other models, the structure is simpler, and the gap doesn't matter as much.

Summary

In short, Byun and Kim proved that for a specific class of complex, speed-dependent mathematical equations, the solution will remain smooth and well-behaved (no jagged cliffs) as long as the difference between the "low-speed" and "high-speed" rules isn't too large. They provided precise formulas for exactly how large that difference can be, using a clever "two-point comparison" technique to force the math to behave.

Technical Summary: Lipschitz Regularity for Fully Nonlinear Elliptic Equations with (p, q)-Growth

Problem Statement
This paper investigates the interior and global Lipschitz regularity of viscosity solutions to fully nonlinear elliptic equations of the form:
$F(D^2u, Du, x) = f(x, Du) \quad \text{in } \Omega,$
where the operator $F$ satisfies a nonuniform ellipticity condition with $(p, q)$ -growth. Specifically, the ellipticity ratio is allowed to grow polynomially with the gradient magnitude $|Du|$, behaving as $1 + |Du|^{q-p}$ . The authors focus on the regime where the gap $q-p$ is small, aiming to establish that solutions are Lipschitz continuous ( $u \in \text{Lip}$ ) under specific constraints on this gap relative to the Hölder continuity of the coefficients and the a priori regularity of the solution.

The study addresses the challenge that when $|Du|$ is large, the equation becomes non-uniformly elliptic, potentially leading to irregular behavior. Proving Lipschitz regularity effectively bounds the gradient, thereby placing the equation back into a uniformly elliptic regime. The work considers two primary structural examples:

Double Phase Problems: Fully nonlinear analogues of the variational double phase functional $\int (|Du|^p + a(x)|Du|^q)dx$ .
Anisotropic $(p, q)$ -Growth Problems: Equations involving different growth rates in different coordinate directions.

Methodology
The core analytical tool employed is the Ishii-Lions method, a doubling-of-variables technique used to derive gradient bounds for viscosity solutions. The proof strategy involves the following key steps:

Reduction to $(0, \gamma)$ -Growth: The authors first transform the original $(p, q)$ -growth problem into a $(0, \gamma)$ -growth problem (where $\gamma = q-p$ ) by dividing the equation by $|Du|^p$ . This simplifies the structural assumptions while preserving the essential non-uniform ellipticity dependent on the gradient.
Scaling Arguments: To handle the norms of the solution, scaling techniques are applied to normalize the solution (e.g., assuming $\|u\|_{L^\infty} \leq 1$ or $\|u\|_{C^\beta} \leq 1$ ) while adjusting the structural constants $\Lambda$ and $C_f$ .
Interior Estimates: For interior regularity, the authors construct a maximum principle argument using a test function of the form:
$\Phi(x, y) = u(x) - u(y) - L_1 \phi(|x-y|) - \frac{L_2}{2}|x-x_0|^2 - \frac{L_2}{2}|y-x_0|^2.$
The localization terms involving $L_2$ are crucial for the interior case to control the difference in gradients at two points. The proof proceeds by contradiction, assuming the maximum is positive and deriving a contradiction for large $L_1$ and $L_2$ by analyzing the viscosity inequalities at the maximum points using the Pucci operators and the structural assumptions (A1)-(A4).
Boundary and Global Estimates: For global regularity, the authors combine the Ishii-Lions method with a barrier construction (following Birindelli and Demengel). They construct explicit barrier functions near the boundary to control the solution's behavior relative to the boundary data $g \in C^{1,1}$ $g \in C^{1, 1}$ .
- A critical distinction in the global argument is the absence of the localization terms ( $L_2$ terms) used in the interior proof. Consequently, the "gradient difference" term (assumption A2) does not appear in the global estimates, allowing for a relaxation of structural assumptions in the global theorem.

Key Results
The paper establishes the following main theorems regarding the admissible gap $q-p$ required for Lipschitz regularity:

Interior Regularity (Theorem 1.1):
- If $u \in L^\infty$ (or merely continuous), Lipschitz continuity holds if $q - p < \alpha$ , where $\alpha$ is the Hölder exponent of the coefficient $x \mapsto F(\cdot, \cdot, x)$ .
- If $u \in C^\beta$ a priori, the admissible gap improves to:
  $q - p < \min \left\{ 1 + \frac{\beta}{2(1-\beta)}, \frac{\alpha}{1-\beta} \right\}.$
- The authors note that the interior bound for $C^\beta$ solutions is slightly more restrictive than the optimal variational bound due to the localization terms required in the Ishii-Lions argument.
Global Regularity (Theorem 1.2):
- For a bounded $C^{1,1}$ domain $\Omega$ with boundary data $g \in C^{1,1}$ , if $u \in L^\infty$ , the condition $q - p < \alpha$ suffices.
- If $u \in C^\beta$ a priori, the global Lipschitz regularity holds under the improved bound:
  $q - p < \frac{\alpha}{1-\beta}.$
- This global bound matches the sharp gap conditions known for variational double phase problems (specifically those in [6, 12]).

Significance and Claims
The authors position their work as an extension of regularity theory from variational (divergence form) settings to fully nonlinear (non-divergence form) equations with $(p, q)$ -growth.

Comparison with Variational Theory: The results recover the sharp gap conditions known for the double phase problem in the variational setting (e.g., $q-p < \alpha/(1-\beta)$ for $C^\beta$ solutions).
Distinction from Product-Type Models: The paper highlights that their model (1.3) differs fundamentally from product-type models (1.11) studied by De Filippis. In product-type models, $C^{1,\alpha}$ regularity can be achieved independently of the gap $q-p$ due to the specific product structure allowing the removal of gradient terms. In contrast, the authors' model does not possess this structure (when $F \neq G$ ), making the gap conditions on $p$ and $q$ essential for regularity.
Novelty in Non-Divergence Setting: The authors state that while irregular counterexamples are well-known for divergence-form equations beyond optimal gaps, no analogous counterexamples are currently available for fully nonlinear non-divergence equations with $(p, q)$ -growth. Thus, their results provide the first systematic Lipschitz regularity theory for this class of non-divergence equations, establishing the necessary gap conditions analogous to the variational case.
Structural Assumptions: The paper clarifies that assumption (A2) (Lipschitz continuity of $F$ with respect to the gradient) is necessary only for the interior estimates due to the localization technique but is not required for the global estimates, demonstrating a nuanced understanding of the regularity mechanisms in non-divergence equations.