On ABP Estimates for a Class of Fully Nonlinear Singular-Degenerate Elliptic Equations and Applications
Abstract
We establish an Alexandrov-Bakelman-Pucci type estimate for a class of fully nonlinear elliptic equations with singular or degenerate coefficients and nonlinear dependence on the gradient. The estimate bounds sub-solutions in terms of their boundary values and weighted integrals of the data. It holds for classical sub-solutions and for viscosity sub-solutions with locally Lipschitz gradients. The assumptions allow general rates of growth in the gradient, with a smallness condition on the data when the associated gradient weight has finite integral. The proof uses the classical weighted contact-set argument. As applications, we obtain sup-norm estimates for viscosity sub-solutions to a class of generalized Monge-Ampère equations, Hessian equations, Hessian quotients under an additional condition on the contact Hessian, and anisotropic mean-curvature-type equations.
Related
- Regularity for degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications
- Gradient rearrangement estimates for singular and degenerate anisotropic p-Laplace equations
- Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
- Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures
- Boundedness and contractive estimates for orthotropic, widely degenerate, doubly nonlinear diffusion equations
BibTeX
@misc{fang2026estimates,
title = {{On ABP Estimates for a Class of Fully Nonlinear Singular-Degenerate Elliptic Equations and Applications}},
author = {Junyuan Fang and Jack Peltier and Tuoc Phan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30112},
eprint = {2610.30112},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30112},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On ABP Estimates for a Class of Fully Nonlinear Singular-Degenerate Elliptic Equations and Applications
AU - Junyuan Fang
AU - Jack Peltier
AU - Tuoc Phan
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30112
UR - https://arxiv.org/abs/2610.30112
AB - We establish an Alexandrov-Bakelman-Pucci type estimate for a class of fully nonlinear elliptic equations with singular or degenerate coefficients and nonlinear dependence on the gradient. The estimate bounds sub-solutions in terms of their boundary values and weighted integrals of the data. It holds for classical sub-solutions and for viscosity sub-solutions with locally Lipschitz gradients. The assumptions allow general rates of growth in the gradient, with a smallness condition on the data when the associated gradient weight has finite integral. The proof uses the classical weighted contact-set argument. As applications, we obtain sup-norm estimates for viscosity sub-solutions to a class of generalized Monge-Ampère equations, Hessian equations, Hessian quotients under an additional condition on the contact Hessian, and anisotropic mean-curvature-type equations.
ER -