Higher integrability for parabolic double phase equations with an improved gap bound

Technical Summary: Higher Integrability for Parabolic Double Phase Equations with an Improved Gap Bound

Problem Statement
The paper investigates the local higher integrability of the gradient for Hölder continuous weak solutions to the parabolic double phase equation:
$$\partial_t u - \text{div}\left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right) = 0$$
in a space-time cylinder $\Omega_T = \Omega \times (0, T]$, where $1 < p \le q$. The modulating coefficient $a(z)$ is a non-negative function that governs the transition between $p$-growth (where $a=0$) and $(p,q)$-growth.

The central challenge addressed is the relationship between the gap between the exponents $q-p$ and the regularity of the coefficient $a$. In the elliptic setting, the regularity of solutions is known to depend critically on whether the gap satisfies a condition related to the Lavrentiev phenomenon. The authors aim to establish a similar higher integrability result for the parabolic case under relaxed assumptions on the coefficient $a$ and an improved gap condition.

Methodology
The authors employ a framework based on intrinsic geometry and reverse Hölder inequalities, adapted to the parabolic setting with non-standard growth conditions. Key methodological components include:

1. Relaxed Coefficient Class ($Z_\kappa$): Instead of assuming $a \in C^{\alpha, \alpha/2}$, the authors work with the class $Z_\kappa(\Omega_T)$. A function $a$ belongs to this class if it satisfies a one-sided pointwise bound:
   $$a(z_1) \le c_a \left( a(z_2) + d_p(z_1, z_2)^\kappa \right)$$
   where $d_p$ is the parabolic metric. This class accommodates coefficients that may vanish and grow at a rate determined by $\kappa$, including non-Hölder continuous functions like $|x|^\kappa$.

2. Slanted Steklov Average: To handle the time regularity of the coefficient $a$ without requiring Hölder continuity in time (which is not guaranteed in $Z_\kappa$ for $\kappa > 1$), the authors introduce a new mollification technique called the "slanted Steklov average." Defined as:
   $$[u]_h^R(z) = \frac{1}{h} \int_t^{t+h} u(x + (s-t)\zeta, s) \, ds$$
   This technique, combined with a mild almost-increasing condition on $a$ (either forward or backward in time), allows for the derivation of energy estimates and Caccioppoli inequalities without the circular dependencies that arise from standard Lipschitz truncation methods in this setting.

3. Intrinsic Geometry and Stopping Time: The proof utilizes intrinsic cylinders ($Q_{\rho, \lambda}$ for the $p$-phase and $G_{\rho, \lambda}$ for the $(p,q)$-phase) where the scaling parameter $\lambda$ is determined by the solution's gradient. A stopping time argument is used to select appropriate radii for these cylinders based on the level sets of the integrand $H(z, |Du|) = |Du|^p + a(z)|Du|^q$.

4. Reverse Hölder Inequalities: The core of the proof involves establishing reverse Hölder inequalities for both the $p$-phase (where $a$ is small) and the $(p,q)$-phase (where $a$ is significant). These inequalities rely on parabolic Sobolev-Poincaré inequalities and Gagliardo-Nirenberg estimates adapted to the intrinsic scaling.

Key Contributions and Results

• Improved Gap Condition: The main result (Theorem 1.3) establishes higher integrability for the gradient under the gap condition:
  $$2 \le p \le q \le p + \frac{q\kappa}{q - 2\gamma}$$
  where $\gamma \in [0, 1)$ is the Hölder exponent of the solution $u$ in space and $\gamma/q$ in time.

  • The authors note that this condition is purely parabolic. It differs from the optimal elliptic gap bound ($q-p \le \kappa/(1-\gamma)$) and is strictly stricter in the sense that the allowable gap is smaller when $p \ge 2$.
  • The bound reflects the time regularity of the solution; the authors observe that if the time regularity could be improved from $\gamma/q$ to $\gamma/2$, the elliptic gap condition would be recovered.

• Extension to $Z_\kappa$ Coefficients: The paper is the first to apply the $Z_\kappa$ class (previously used in elliptic double phase problems) to the parabolic context. This allows for the treatment of coefficients that are not necessarily Hölder continuous in time, provided they satisfy the one-sided bound and the mild monotonicity condition.

• New Mollification Technique: The introduction of the slanted Steklov average provides a mechanism to derive energy estimates without assuming $a$ is Hölder continuous in time, overcoming limitations of standard space-time mollification when $\kappa > 1$.

• Main Theorem: For a weak solution $u$ that is locally $\gamma$-Hölder continuous in space and $\gamma/q$-Hölder continuous in time, there exists an $\epsilon_0 > 0$ such that for any $\epsilon \in (0, \epsilon_0)$:
  $$\fint_{Q_r(z_0)} H(z, |Du|)^{1+\epsilon} \, dz \le c \left( \frac{2 + \|a\|_\infty}{r^{q-2\gamma}} \right)^{1 + \frac{\epsilon q}{2}}$$
  This confirms the local higher integrability of the gradient in the Orlicz-Sobolev space associated with the double phase functional.

Significance and Claims
The paper claims to advance the regularity theory of non-uniformly elliptic parabolic problems by:

1. Relaxing Regularity Assumptions: Moving beyond the standard $C^{\alpha, \alpha/2}$ assumption on the coefficient $a$ to the broader $Z_\kappa$ class, which better models materials with varying ellipticity.
2. Refining the Gap Condition: Providing a gap bound that is intrinsic to the parabolic nature of the equation, explicitly linking the allowable gap between $p$ and $q$ to the time-Hölder regularity of the solution.
3. Methodological Innovation: Demonstrating that the slanted Steklov average is a viable alternative to Lipschitz truncation for deriving energy estimates in settings where the coefficient's regularity is insufficient for standard methods.

The authors explicitly state that their approach extends to systems and the singular regime ($p < 2$), though the current work restricts itself to scalar homogeneous equations because Hölder regularity for systems and the singular case has not yet been fully established in the literature. They also note that the derived gap bound may not be sharp, as the phenomenon is governed entirely by time regularity in the parabolic setting.