A priori estimates for solutions of degenerate fully nonlinear elliptic equations with $L^p$ data

Technical Summary: A Priori Estimates for Solutions of Degenerate Fully Nonlinear Elliptic Equations with $L^p$ Data

Problem Statement
This paper investigates the interior regularity of viscosity solutions $u$ to degenerate fully nonlinear elliptic equations of the form:
$$|Du|^\gamma F(D^2u, x) = f \quad \text{in } B_1 \subset \mathbb{R}^n,$$
where $\gamma > 0$, $F$ is a uniformly elliptic fully nonlinear operator, and the right-hand side $f$ belongs to integrable spaces. The degeneracy arises because the coefficient $|Du|^\gamma$ vanishes at critical points of $u$, causing the equation to lose uniform ellipticity. The primary objective is to establish a priori estimates for $u$ based solely on the integrability norms of $f$, specifically addressing the cases where $f \in L^p$ for $p > n$ (supercritical) and $f \in L^{n,1}$ (the critical Lorentz space).

Methodology
The authors employ a combination of perturbative techniques, compactness arguments, and geometric measure-theoretic methods adapted to the degenerate structure.

1. Uniform Hölder Estimates via Sliding Methods:
   A central technical step involves establishing uniform Hölder estimates for equations that are only uniformly elliptic in specific gradient regimes. By subtracting an affine function and scaling, the authors analyze equations of the form $|Du - q|^\gamma F(D^2u, x) = f$. They utilize a dichotomy based on the gradient size:

   • Small Gradients ($|Du - q| \leq A_0$): The equation behaves like a uniformly elliptic equation. The authors apply the sliding paraboloid method (inspired by Savin, Mooney, and Imbert–Silvestre) to derive measure-type estimates.
   • Large Gradients ($|Du - q| > A_0$): The degeneracy is less severe. Here, the authors utilize a sliding cusp method to obtain similar measure estimates.
     These estimates are combined using a "growing ink-spots" lemma to prove that solutions satisfy a uniform Hölder estimate independent of the shift vector $q$, provided $f \in L^n$.

2. Approximation Lemma and Compactness:
   To bridge the gap between the uniform Hölder estimates and higher regularity ($C^{1,\alpha}$ or Log-Lipschitz), the authors develop an approximation lemma. This lemma asserts that if the oscillation of the operator $F$ (measured by $\beta_F$) and the norm of $f$ are sufficiently small in the Lorentz space $L^{n,1}$, the solution $u$ is close to a solution $h$ of a frozen, homogeneous, uniformly elliptic equation $F(D^2h) = 0$.

   • Corrector Argument: A critical difficulty arises in proving the stability of this approximation when $f$ is merely integrable. The authors construct corrector functions $\psi_j$ to handle the perturbation. To ensure these correctors converge to zero in the $C^1$ sense (specifically $D\psi_j \to 0$), they rely on borderline potential estimates for the Lorentz space $L^{n,1}$ developed by Daskalopoulos–Kuusi–Mingione. This step is noted as the reason why the $L^{n,1}$ condition is necessary rather than the weaker $L^n$ condition for the approximation lemma.

3. Iteration Schemes:

   • For $p > n$: An iterative scheme is used to approximate $u$ by affine functions $l_k(x) = a_k + b_k \cdot x$. The decay rate of the error term is controlled to establish $C^{1,\alpha}$ regularity.
   • For $p = n$ (Critical Case): A similar iteration is performed, but the convergence of the gradient coefficients $b_k$ is not guaranteed. Instead, the growth of $b_k$ is controlled linearly with the iteration step $k$, leading to a logarithmic modulus of continuity.

Key Contributions and Results

1. Optimal $C^{1,\alpha}$ Regularity for $p > n$:
   The paper establishes that if $f \in L^p(B_1)$ with $p > n$, the viscosity solution $u$ belongs to $C^{1,\alpha}_{loc}(B_1)$ for any $\alpha$ in the range:
   $$\alpha \in (0, \alpha_0) \cap \left(0, \frac{1 - n/p}{1 + \gamma}\right],$$
   where $\alpha_0$ is the exponent from the classical Krylov–Safonov theory. The authors provide a heuristic model ($u(x) = |x|^{1+\alpha}$) to demonstrate that this exponent is sharp and cannot be improved under the sole assumption $f \in L^p$. The estimate takes the form:
   $$\|u\|_{C^{1,\alpha}(B_{1/2})} \leq C(\|u\|_{L^\infty(B_1)} + \|f\|_{L^p(B_1)}^{\frac{1}{1+\gamma}}).$$

2. Log-Lipschitz Regularity for the Critical Case ($f \in L^{n,1}$):
   In the critical case where $p=n$, $C^{1,\alpha}$ regularity is generally not expected. Instead, the authors prove that if $f \in L^{n,1}(B_1)$ (the Lorentz space slightly stronger than $L^n$), the solution $u$ is locally Log-Lipschitz. Specifically, for $x, y \in B_{1/2}$:
   $$|u(x) - u(y)| \leq C(\|u\|_{L^\infty} + \|f\|_{L^{n,1}}^{\frac{1}{1+\gamma}}) \cdot (-|x-y|\log|x-y|).$$
   This result relies on the specific properties of the Lorentz space to control the corrector functions in the approximation lemma.

3. Robustness to Degeneracy:
   The work extends regularity theory to equations where ellipticity degenerates at critical points, a setting where classical methods (like the Ishii–Lions method) fail for merely integrable data. The authors replace pointwise methods with measure-theoretic approaches (sliding paraboloids and cusps) that remain robust under $L^n$ assumptions for the intermediate uniform Hölder estimates.

Significance and Scope
The paper claims to provide the first optimal interior regularity estimates for this class of degenerate equations in terms of the integrability of the source term $f$. It clarifies the precise threshold of regularity ($C^{1,\alpha}$ vs. Log-Lipschitz) dictated by the integrability of $f$ and the degeneracy parameter $\gamma$.

The authors explicitly note that their results are a priori estimates; the proofs assume a priori Hölder continuity of $f$ and the operator coefficients to justify the construction of test functions and correctors, but the final constants depend only on the $L^p$ or $L^{n,1}$ norms.

The paper does not claim to solve boundary value problems (though it references existing literature on boundary regularity) or parabolic counterparts. It focuses strictly on the interior regularity of viscosity solutions. The authors acknowledge that the subcritical range $p < n$ (where gradient Hölder regularity is not expected) remains an open question for future work, noting that solutions may still satisfy uniform Hölder estimates in that regime.