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

Abstract

We prove optimal boundary $C^{1,\hat{I}\pm }$ regularity for viscosity solutions of degenerate fully nonlinear uniformly elliptic equations with oblique boundary conditions and Hamiltonian terms of the form \[ \begin{cases} |Du|^γF(D^2 u) + \varrho(x)|Du|^σ = f(x) & \text{in } Ω,\\ β(x)\cdot Du+ζ(x)u = g(x) & \text{on } \partial Ω, \end{cases} \] where ${\hat{I}}^3>0$ and $0<\ddot{I}ƒ\le 1+{\hat{I}}^3$. 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.

18 pages