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 $F(D^{2}u)=f$, 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 Hölder exponent for gradient regularity of $F(D^{2}h)=0$, 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.