Endpoint Differentiability Moduli for Fully Nonlinear Elliptic Equations

Abstract

A classical consequence of Caffarelli's fully nonlinear regularity theory [L. A. Caffarelli, Ann. of Math. (2) 130 (1989), no. 1, 189-213] is that viscosity solutions of uniformly elliptic equations $F(D^2u)=f$, with $f\in L^p$, $p>n$, are locally $C^{1,\gamma}$ for every $\gamma<\min\{\alpha_H,\sigma_p\}$. Here $\alpha_H=\alpha_H(n,\lambda,\Lambda)\in(0,1)$ denotes the universal H\"older exponent for gradient regularity of $F(D^2h)=0$, while $\sigma_p=1-n/p$ is the scaling exponent of the source term. In the source-limited regime $\sigma_p<\alpha_H$, the singularity of $f$ is the decisive obstruction and the endpoint $\gamma=\sigma_p$ is attainable. In the homogeneous-limited regime $\alpha_H\le\sigma_p$, however, classical theory only yields $\gamma<\alpha_H$, leaving the limiting differentiability estimate unquantified. This is the endpoint gap addressed here. When $\alpha_H<\sigma_p$, we prove that solutions admit pointwise Taylor expansions satisfying $|u(x)-u(x_0)-Du(x_0)\cdot(x-x_0)|\lesssim |x-x_0|^{1+\alpha_H}\left(1+\log\frac{1}{|x-x_0|}\right)^m$. Thus the homogeneous differentiability scale is reached up to an explicit logarithmic defect. At the critical threshold $\alpha_H=\sigma_p$, finite logarithmic powers no longer close the iteration; nevertheless, a slower selection of scales yields $|u(x)-u(x_0)-Du(x_0)\cdot(x-x_0)|=O\left(|x-x_0|^{1+\alpha_H}\exp\left(A\sqrt{1+\log\frac{1}{|x-x_0|}}\right)\right)$. Both estimates improve the full family of classical sub-endpoint $C^{1,\gamma}$ bounds, $\gamma<\alpha_H$, by quantifying differentiability at the limiting homogeneous exponent. The proof introduces a new scale-selection mechanism for endpoint Campanato-type recurrences, suggesting a flexible tool whenever the limiting smoothness is dictated by the homogeneous theory itself. We also discuss the role and possible optimality of the resulting logarithmic defects.