Regularity theory for a class of 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, \[ \left\{ \begin{alignedat}{2} \Phi(|Du|,x)F(D^{2}u,x)+H(|Du|,x) &=f(x) \quad && \text{in } \Omega,\\ u&=g \quad && \text{on } \partial\Omega. \end{alignedat} \right. \] Here, $F$ is uniformly elliptic, while $\Phi$ and $H$ satisfy suitable structural and growth conditions allowing for both degenerate and singular regimes. Our first result establishes sharp global $C^{1,\alpha}$ 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 H\"older-type decay assumption on $ f $, we derive sharp boundary $C^{1,\beta'}$ 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.