Gradient bounds for viscosity solutions to certain elliptic equations

Our principal object of study is the modulus of continuity of a periodic or uniformly vanishing function $u:{\mathbb{R}}^n\to \mathbb{R}$ which satisfies a degenerate elliptic equation $F(x,u,\mathrm{\nabla }u,D^{2}u)=0$ in the viscosity sense. The equations under consideration here have second-order terms of the form $-\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\mathcal{A}(\mathrm{\parallel }\mathrm{\nabla }u\mathrm{\parallel })\cdot D^{2}u),$ where $\mathcal{A}$ is an $n\times n$ matrix which is symmetric and positive semi-definite. Following earlier work, \cite{Li21}, of the second author, which addressed the parabolic case, we identify a one-dimensional equation for which the modulus of continuity is a subsolution. In favorable cases, this one-dimensional operator can be used to derive a gradient bound on $u$ or to draw other conclusions about the nature of the solution.