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

TL;DR

This paper establishes optimal boundary regularity for solutions of certain degenerate fully nonlinear elliptic equations with oblique boundary conditions, introducing a new compactness framework and boundary improvement technique.

Contribution

It introduces a novel compactness framework and boundary improvement-of-flatness method for degenerate fully nonlinear equations with oblique boundary conditions.

Findings

Proves optimal boundary $C^{1,eta}$ regularity for viscosity solutions.

Develops a new compactness framework linking affine translations to Hamiltonian structure.

Adapts boundary improvement-of-flatness to oblique boundary data.

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.