Borderline gradient continuity for degenerate/singular fully nonlinear elliptic equations with Hamiltonian terms

Abstract

This paper focuses on a class of fully nonlinear elliptic equations with general double phase degeneracy/singularity law and Hamiltonian terms of the form $$\Phi(|Du|,x)F(D^2 u, x)+H(Du,x) =f(x) \quad \text{in} \quad B_{1},$$ where $\Phi$ takes one of two typical forms: $$\Phi(|Du|,x)=\sigma_{1}(|Du|)+a(x)\sigma_{2}(|Du|)\quad {\rm or}\quad \Phi(|Du|,x)=\frac{\sigma_{1}(|Du|)}{|Du|}+a(x)\frac{\sigma_{2}(|Du|)}{|Du|}.$$ Under suitable assumptions on the operator $F$, Hamiltonian term $H$, source term $f$ and modulating coefficient $a$, we establish $C^{1}$ regularity for viscosity solutions, provided that $\sigma_{1},\sigma_{2}$ are moduli of continuity and their inverses are Dini continuous. Our argument is based on a tangential analysis via approximating hyperplanes combined with a new recursive renormalization algorithm adapted to the present framework. It is noteworthy that our results are new even for the case $a(x)\equiv 0$.