An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞

1 History, Examples, Motivation and First Definitions.- 2 Second Definitions and Basic Analytic Properties of the Notions.- 3 Stability Properties of the Notions and Existence via Approximation.- 4 Mollification of Viscosity Solutions and Semi convexity.- 5 Existence of Solution to the Dirichlet Problem via Perron's Method.- 6 Comparison results and Uniqueness of Solution to the Dirichlet Problem.- 7 Minimisers of Convex Functionals and Viscosity Solutions of the Euler-Lagrange PDE.- 8 Existence of Viscosity Solutions to the Dirichlet Problem for the Laplacian.- 9 Miscellaneous topics and some extensions of the theory.

"In this small book, the author, after introducing basic and non-basic concepts of the theory of viscosity solutions for first and second order PDEs, applies the theory to two specific problems such as existence of viscosity solution for the Euler-Lagrange PDE and for the -Laplacian. ... The book can be certainly used as text for an advanced course and also as manual for researchers." (Fabio Bagagiolo, zbMATH, Vol. 1326.35006, 2016)

"The book under review is a nice introduction to the theory of viscosity solutions for fully nonlinear PDEs ... . The book, which is addressed to a public having basic knowledge in PDEs, is based on a course given by the author ... . The explanations are very clear, and the reader is introduced to the theory step by step, the author taking the time to explain several technical details, but without making the exposition too heavy."(Enea Parini, Mathematical Reviews, November, 2015)