Regularity for degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications
Abstract
In this paper, we investigate regularity properties for viscosity solutions to a general class of degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms, { 2 Φ(|Du|,x)F(D2u,x)+H(|Du|,x) & =f(x) && in Ω, u(x) & = g(x) && on ∂Ω. . Here, F is uniformly elliptic, while Φ and H satisfy suitable structural and growth conditions allowing for both degenerate and singular regimes. Our first result establishes sharp global C1,α regularity for this general class of equations, thereby providing a unified regularity framework for both degenerate and singular regimes. We next develop an oscillation-based approach for fully nonlinear equations with unbalanced variable degeneracy and Hamiltonian terms. Under a Hölder-type decay assumption on $ f $, we derive sharp boundary C1,β' estimates with an explicit exponent, and establish a quantitative non-degeneracy estimate in the singular region. Finally, as applications of the general theory, under a suitable viscosity curvature condition on the level sets of the solution, we establish global $ C^{1,\frac{1}{3}} $ regularity for the infinity-Poisson and global $ C^{1,\frac{1}{p-1}} $ regularity for the $ p $-Poisson with $ p>2 $. These results may provide a new perspective on these two long-standing open problems.
Related
- Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
- On ABP Estimates for a Class of Fully Nonlinear Singular-Degenerate Elliptic Equations and Applications
- Global regularity for a class of degenerate/singular Monge-Ampère equations
- Local regularity for solutions to singular and degenerate quasilinear parabolic equations
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BibTeX
@misc{wang2026regularity,
title = {{Regularity for degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications}},
author = {Jiangwen Wang and Feida Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.11607},
eprint = {2606.11607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11607},
}
RIS
TY - UNPB
TI - Regularity for degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications
AU - Jiangwen Wang
AU - Feida Jiang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2606.11607
UR - https://arxiv.org/abs/2606.11607
AB - In this paper, we investigate regularity properties for viscosity solutions to a general class of degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms, { 2 Φ(|Du|,x)F(D^2u,x)+H(|Du|,x) & =f(x) && in Ω, u(x) & = g(x) && on ∂Ω. . Here, F is uniformly elliptic, while Φ and H satisfy suitable structural and growth conditions allowing for both degenerate and singular regimes. Our first result establishes sharp global C^1,α regularity for this general class of equations, thereby providing a unified regularity framework for both degenerate and singular regimes. We next develop an oscillation-based approach for fully nonlinear equations with unbalanced variable degeneracy and Hamiltonian terms. Under a Hölder-type decay assumption on $ f $, we derive sharp boundary C^1,β' estimates with an explicit exponent, and establish a quantitative non-degeneracy estimate in the singular region. Finally, as applications of the general theory, under a suitable viscosity curvature condition on the level sets of the solution, we establish global $ C^{1,\frac{1}{3}} $ regularity for the infinity-Poisson and global $ C^{1,\frac{1}{p-1}} $ regularity for the $ p $-Poisson with $ p>2 $. These results may provide a new perspective on these two long-standing open problems.
ER -