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

Nikos Katzourakis (Autor)

eBook Download: PDF

2014 | 2015
XII, 123 Seiten
Springer International Publishing (Verlag)
978-3-319-12829-0 (ISBN)

Infos

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.

Inhalt

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.

Produkt-Details

• Erscheint lt. Verlag: 26.11.2014
• Reihe/Serie: SpringerBriefs in Mathematics
• Reihe/Serie: SpringerBriefs in Mathematics
• Zusatzinfo: XII, 123 p. 25 illus., 1 illus. in color.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Schlagworte: Calculus of Variations • Elliptic PDEs • Game Theory • Geometric evolution • Partial differential equations • Viscosity Solutions
• ISBN-10: 3-319-12829-9 / 3319128299
• ISBN-13: 978-3-319-12829-0 / 9783319128290