Within this talk, we restrict ourselves to the fully nonlinear uniformly elliptic equation of the form $F(D^2u)=0$, which is invariant under translation. We will introduce a useful approximation of the viscosity solution, called Jenson Approximation, which effectively captures the translation invariance of our PDE. Using Jenson Approximation, we establish a variation of Jenson’s uniqueness theorem for Dirichlet problem, which concerns the difference of two viscosity solutions. And again, due to the translation invariance of the PDE, we can apply the same idea to study the difference quotient of a viscosity solution, leading to interior $C^{1,\alpha}$ estimate. If time permits, we will also cover the application to concave $F$, in which we basically apply the same idea to study second order difference quotient (hence second derivatives) of a solution. (Notice: the additional concavity condition is crucial for discussing second order difference quotient.) References: [Caffarelli-Cabre Fully Nonlinear Elliptic Equations]
