Partial C2,α and W2, VMO regularity for nondifferentiable fully nonlinear elliptic equations and Isaacs equations
Abstract
We investigate the partial regularity of viscosity solutions to fully nonlinear elliptic equations of the form F(D2u, x) = f(x) in previously unexplored settings: (i) When F is non-differentiable with respect to D2u. (ii) When either F or f lacks Dini continuity in x. In the first scenario, we introduce a tangent operator property for F such that, for a fixed point x0 and a given matrix M0, F(M, x0) is tangential to a convex or concave operator at M0. We establish partial C2,α regularity for equations exhibiting this property. In the second scenario, we establish new partial W2, VMO regularity provided that the oscillation of F in x under the Ln-norm is of order o(log−1(1/r)). We address the Hausdorff dimensions of the singular sets of solutions for equations of the form F(D2u)=f(x) with data f in W1,n. Additionally, we demonstrate that the tangent operator property holds for various classes of equations, including nondifferentiable twisted elliptic equations, infimum of C1 equations, semiconvex equations, and piecewise differentiable equations. Finally, we tackle the issue of smooth enveloping and obtain partial regularity of solutions for three subclasses of Isaacs equations.
Related
- Potential estimates and improved Sobolev regularity for nonlocal fully nonlinear elliptic equations
- Boundary Regularity for Fully Nonlinear Singular Elliptic Equations
- Regularity for degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications
- Local and global C1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type
- Regularity Estimates for Quasilinear Elliptic Equations with Measure Data and Matrix-Valued Weight
BibTeX
@misc{gutierrez2026partial,
title = {{Partial $C^2,α$ and $W^2, VMO$ regularity for nondifferentiable fully nonlinear elliptic equations and Isaacs equations}},
author = {Cristian E. Gutiérrez and Qingbo Huang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00137},
eprint = {2611.00137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00137},
}
RIS
TY - UNPB
TI - Partial C^2,α and W^2, VMO regularity for nondifferentiable fully nonlinear elliptic equations and Isaacs equations
AU - Cristian E. Gutiérrez
AU - Qingbo Huang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00137
UR - https://arxiv.org/abs/2611.00137
AB - We investigate the partial regularity of viscosity solutions to fully nonlinear elliptic equations of the form F(D^2u, x) = f(x) in previously unexplored settings: (i) When F is non-differentiable with respect to D^2u. (ii) When either F or f lacks Dini continuity in x. In the first scenario, we introduce a tangent operator property for F such that, for a fixed point x_0 and a given matrix M_0, F(M, x_0) is tangential to a convex or concave operator at M_0. We establish partial C^2,α regularity for equations exhibiting this property. In the second scenario, we establish new partial W^2, VMO regularity provided that the oscillation of F in x under the L^n-norm is of order o(log^−1(1/r)). We address the Hausdorff dimensions of the singular sets of solutions for equations of the form F(D^2u)=f(x) with data f in W^1,n. Additionally, we demonstrate that the tangent operator property holds for various classes of equations, including nondifferentiable twisted elliptic equations, infimum of C^1 equations, semiconvex equations, and piecewise differentiable equations. Finally, we tackle the issue of smooth enveloping and obtain partial regularity of solutions for three subclasses of Isaacs equations.
ER -