C^{1,\alpha} regularity for a class of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions

1. Problem Statement

The paper investigates the global regularity of viscosity solutions to a class of fully nonlinear elliptic equations subject to oblique boundary conditions on bounded $C^1$ domains. The specific equation under study is:

$$\begin{cases} \Phi(x, |Du|)F(D^2u) = f & \text{in } \Omega, \\ \beta \cdot Du = g & \text{on } \partial\Omega, \end{cases}$$

where:

• $\Omega \subset \mathbb{R}^n$ is a bounded $C^1$ domain.
• $F: S^n \to \mathbb{R}$ is a uniformly elliptic operator (in the sense of Pucci extremal operators).
• $\Phi(x, t)$ is a continuous function exhibiting singular or degenerate behavior with respect to the gradient magnitude $|Du|$.
• $\beta$ is a vector field on the boundary satisfying an oblique condition ($\beta \cdot n \geq \delta_0 > 0$).
• $f$ and $g$ are given source and boundary data.

Key Challenge: The function $\Phi$ allows for both:

1. Degeneracy: $\Phi(x, t) \to 0$ as $t \to 0$ (e.g., $t^\gamma$ with $\gamma > 0$).
2. Singularity: $\Phi(x, t) \to \infty$ as $t \to 0$ (e.g., $t^\gamma$ with $-1 < \gamma < 0$).

The authors aim to establish global $C^{1,\alpha}$ regularity (Hölder continuity of the gradient) for solutions, extending previous results that were limited to specific power-type nonlinearities or homogeneous boundary conditions.

2. Methodology

The proof relies on a compactness argument combined with an iterative improvement of flatness technique. The core strategy involves approximating the solution by a sequence of affine functions at shrinking scales.

A. Unified Framework via Transformation

A major technical hurdle is handling the singular case ($-1 < i(\Phi) < 0$) and the degenerate case ($i(\Phi) \geq 0$) simultaneously.

• The authors utilize a transformation (Lemma 2.2) to convert the singular equation into a degenerate one.
• By defining $\tilde{\Phi}(x, t) = \Phi(x, t)t^{-i(\Phi)}$, the equation is rewritten in a form where the singularity is absorbed into a source term involving $|Du|^\gamma$ (with $\gamma \in [0, 1)$).
• This allows the analysis to focus on a unified problem where $i(\Phi) \geq 0$ and a gradient-dependent source term exists.

B. Scaling and Uniform Estimates

The proof employs a scaling argument to analyze the behavior of solutions at different scales $r_k$.

• The solution $u$ is rescaled to define a sequence of functions $v_k$.
• A critical difficulty arises because the rescaled equations depend on a parameter $q$ (related to the gradient of the approximating affine function). As $|q|$ becomes large, the source term in the rescaled equation may grow unbounded.
• Resolution: The authors restrict their analysis to regimes where the quantity $(1 + |q|^{(\gamma - i(\Phi))^+})\|f\|_{L^\infty}$ is bounded. They prove that under this condition, the necessary Hölder estimates remain uniform with respect to $q$.

C. Compactness and Improvement of Flatness

• Boundary Hölder Regularity: Using the "sliding paraboloid and cusp methods" (adapted from previous works on oblique conditions), they establish that solutions are Hölder continuous where the gradient is far from a specific point.
• Compactness Lemma (Lemma 3.2): They prove that if the data ($f, g, \beta, \phi$) and the parameter $q$ are sufficiently small, the solution $u$ can be approximated by a $C^{1,\alpha_0}$ function $h$ solving a homogeneous equation with constant oblique boundary conditions.
• Iterative Step (Proposition 3.1): By iterating the compactness argument, they construct a sequence of affine functions $\{l_k\}$ such that the error $\|u - l_k\|_{L^\infty}$ decays at a rate of $r_k^{1+\alpha}$. This decay rate implies $C^{1,\alpha}$ regularity.

3. Key Contributions

1. Unified Treatment of Singular and Degenerate Cases: Unlike previous works that treated singular ($\gamma < 0$) and degenerate ($\gamma > 0$) cases separately or restricted to power-type nonlinearities, this paper provides a unified framework for a broad class of nonlinearities $\Phi(x, t)$ satisfying structural conditions (A2).
2. Oblique Boundary Conditions on $C^1$ Domains: The results extend global regularity theory to domains with only $C^1$ boundaries (not requiring $C^2$) under non-homogeneous oblique boundary conditions.
3. Handling Non-Homogeneous Singularities: The paper addresses the significant difficulty of non-homogeneous boundary conditions in the singular regime, where standard methods (like the Ishii-Lions method) are not straightforward to apply. The authors circumvent this by using the compactness method and careful scaling estimates.
4. Optimal Regularity Exponents: The paper derives explicit ranges for the Hölder exponent $\alpha$ depending on the ellipticity constants, the domain geometry, and the growth rates $i(\Phi)$ and $s(\Phi)$ of the function $\Phi$.

4. Main Results

Theorem 1.1 (Global $C^{1,\alpha}$ Regularity):
Let $u$ be a viscosity solution to the problem (1.1). Under assumptions (A1)–(A3), $u \in C^{1,\alpha}(\Omega \cap B_{1/2})$ with the estimate:
$$\|u\|_{C^{1,\alpha}} \leq C \left( \|u\|_{L^\infty} + \|f\|_{L^\infty}^{1/(1+i(\Phi))} + \|g\|_{C^\alpha} \right).$$
The admissible exponent $\alpha$ is given by:

• Degenerate Case ($i(\Phi) \geq 0$): $\alpha \in (0, \alpha_0) \cap \left(0, \frac{1}{1+s(\Phi)}\right] \cap (0, \bar{\alpha}]$.
• Singular Case ($-1 < i(\Phi) < 0$): $\alpha \in (0, \alpha_0) \cap \left(0, \frac{1}{1+s(\Phi)-i(\Phi)}\right] \cap (0, \bar{\alpha}]$.

Here, $\alpha_0$ is the optimal exponent for the homogeneous equation with constant oblique boundary conditions, and $C$ depends on structural constants, the modulus of continuity of the boundary, and the data.

Corollary 1.1: The result is extended to Robin-type boundary conditions ($\beta \cdot Du + hu = g$).

5. Significance

This work represents a significant advancement in the regularity theory of fully nonlinear elliptic equations.

• Generality: It moves beyond the restrictive "power-type" assumptions ($\Phi(t) = t^\gamma$) to include more complex nonlinearities, such as those involving variable exponents or sums of different powers.
• Robustness: By successfully handling the singular case with oblique conditions on $C^1$ domains, the paper fills a gap in the literature where such combinations were previously unresolved or required stronger domain regularity ($C^2$).
• Methodological Impact: The combination of the sliding cusp method with a refined scaling argument to handle the parameter $q$ in the singular regime offers a robust template for future studies on singular/degenerate PDEs with complex boundary conditions.

The paper effectively bridges the gap between the well-understood uniformly elliptic case and the challenging singular/degenerate regimes, providing a comprehensive regularity theory for a wide class of physical and geometric models governed by such equations.