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.
Author(s): Nikos Katzourakis (auth.)
Series: SpringerBriefs in Mathematics
Edition: 1
Publisher: Springer International Publishing
Year: 2015
Language: English
Pages: 123
Tags: Partial Differential Equations; Calculus of Variations and Optimal Control; Optimization
Front Matter....Pages i-xii
Front Matter....Pages 1-1
History, Examples, Motivation and First Definitions....Pages 3-17
Second Definitions and Basic Analytic Properties of the Notions....Pages 19-33
Stability Properties of the Notions and Existence via Approximation....Pages 35-48
Mollification of Viscosity Solutions and Semiconvexity....Pages 49-61
Existence of Solution to the Dirichlet Problem via Perron’s Method....Pages 63-71
Comparison Results and Uniqueness of Solution to the Dirichlet Problem....Pages 73-87
Front Matter....Pages 89-89
Minimisers of Convex Functionals and Existence of Viscosity Solutions to the Euler-Lagrange PDE....Pages 91-99
Existence of Viscosity Solutions to the Dirichlet Problem for the $$\infty $$ ∞ -Laplacian....Pages 101-109
Miscellaneous Topics and Some Extensions of the Theory....Pages 111-123
