Juan Manfredi, University of Pittsburgh
We review the theory of viscosity solutions to non-linear elliptic partial differential equations, including the Theorem of Sums. We then provide a detailed exposition of two distinct regularity methodologies and explore their interrelation. We examine the coupling method within the framework of tug-of-war stochastic games augmented by noise and address the regularity of viscosity solutions to the $p$-Laplace equation using the Ishii-Lions method.