An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L8
Published on 10. December 2014
Book
Paperback/Softback
XII, 123 pages
978-3-319-12828-3 (ISBN)
Description
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.
Reviews / Votes
"In this small book, the author, after introducingbasic 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)
More details
Product info
Book
Series
Edition
2015
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
Professional and scholarly
Graduate
Illustrations
1
24 s/w Abbildungen, 1 farbige Abbildung
XII, 123 p. 25 illus., 1 illus. in color.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 8 mm
Weight
219 gr
ISBN-13
978-3-319-12828-3 (9783319128283)
DOI
10.1007/978-3-319-12829-0
Schweitzer Classification
Other editions
Additional editions
Nikos Katzourakis
An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L8
E-Book
11/2014
1st Edition
Springer
€69.54
Available for download
Content
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.