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(D2u)=f, with f∈ Lp, p > n, are locally C1,γ for every γ< min{αH,σp}. Here αH=αH(n,λ,Λ)∈(0,1) denotes the universal Hölder exponent for gradient regularity of F(D2h)=0, while σp=1−n/p is the scaling exponent of the source term. In the source-limited regime σp < αH, the singularity of f is the decisive obstruction and the endpoint γ=σp is attainable. In the homogeneous-limited regime αH≤σp, however, classical theory only yields γ< αH, leaving the limiting differentiability estimate unquantified. This is the endpoint gap addressed here. When αH < σp, we prove that solutions admit pointwise Taylor expansions satisfying |u(x)−u(x0)−Du(x0)·(x−x0)|≲ |x−x0|1+αH(1+log1/(|x−x0|))m. Thus the homogeneous differentiability scale is reached up to an explicit logarithmic defect. At the critical threshold αH=σp, finite logarithmic powers no longer close the iteration; nevertheless, a slower selection of scales yields |u(x)−u(x0)−Du(x0)·(x−x0)|=O(|x−x0|1+αHexp(A√(1+log1/(|x−x0|)))). Both estimates improve the full family of classical sub-endpoint C1,γ bounds, γ< αH, 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.
Related
- Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices
- Calabi-Bernstein type theorems for entire solutions of certain nonlinear elliptic equations in the plane
- A new approach to interior estimates for fully nonlinear elliptic equations
- Estimates on Fully Nonlinear Elliptic Equations with Unrectifiable Exceptional Gradient Sets
- Spectral-Edge Continuation, Dimensional Crossover, and Endpoint Stability for Partially Confined Nonlinear Schrödinger Equations
BibTeX
@misc{sobral2026endpoint,
title = {{Endpoint Differentiability Moduli for Fully Nonlinear Elliptic Equations}},
author = {Aelson Sobral and Eduardo V. Teixeira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16326},
eprint = {2609.16326},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16326},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Endpoint Differentiability Moduli for Fully Nonlinear Elliptic Equations
AU - Aelson Sobral
AU - Eduardo V. Teixeira
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16326
UR - https://arxiv.org/abs/2609.16326
AB - 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∈ L^p, p > n, are locally C^1,γ for every γ< min{α_H,σ_p}. Here α_H=α_H(n,λ,Λ)∈(0,1) denotes the universal Hölder exponent for gradient regularity of F(D^2h)=0, while σ_p=1−n/p is the scaling exponent of the source term. In the source-limited regime σ_p < α_H, the singularity of f is the decisive obstruction and the endpoint γ=σ_p is attainable. In the homogeneous-limited regime α_H≤σ_p, however, classical theory only yields γ< α_H, leaving the limiting differentiability estimate unquantified. This is the endpoint gap addressed here. When α_H < σ_p, we prove that solutions admit pointwise Taylor expansions satisfying |u(x)−u(x_0)−Du(x_0)·(x−x_0)|≲ |x−x_0|^1+α_H(1+log1/(|x−x_0|))^m. Thus the homogeneous differentiability scale is reached up to an explicit logarithmic defect. At the critical threshold α_H=σ_p, finite logarithmic powers no longer close the iteration; nevertheless, a slower selection of scales yields |u(x)−u(x_0)−Du(x_0)·(x−x_0)|=O(|x−x_0|^1+α_Hexp(A√(1+log1/(|x−x_0|)))). Both estimates improve the full family of classical sub-endpoint C^1,γ bounds, γ< α_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.
ER -