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 $$\hat{I}¦(|Du|,x)F(D^{2}u,x)+H(Du,x)=f(x)\text{in}{B}_1,$$ where $\hat{I}¦$ takes one of two typical forms: $$\hat{I}¦(|Du|,x)=\ddot{I}{ƒ}_1(|Du|)+a(x)\ddot{I}{ƒ}_2(|Du|)\mathrm{or}\hat{I}¦(|Du|,x)=\frac{\ddot{I}{ƒ}_1(|Du|)}{|Du|}+a(x)\frac{\ddot{I}{ƒ}_2(|Du|)}{|Du|}.$$ 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 $\ddot{I}{ƒ}_1,\ddot{I}{ƒ}_2$ 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 $a(x)\equiv 0$.

27 pp