Regularity for monotone Operators and applications to homogenization of $p$-Laplace type equations

Technical Summary: Regularity for Monotone Operators and Applications to Homogenization of p-Laplace Type Equations

Problem Statement
The manuscript addresses the regularity theory for solutions to quasilinear elliptic equations in divergence form with rapidly oscillating periodic coefficients. Specifically, it investigates equations of the form:
$$\nabla \cdot a\left(\frac{x}{\varepsilon}, \nabla u\right) = \nabla \cdot F$$
where $\varepsilon \in (0, 1]$ is a small parameter, and the operator $a(x, \xi)$ satisfies growth and monotonicity conditions modeled on the $p$-Laplace operator. The primary focus is on establishing uniform large-scale regularity estimates (specifically $L^q$-estimates for the gradient and Lipschitz estimates) for solutions when $p \neq 2$.

A central difficulty identified is that while classical homogenization theory allows the oscillating equation to converge to a homogenized autonomous equation $\nabla \cdot \bar{a}(\nabla u) = \nabla \cdot F$, the structural properties of the homogenized operator $\bar{a}$ are not guaranteed to match those of the original operator $a$ when $p \neq 2$. In the linear case ($p=2$) or when $p=2$ in nonlinear settings, the homogenized operator retains strong monotonicity and uniform ellipticity, allowing the application of classical regularity theory. However, for $p \neq 2$, the homogenized operator $\bar{a}$ typically satisfies weaker monotonicity conditions (specifically, the strong monotonicity condition (1.3) may fail, replaced by the weaker condition (1.7)). Consequently, classical regularity results cannot be directly applied to the homogenized equation, creating a barrier to deriving uniform estimates for the original oscillating problem.

Methodology
The authors employ a multi-step strategy combining perturbation arguments, homogenization theory, and a novel analysis of non-uniformly elliptic equations:

1. Local Regularity for Degenerate Operators: The authors first establish local Calderón-Zygmund type estimates for autonomous equations $\nabla \cdot a(\nabla u) = \nabla \cdot F$ under a weaker monotonicity assumption (Assumption 2) than the standard strong monotonicity (Assumption 1). This is achieved via a variant of Moser's iteration method adapted for non-uniformly elliptic operators, utilizing difference quotients and fractional Sobolev embeddings. This step proves that solutions to the homogenized equation possess the necessary regularity despite the weaker structural conditions.
2. Homogenization and Correctors: The paper constructs correctors and flux correctors for the oscillating problem. A critical theoretical contribution is the proof that the homogenized operator $\bar{a}$ satisfies the weaker Assumption 2 (and Assumption 3 in the non-degenerate case), even if the original operator $a$ satisfies the stronger Assumption 1. This stability under homogenization is established using cell formulas and properties of monotone operators.
3. Large-Scale Perturbation: To transfer regularity from the homogenized equation back to the oscillating equation, the authors utilize a two-scale expansion and a perturbation argument. They compare the solution $u$ of the oscillating equation to a solution $v$ of the homogenized equation. By establishing quantitative convergence rates (via estimates on the remainder terms involving correctors), they show that the regularity of the homogenized solution lifts to the oscillating solution on large scales.
4. Maximal Function Techniques: The derivation of global $L^q$ estimates relies on maximal function techniques (specifically truncated maximal functions) and a generalized Calderón-Zygmund theory adapted to the $p$-Laplace setting, following the spirit of works by Avellaneda and Lin but extended to the nonlinear, degenerate case.

Key Contributions and Results

• Theorem 1.1 (Local Regularity): The authors prove local $L^q$-estimates for the gradient of solutions to autonomous equations satisfying the weaker monotonicity condition (Assumption 2). This result is significant because it removes the requirement for strong monotonicity (1.3), which is often lost in homogenization for $p \neq 2$. The estimate holds for all $q \in [p, \infty)$.
• Theorem 1.2 (Uniform Large-Scale $L^q$ Estimates): For the oscillating equation with periodic coefficients, the authors establish uniform Calderón-Zygmund estimates. Crucially, the constant $C$ in the estimate is independent of $\varepsilon$. This result is new for $p$-Laplace type equations with oscillating coefficients where $p \neq 2$.
• Theorem 1.3 (Uniform Lipschitz Estimates): In the non-degenerate case ($\mu = 1$), the authors derive uniform Lipschitz estimates for solutions to the homogeneous equation ($F \equiv 0$). This extends the seminal results of Avellaneda and Lin from linear systems to nonlinear, non-degenerate elliptic equations.
• Stability of Assumptions: The paper demonstrates that the class of operators satisfying the weaker Assumption 2 (and Assumption 3) is closed under homogenization. This justifies the regularity assumptions required for the homogenized solution in previous works (e.g., [23]) and allows the application of the developed regularity theory to a broader class of operators, including Finsler $p$-Laplacians and orthotropic $p$-Laplace operators.

Significance and Claims
The paper claims to provide the first general unconditional uniform/large-scale regularity results of the Calderón-Zygmund type for $p$-Laplace type equations with oscillating coefficients when $p \neq 2$. Prior to this work, such results were either restricted to the linear case ($p=2$) or required conditional assumptions (e.g., assuming the homogenized operator retains strong monotonicity, which is not generally true).

The authors emphasize that their approach overcomes the lack of strong monotonicity in the homogenized operator by developing a regularity theory for non-uniformly elliptic equations that does not rely on linearization techniques. This allows them to recover large-scale regularity results previously obtained in [59] under conditional assumptions, but now without those additional restrictions. The results are applicable to a wide range of degenerate elliptic operators, including the orthotropic $p$-Laplace operator, which satisfies the weaker assumptions but not the standard strong monotonicity.

The manuscript does not claim to solve the problem for random coefficients in the general $p \neq 2$ case without further assumptions, noting that such results (e.g., in [23]) currently rely on specific model restrictions. However, it establishes the necessary deterministic framework (regularity of the homogenized operator) that would be required for such stochastic extensions.