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

New York: Springer, 2015. - 123p.The purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.History, Examples, Motivation and First Definitions
Second Definitions and Basic Analytic Properties of the Notions
Stability Properties of the Notions and Existence via Approximation
Mollification of Viscosity Solutions and Semiconvexity
Existence of Solution to the Dirichlet Problem via Perron’s Method
Comparison Results and Uniqueness of Solution to the Dirichlet ProblemMinimisers of Convex Functionals and Existence of Viscosity Solutions to the Euler-Lagrange PDE
Existence of Viscosity Solutions to the Dirichlet Problem for the $$\infty $$ ∞ -Laplacian
Miscellaneous Topics and Some Extensions of the Theory