Calderon-Zygmund estimates for generalized double phase equations with matrix weights

1. Problem Statement

The paper investigates the regularity theory for a class of generalized double-phase elliptic equations involving Orlicz growth and variable matrix weights. The prototype equation studied is:

$$\text{div} \left( M \left[ \frac{G'(|MDu|)}{|MDu|} MDu + a(x) \frac{H'(|MDu|)}{|MDu|} MDu \right] \right) = \text{div} \left( M \left[ \frac{G'(|MF|)}{|MF|} MF + a(x) \frac{H'(|MF|)}{|MF|} MF \right] \right)$$

in a bounded domain $\Omega \subset \mathbb{R}^n$, where:

• $M(x)$: A symmetric, positive-definite matrix weight satisfying $|M(x)||M^{-1}(x)| \le \Lambda$.
• $G, H$: Young functions (Orlicz functions) representing the growth of the operator.
• $a(x)$: A non-negative Hölder continuous coefficient ($C^{0,\alpha}$) modulating the transition between the two phases.
• Structural Condition: The ratio of the growth functions is controlled by $a(x)$ and the dimension, specifically $\sup_{t \ge 0} \frac{H(t)}{G(t) + G^{\min\{1+\alpha/n, 1+i_0\}}(t)} < \infty$.

The Core Objective:
To prove Calderón–Zygmund estimates in this setting. Specifically, the authors aim to show that if the weighted datum (the right-hand side) possesses higher integrability in a generalized Orlicz space, then the weighted gradient of the weak solution possesses the same higher integrability.
$$\Psi(x, MF) \in L^\Upsilon_{\text{loc}} \implies \Psi(x, MDu) \in L^\Upsilon_{\text{loc}}$$
where $\Psi(x, z) = G(|z|) + a(x)H(|z|)$ and $\Upsilon$ is an arbitrary Young function.

2. Methodology

The proof employs a sophisticated combination of techniques from Orlicz space theory, weighted elliptic regularity, and double-phase analysis. The methodology is structured as follows:

A. Overcoming the Lavrentiev Phenomenon

A critical prerequisite for the analysis is the absence of the Lavrentiev phenomenon for the weighted functional. The authors rely on a density result (from [11]) which guarantees that smooth functions are dense in the weighted Musielak–Orlicz space, provided the matrix weight $M$ satisfies a small log-BMO condition ($|\log M|_{\text{BMO}} \le \delta$). This allows the use of smooth test functions in weak formulations.

B. Three-Step Comparison Scheme

The core of the proof follows the standard "exit-time" argument used in Calderón–Zygmund theory, adapted for this complex structure. The authors compare the solution $u$ to a sequence of reference problems on nested balls:

1. First Comparison ($u$ vs. $h$):

   • $h$ solves the homogeneous equation with the original variable weight $M$ and boundary data $u$.
   • Goal: Establish higher integrability for $h$ using the self-improving properties of the equation and the small log-BMO condition.

2. Second Comparison ($h$ vs. $k$):

   • $k$ solves the homogeneous equation with the averaged matrix weight $\bar{M} = (\exp(\fint_B \log M))$ and boundary data $h$.
   • Goal: Control the difference $|Dh - Dk|$. The error is bounded by the oscillation of $M$ (measured by the log-BMO norm). The smallness of the log-BMO condition ensures this difference is negligible.

3. Third Comparison ($k$ vs. $v$):

   • $v$ solves the equation with frozen coefficients (constant $a(x_0)$) and the averaged weight $\bar{M}$.
   • Goal: Utilize the regularity of autonomous Orlicz double-phase equations (Lipschitz estimates) to show that $v$ has bounded gradients.
   • Phase Splitting: The analysis splits into two cases based on the coefficient $a(x)$:
     • $(G, H)$-phase: Where $a(x)$ is large enough to activate the $H$-growth.
     • $G$-phase: Where $a(x)$ is small, and the equation behaves like a single-phase Orlicz equation.

C. Level Set Estimates

Using the comparison estimates, the authors derive a reverse Hölder inequality for the level sets of the gradient. They employ Vitali's covering lemma and a truncation argument involving the Young function $\Upsilon$ to integrate these estimates and establish the final $L^\Upsilon$ bound.

3. Key Contributions

1. Unification of Frameworks: The paper successfully unifies two distinct areas of regularity theory:

   • Double Phase/Orlicz Growth: Previously studied for constant weights (e.g., $M=I$) in works like [3].
   • Matrix-Weighted Elliptic Equations: Previously studied for power-type growth ($p$-Laplacian) in works like [5].
     The authors bridge these to handle generalized Orlicz growth with variable matrix weights.

2. Extension to Variable Matrix Weights: While previous results (e.g., [10]) handled double phase with matrix weights only for power-type growth ($t^p, t^q$), this work extends the theory to general Young functions $G$ and $H$.

3. Optimal Structural Conditions: The authors identify that the small log-BMO condition on the matrix weight $M$ is the correct quantitative regime to ensure regularity. They show that this condition, combined with the structural gap condition on $G$ and $H$, is sufficient to prevent the Lavrentiev phenomenon and derive Calderón–Zygmund estimates.

4. Handling Non-Uniform Ellipticity: The proof carefully manages the interaction between the non-uniform ellipticity introduced by $a(x)$ and the anisotropy introduced by the matrix $M$.

4. Main Results

Theorem 2.1 (Main Result):
Let $u$ be a distributional solution to the generalized double-phase equation with matrix weight $M$. If:

• The matrix weight $M$ satisfies the structural bound $|M||M^{-1}| \le \Lambda$ and the small log-BMO condition ($|\log M|_{\text{BMO}} \le \delta$).
• The growth functions $G, H$ satisfy the structural gap condition (1.1).
• The datum $F$ satisfies $\Psi(x, MF) \in L^\Upsilon_{\text{loc}}(\Omega)$ for a Young function $\Upsilon$.

Then, the gradient satisfies:
$$\Psi(x, MDu) \in L^\Upsilon_{\text{loc}}(\Omega)$$
Furthermore, a quantitative estimate holds:
$$\fint_{B_{R/2}} \Upsilon(\Psi(x, MDu)) \, dx \le c \Upsilon\left( \fint_{B_R} \Psi(x, MDu) \, dx \right) + c \fint_{B_R} \Upsilon(\Psi(x, MF)) \, dx$$
for sufficiently small balls $B_R$.

5. Significance

• Theoretical Advancement: This work represents a significant step forward in the regularity theory of non-standard growth problems. It moves beyond the classical $p$-Laplacian and standard double-phase ($t^p + a(x)t^q$) settings to a fully generalized Orlicz framework with matrix weights.
• Robustness of Log-BMO: The paper reinforces the importance of the log-BMO condition in matrix-weighted problems, showing it is robust enough to handle the complexities of double-phase transitions.
• Applications: The results are applicable to models in non-homogeneous elasticity and composite materials where the material properties vary spatially (matrix weights) and the energy density exhibits different growth rates in different regions (double phase).
• Foundation for Future Work: By establishing the density of smooth functions and the necessary comparison estimates, this paper lays the groundwork for studying higher regularity (e.g., $C^{1,\alpha}$ or Lipschitz continuity) for solutions to such generalized equations.

In summary, Byun and Kim provide a rigorous and unified framework that extends Calderón–Zygmund theory to the most general setting of double-phase equations with matrix weights, resolving the interplay between Orlicz growth, variable coefficients, and matrix-valued weights.