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

 eBook
Sofort lieferbar | Lieferzeit: Sofort lieferbar I
ISBN-13:
9783319128290
Einband:
eBook
Seiten:
123
Autor:
Nikos Katzourakis
Serie:
SpringerBriefs in Mathematics
eBook Typ:
PDF
eBook Format:
eBook
Kopierschutz:
Digital Watermark [Social-DRM]
Sprache:
Englisch
Beschreibung:

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.

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.