Gradient higher integrability for degenerate parabolic double phase systems with two modulating coefficients

1. Problem Statement

The paper investigates the regularity of weak solutions to degenerate parabolic double phase systems involving two modulating coefficients. The system is given by:
$$u_t - \text{div} \left( a(z)|Du|^{p-2}Du + b(z)|Du|^{q-2}Du \right) = -\text{div} \left( a(z)|F|^{p-2}F + b(z)|F|^{q-2}F \right)$$
in a space-time cylinder $\Omega_T = \Omega \times (0, T)$, where:

• $2 \le p \le q < \infty$.
• $a(z)$ and $b(z)$ are non-negative modulating coefficients.
• Key Assumptions:
  • $a(z)$ is uniformly continuous.
  • $b(z)$ is Hölder continuous ($C^{\alpha, \alpha/2}$).
  • The sum of coefficients is bounded from below: $a(z) + b(z) \ge \delta > 0$ (preventing simultaneous vanishing).
  • The exponents satisfy the standard growth condition for double phase problems: $q \le p + \frac{2\alpha}{n+2}$.

The primary goal is to establish local higher integrability of the spatial gradient $Du$. Specifically, the authors aim to prove that if the source term $F$ and the gradient $Du$ have integrability related to the energy density $H(z, \xi) = a(z)|\xi|^p + b(z)|\xi|^q$, then $Du$ actually possesses higher integrability (i.e., $H(z, |Du|) \in L^{1+\varepsilon}_{loc}$).

2. Methodology

The authors employ a sophisticated intrinsic geometry approach combined with a phase separation strategy. This is a significant departure from previous works on single-modulating coefficient systems.

A. Intrinsic Geometry

The system exhibits different growth behaviors depending on the relative magnitude of the coefficients $a(z)$ and $b(z)$. To handle this, the authors define three types of intrinsic cylinders scaled by a parameter $\lambda \ge 1$:

1. $p$-intrinsic cylinders ($Q_{\rho, \lambda}$): Where the $p$-growth term dominates ($a(z)\lambda^p \gtrsim b(z)\lambda^q$). The time scaling is $\sim \lambda^{2-p}$.
2. $q$-intrinsic cylinders ($J_{\rho, \lambda}$): Where the $q$-growth term dominates ($b(z)\lambda^q \gtrsim a(z)\lambda^p$). The time scaling is $\sim \lambda^{2-q}$.
3. $(p, q)$-intrinsic cylinders ($G_{\rho, \lambda}$): Where both terms are comparable. The time scaling is determined by the sum $\Lambda = a(z)\lambda^p + b(z)\lambda^q$.

B. Phase Separation and Comparison Scheme

The core novelty lies in the delicate comparison scheme used to separate the phases. Unlike previous elliptic or single-coefficient parabolic studies, the presence of two coefficients and the parabolic nature create nine distinct cases based on the size of $a(z)$ and $b(z)$ within the intrinsic cylinders:

• $p$-phase: $a(z)$ is large, $b(z)$ is small.
• $q$-phase: $b(z)$ is large, $a(z)$ is small.
• $(p, q)$-phase: Both coefficients are significant.

The authors must handle the transition between these phases carefully, ensuring that the intrinsic scaling parameters ($\lambda$) and the radii of the cylinders are chosen consistently to satisfy the necessary reverse Hölder inequalities.

C. Key Technical Steps

1. Energy Estimates: Derivation of Caccioppoli-type inequalities adapted to the double-phase structure.
2. Reverse Hölder Inequalities: Proving these inequalities separately for each of the three phases ($p$, $q$, and mixed). This involves intricate estimates using the continuity of $a(z)$ and Hölder continuity of $b(z)$ to control the "cross-terms" where one coefficient is small but non-zero.
3. Stopping Time Argument: A refined argument is used to select the appropriate intrinsic radius for any point in the superlevel set of the gradient. This ensures that the point falls into one of the three defined phases.
4. Vitali Covering Lemma: A generalized covering lemma is constructed to handle the nine possible combinations of intrinsic cylinders (e.g., a $p$-cylinder covering a $q$-cylinder, etc.). The authors prove that the scaling factors $\lambda$ are comparable across intersecting cylinders, allowing for a unified covering argument.
5. Gehring's Lemma: The final step combines the reverse Hölder inequalities with the covering lemma and Fubini's theorem to bootstrap the integrability from $L^1$ to $L^{1+\varepsilon}$.

3. Key Contributions

• First Parabolic Result for Two Coefficients: This is the first regularity result for parabolic double phase systems with two modulating coefficients. Previous results were limited to the elliptic case or parabolic cases with only one coefficient (e.g., $a(z) \equiv 1$).
• Unified Intrinsic Geometry: The paper successfully extends the intrinsic geometry method to a system where the dominant phase can switch between $p$ and $q$ (or be mixed) depending on the local values of two distinct coefficients.
• Handling Uniform Continuity: The authors relax the regularity requirement on $a(z)$ to uniform continuity (rather than Hölder continuity), which is a more general and technically challenging assumption in the context of double phase problems.
• Nine-Case Analysis: The rigorous treatment of the nine distinct geometric cases arising from the interaction of the two coefficients and the three intrinsic phases represents a significant technical advancement over existing literature.

4. Main Results

The main theorem (Theorem 1.1) establishes the local higher integrability of the gradient. Specifically, for a weak solution $u$, there exist constants $\varepsilon_0 > 0$ and $c \ge 1$ such that for any cylinder $Q_{2r}(z_0) \subset \Omega_T$:

$$\iint_{Q_r(z_0)} H_1(z, |Du|)^{1+\varepsilon} \, dz \le c \left( \iint_{Q_{2r}(z_0)} H_1(z, |Du|) \, dz \right)^{1 + \frac{\varepsilon q}{2}} + c \left( \iint_{Q_{2r}(z_0)} H_1(z, |F|)^{1+\varepsilon} \, dz \right)^{\frac{q}{2}}$$

where $H_1(z, \kappa) = a(z)\kappa^p + b(z)\kappa^q + 1$ and $\varepsilon \in (0, \varepsilon_0)$.

This result implies that the gradient $Du$ belongs to a higher Lebesgue space than initially assumed, which is a crucial step toward proving further regularity properties like Hölder continuity or Calderón-Zygmund estimates.

5. Significance

• Modeling Anisotropic Materials: Double phase systems model materials with strongly anisotropic properties (e.g., composites with different hardening behaviors). The inclusion of two modulating coefficients provides a more realistic framework for such materials than single-coefficient models.
• Theoretical Foundation: By establishing higher integrability, this work lays the necessary groundwork for future research into the partial regularity (Hölder continuity) and Calderón-Zygmund estimates for parabolic double phase systems with two coefficients.
• Methodological Advancement: The techniques developed, particularly the handling of the "mixed" phase and the nine-case Vitali covering, provide a robust toolkit for analyzing other degenerate parabolic systems with non-standard growth conditions.

In summary, this paper resolves a major open problem in the regularity theory of parabolic PDEs by successfully adapting intrinsic scaling methods to the complex, multi-phase environment created by two modulating coefficients.