Gradient bounds for solutions of nonlinear strictly elliptic equations, applications and extensions

In this short manuscript, we briefly recall some well-known methods for obtaining gradient bounds of viscosity solutions for elliptic and parabolic equations. The two methods we focus here are the idea coming from Ishii- Lions’ method applies for strictly elliptic equations and the other one is socalled weak Bernstein’s method developed by Barles applies for degenerate equations. We present some results based on these methods and their applications. We also discuss some promising extensions in the future.