Boundary regularity for parabolic systems with nonstandard $(p,q)$-growth conditions in smooth convex domains

Technical Summary of "Boundary Regularity for Parabolic Systems with Nonstandard (p, q)-Growth Conditions in Smooth Convex Domains"

Problem Statement
This manuscript investigates the boundary regularity of local weak solutions to nonlinear parabolic systems of the form:
$$\partial_t u_i - \text{div}(a(|Du|)Du_i) = f_i, \quad i = 1, \dots, N,$$
defined in a space-time cylinder $\Omega_T = \Omega \times (0, T)$. The domain $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) is a bounded, convex $C^2$-set. The coefficients $a: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ are of Uhlenbeck-type and satisfy nonstandard $(p, q)$-growth conditions with $2 \le p \le q < p + \frac{4}{n+2}$. The inhomogeneity $f$ belongs to $L^{n+2+\sigma}(\Omega_T, \mathbb{R}^N)$ for some $\sigma > 0$.

The primary objective is to establish a local $L^\infty$-gradient estimate (Lipschitz regularity) for weak solutions $u$ that vanish on the lateral boundary $(\partial \Omega)_T$, extending previous interior results to the boundary in the context of nonstandard growth.

Methodology
The proof strategy follows a three-step approach, adapting techniques from elliptic and standard parabolic regularity theory to the nonstandard growth setting:

1. A Priori Estimates for Smooth Solutions:
   The authors first derive a quantitative $L^\infty$-gradient estimate for smooth classical solutions ($u \in C^3$) with smooth coefficients and data. This involves:

   • Establishing a Caccioppoli-type inequality involving second-order derivatives, utilizing the convexity of $\Omega$ and homogeneous Dirichlet boundary conditions to discard unfavorable boundary terms via a differential geometric identity (Grisvard).
   • Deriving a reverse Hölder-type inequality. This step relies on the specific parameter relation $q < p + \frac{4}{n+2}$, which ensures that the Sobolev embedding in the parabolic setting (improving integrability from $p$ to $p + \frac{4}{n}$) is sufficient to close the estimate.
   • Executing a Moser iteration procedure to obtain an $L^\infty$ bound. A critical component is handling the "scaling deficit" inherent to $(p, q)$-growth systems, which prevents the estimate from being homogeneous.

2. Approximation Procedure:
   Since the main result applies to weak solutions with non-smooth data and coefficients, an approximation scheme is constructed:

   • The coefficients $a$ and the datum $f$ are mollified to obtain smooth approximations $a_\varepsilon$ and $f_\varepsilon$.
   • A non-degenerate $q$-Laplacian-like term is added to the coefficients to define $b_\varepsilon$, ensuring standard $q$-growth conditions that guarantee the existence of unique, regular ($C^3$) weak solutions $u_\varepsilon$.
   • A comparison estimate is proven between the approximating solutions $u_\varepsilon$ and the original weak solution $u$, demonstrating strong convergence of the gradients $Du_\varepsilon \to Du$ in $L^p$.

3. Limit Passage:
   The quantitative $L^\infty$-gradient estimate derived for the smooth approximations is applied to $u_\varepsilon$. By passing to the limit $\varepsilon \to 0$ and utilizing the lower semicontinuity of the $L^\infty$-norm and the convergence of the data, the estimate is transferred to the original weak solution $u$.

Key Contributions and Results
The main result is Theorem 1.3, which establishes that for any local weak solution $u$ vanishing on the lateral boundary, the gradient $Du$ is essentially bounded in a neighborhood of the boundary. Specifically, for any parabolic cylinder $Q_\rho(z_0)$ centered at a boundary point $x_0 \in \partial \Omega$:
$$Du \in L^\infty(\Omega_T \cap Q_\rho(z_0), \mathbb{R}^{Nn}).$$
The paper provides a precise quantitative estimate (Equation 1.6):
$$\sup_{\Omega_T \cap Q_\rho(z_0)} |Du| \le C \left( \left( 1 + \|f\|_{L^{n+2+\sigma}}^{(n+2)(n+2+\sigma)/\sigma} \rho^{n+2} \right) \fint_{\Omega_T \cap Q_{2\rho}(z_0)} (1 + |Du|^p) \, dxdt \right)^{\frac{2p}{4+(p-q)(n+2)}}.$$

Key technical features of this result include:

• Parameter Range: The result holds for the range $2 \le p \le q < p + \frac{4}{n+2}$. This range coincides with recent interior results for nonstandard growth but is adapted for the boundary setting.
• Scaling Deficit: The exponent $\frac{2p}{4+(p-q)(n+2)}$ represents the scaling deficit characteristic of parabolic systems with nonstandard growth. The paper notes that this exponent blows up as $q$ approaches the upper bound $p + \frac{4}{n+2}$, highlighting the sharpness of the condition.
• Domain Regularity: The proof requires $\Omega$ to be a $C^2$-domain. This regularity is essential for the application of the differential geometric identity used to handle boundary integrals in the Caccioppoli estimate.
• Data Regularity: The inhomogeneity $f$ is assumed to be in $L^{n+2+\sigma}$, which is sufficient for the derivation of the estimate without requiring weak differentiability of $f$.

Significance and Claims
The paper positions its work as a boundary counterpart to existing interior regularity results for nonstandard $(p, q)$-growth parabolic systems (citing [6, 42]) and extends the boundary regularity theory known for standard $p$-growth parabolic systems (citing [5]) to the nonstandard case.

The authors claim that this work fills a gap in the literature regarding global (up to the boundary) regularity for parabolic systems with nonstandard growth, a topic where results are significantly scarcer than in the elliptic setting. The estimate provided is consistent with previous boundary results in the elliptic $(p, q)$-setting [7] and the parabolic $(p, p)$-case [5]. The paper does not claim to solve the problem for the degenerate case ($\mu=0$) or for general non-Uhlenbeck structures, noting that the latter typically only allows for partial regularity. The focus remains strictly on establishing the local Lipschitz bound under the specified structural and geometric assumptions.