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arXiv · 2610.30112

On ABP Estimates for a Class of Fully Nonlinear Singular-Degenerate Elliptic Equations and Applications

Junyuan Fang·Jack Peltier·Tuoc Phan
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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.

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Junyuan Fang, Jack Peltier, Tuoc Phan. 2026-10-21. On ABP Estimates for a Class of Fully Nonlinear Singular-Degenerate Elliptic Equations and Applications. https://arxiv.org/abs/2610.30112

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