Regularity for degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications
Year: 2026
Type: preprint
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, \[ \left\{ \begin{alignedat}{2} Φ(|Du|,x)F(D^{2}u,x)+H(|Du|,x) & =f(x) \quad && \text{in } \quad Ω,\\ u(x) & = g(x) \quad && \text{on } \quad \partialΩ. \end{alignedat} \right. \] 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.
Source: arXiv (Cornell University)
Authors: Wang, Jiangwen, Jiang, Feida
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G283934269
OpenAlex ID: W7164353397