A beginner's guide to the theory of viscosity solutions / Shigeaki Koike

Publisher's description: "The notion of viscosity solutions was first introduced by M. G. Crandall and P.-L. Lions in 1981 to study first-order partial differential equations in nondivergence form, typically, Hamilton-Jacobi equations. Later, the study of viscosity solutions was extended to second-order elliptic/parabolic equations. It has turned out that viscosity solution theory is a powerful tool used by many researchers to investigate fully nonlinear second-order (degenerate) elliptic/parabolic equations arising in optimal control problems, differential games, mean curvature flow, phase transitions, mathematical finance, conservation laws, variational problems, etc. This text is an introduction to viscosity solution theory as indicated by the title.
"After a brief history of weak solutions, it presents several uniqueness (comparison principle) and existence results, which are the main issues. For further topics, it chooses generalized boundary value problems and regularity results. In the Appendix, which is the hardest part, it provides proofs of several important propositions.''