An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calcul...
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※上記表示の販売価格は割引適用後の価格です 出版済み 3週間でお届けいたします。 2015 ed. Series: SpringerBriefs in Mathematics Author: Katzourakis, Nikos Publisher: Springer ISBN: 9783319128283 Cover: PAPERBACK Date: 2014年12月 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. Table of Contents 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. TABLE OF CONTENTS 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.
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