SpringerBriefs in Mathematics

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

Authors: Katzourakis, Nikos

  • Serves as a suitable first reading on the theory of Viscosity Solutions
  • Offers an elementary overview of the topic being specifically addressed to students and non-experts
  • Can be used for a post-graduate course on the theory of Viscosity Solutions
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eBook $39.99
price for USA (gross)
  • ISBN 978-3-319-12829-0
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA
  • ISBN 978-3-319-12828-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

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

“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)


Table of contents (9 chapters)

  • History, Examples, Motivation and First Definitions

    Katzourakis, Nikos

    Pages 3-17

  • Second Definitions and Basic Analytic Properties of the Notions

    Katzourakis, Nikos

    Pages 19-33

  • Stability Properties of the Notions and Existence via Approximation

    Katzourakis, Nikos

    Pages 35-48

  • Mollification of Viscosity Solutions and Semiconvexity

    Katzourakis, Nikos

    Pages 49-61

  • Existence of Solution to the Dirichlet Problem via Perron’s Method

    Katzourakis, Nikos

    Pages 63-71

Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-3-319-12829-0
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA
  • ISBN 978-3-319-12828-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞
Authors
Series Title
SpringerBriefs in Mathematics
Copyright
2015
Publisher
Springer International Publishing
Copyright Holder
The Author(s)
eBook ISBN
978-3-319-12829-0
DOI
10.1007/978-3-319-12829-0
Softcover ISBN
978-3-319-12828-3
Series ISSN
2191-8198
Edition Number
1
Number of Pages
XII, 123
Number of Illustrations and Tables
24 b/w illustrations, 1 illustrations in colour
Topics