The Essentials of Viscosity Solutions in R^n III

(Part III) In their 1992 paper, authors Michael Crandall, Hitoshi Ishii, and Pierre-Louis Lions describe a class of solutions to certain 2nd order PDEs in R^n which they call "viscosity solutions". In brief, a viscosity solution to an appropriate PDE is a continuous function possessing pointwise estimates for derivatives; they also possess a comparison principle which enables one to prove uniqueness of solutions to Dirichlet and Cauchy-Dirichlet problems. In this talk we will introduce the correct family of PDE for viscosity solutions; define viscosity solutions and prove their basic properties; discuss various results comparing viscosity and other notions of solutions; and, time permitting, we will also introduce a semicontinuous Perron's method for existence of viscosity solutions and discuss parabolic viscosity solutions.