Gradient regularity for viscosity solutions to quasilinear parabolic equations with mixed singular-degenerate structure

Abstract: We establish regularity results for viscosity solutions to a class of quasilinear parabolic equations exhibiting nonhomogeneous degeneracy or singularity (a double phase regime) of the form [ u_t - \big(|Du|^{\mathfrak{p}} + \mathfrak{a}(x,t)|Du|^{\mathfrak{q}}\big)Δ_p^{\mathrm N} u = f(x,t) \quad \text{in } Q_1, ] where $-1 < \mathfrak{p} < 0$, $\mathfrak{p} \leq \mathfrak{q}$, and $\mathfrak{a}, f : Q_1 \to \mathbb{R}$ are prescribed functions. Using the Jensen--Ishii method, we prove Lipschitz regularity for appropriately translated solutions. Moreover, combining this approach with intrinsic scaling techniques, we establish interior Hölder continuity estimates for the gradient. Our results extend recent work of Fang and Zhang on the homogeneous case via a different approach.

• The paper demonstrates gradient Hölder continuity for viscosity solutions under minimal structural assumptions by explicitly quantifying the regularity exponent.
• It employs intrinsic scaling, improvement-of-flatness iterations, and Jensen-Ishii viscosity methods to tackle challenges from mixed singular-degenerate growth.
• The results extend prior parabolic PDE techniques, paving the way for advances in numerical schemes, stochastic games, and image processing applications.

Gradient Regularity for Viscosity Solutions to Quasilinear Parabolic Equations with Mixed Singular-Degenerate Structure

Problem Formulation and Motivation

The paper investigates gradient regularity for viscosity solutions to a class of quasilinear parabolic equations in non-divergence form, characterized by mixed singular-degenerate structure. The prototype equation considered is: $$u_t - \big( |Du|^{\mathfrak{p}} + \mathfrak{a}(x,t)|Du|^{\mathfrak{q}} \big)\Delta_p^{\mathrm N} u = f(x,t) \quad \text{in } Q_1 := B_1 \times (-1,0],$$ where $\Delta_p^{\mathrm N}$ denotes the normalized $p$-Laplacian, and $\mathfrak{p}, \mathfrak{q}$ are exponents with $-1 < \mathfrak{p} < 0 \leq \mathfrak{q}$, representing the interplay of singular and degenerate gradient growth. The diffusion coefficient $$\mathcal{H}(x,t,\xi) = |\xi|^{\mathfrak{p}} + \mathfrak{a}(x,t)|\xi|^{\mathfrak{q}}$$ embodies a "double phase" regime that models transitions between different types of regularity, relevant in applications such as stochastic games and image processing.

Prior Results and Challenges

Recent advances have addressed regularity for quasilinear parabolic equations with either purely degenerate or homogeneous singular structure, relying on methods such as energy estimates, variational techniques, and the Ishii-Lions viscosity framework. However, the regime $\mathfrak{p} < 0$, corresponding to simultaneous singular and degenerate growth in non-divergence form, had remained largely unresolved. Standard approaches break down due to the lack of affine invariance and failure of uniform Lipschitz controls for translated profiles. Specifically, the leading singularity destabilizes the gradient estimates under translations, which is critical for improvement-of-flatness iterations.

Main Results

Structural Assumptions

The analysis proceeds under minimal regularity and boundedness assumptions:

• $\mathfrak{p}, \mathfrak{q}$ satisfy $-1 < \mathfrak{p} < 0$, $\mathfrak{p} \leq \mathfrak{q}$,
• $\Delta_p^{\mathrm N}$0, $\Delta_p^{\mathrm N}$1,
• $\Delta_p^{\mathrm N}$2.

Theorem (Gradient Hölder Regularity)

If $\Delta_p^{\mathrm N}$3 is a bounded, continuous viscosity solution, then $\Delta_p^{\mathrm N}$4 is locally Hölder continuous, with explicit exponent $\Delta_p^{\mathrm N}$5 and constant $\Delta_p^{\mathrm N}$6 depending similarly. Quantitatively,

$\Delta_p^{\mathrm N}$7

Additional spatial and temporal estimates are provided, with optimal scaling in the singular regime.

The proof employs a combination of the Jensen-Ishii viscosity method, intrinsic scaling on cylinders adapted to degeneracy, and an improvement-of-flatness iteration governed by alternative regimes: either the degenerate case permits iterative affine approximation at finer scales, or the smooth regime allows the use of classical PDE regularity results.

Analytical Framework and Technical Contributions

Key advances include:

• Development of intrinsic scaling and measure-theoretic "method of alternatives," yielding improvement-of-flatness at all scales until the smooth regime is encountered.
• Translation analysis for the singular/degenerate equation, circumventing the loss of Lipschitz bounds by careful control of the translated profiles via viscosity theory.
• Establishing gradient Hölder continuity for viscosity solutions under structural smallness assumptions on $\Delta_p^{\mathrm N}$8 and the modulating coefficient $\Delta_p^{\mathrm N}$9, overcoming the lack of uniform affine invariance.
• Extension of prior results for homogeneous and doubly degenerate parabolic equations (cf.~Fang-Zhang [FZ23], Bessa-da Silva-Sá [BessaDaSilvaSa25]) to the mixed-growth setting, with new analytic tools applicable to broader nonlinear parabolic problems.

Strong numerical bounds are exhibited in the form of explicit Hölder exponents and radius/scale dependencies; these are sharp in view of the singularity and degeneracy, and match or improve prior bounds in related settings.

Practical and Theoretical Implications

The findings:

• Provide a rigorous foundation for regularity theory in parabolic PDEs exhibiting nonhomogeneous double phase growth, reinforcing the viscosity framework's robustness against singular structure.
• Enable further analysis of stochastic game-theoretic models and nonlinear diffusion equations in image processing, where mixed-gradient regimes naturally occur.
• Offer structural criteria for Hölder regularity, informing numerical schemes and approximation theory for nonstandard PDEs.
• Set the stage for boundary regularity investigations, optimal exponent derivation, and stochastic interpretations of tug-of-war models.

The explicit analytic bounds and iterative techniques may inspire advances for fully nonlinear analogs, systems, and equations with variable exponents or anisotropy.

Directions for Further Research

Possible extensions and open questions include:

• Boundary regularity and global estimates in irregular domains.
• Application to fully nonlinear, possibly stochastic, counterparts.
• Improvement or sharpness analysis of Hölder exponents and constants.
• Adaptation to nonstandard nonlinear diffusion models relevant for computational imaging and game theory.
• Investigation of the equivalence between viscosity and weak solution concepts in the mixed-growth context.

Conclusion

The paper rigorously establishes local Hölder gradient regularity for viscosity solutions to quasilinear parabolic equations with mixed singular-degenerate structure, confirming gradient continuity under minimal structural assumptions and providing explicit, quantifiable bounds. It bridges a gap in regularity theory for non-divergence form equations with double phase growth and develops new analytic tools that may inform future research in nonlinear parabolic PDEs and associated stochastic models.