Sharp regularity for degenerate fully nonlinear equations with oblique boundary conditions and Hamiltonian terms

Abstract

We prove optimal boundary $C^{1,\alpha}$ regularity for viscosity solutions of degenerate fully nonlinear uniformly elliptic equations with oblique boundary conditions and Hamiltonian terms of the form \[ \begin{cases} |Du|^{\gamma}F(D^2 u) + \varrho(x)|Du|^{\sigma} = f(x) & \text{in } \Omega,\\ \beta(x)\cdot Du+\zeta(x)u = g(x) & \text{on } \partial \Omega, \end{cases} \] where $\gamma>0$ and $0<\sigma\le 1+\gamma$. We develop a compactness framework for affine translations, linking the size of the translation to the Hamiltonian structure. This is combined with a boundary improvement-of-flatness argument adapted to oblique boundary data, yielding the optimal boundary regularity.