Postmodern Analysis

Jürgen Jost

384 pages, parution le 12/11/2002 (2^eme édition)

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Résumé

This is an introduction to advanced analysis at the beginning graduate level that blends a modern presentation with concrete examples and applications, in particular in the areas of calculus of variations and partial differential equations. The book does not strive for abstraction for its own sake, but tries rather to impart a working knowledge of the key methods of contemporary analysis, in particular those that are also relevant for application in physics. It provides a streamlined and quick introduction to the fundamental concepts of Banach space and Lebesgue integration theory and the basic notions of the calculus of variations, including Sobolev space theory.

The new edition contains additional material on the qualitative behavior of solutions of ordinary differential equations, some further details on Lp and Sobolev functions, partitions of unity and a brief introduction to abstract measure theory.

Contents

Chapter I. Calculus for Functions of One Variable

Prerequisites
Limits and Continuity of Functions
Differentiability
Characteristic Properties of Differentiable Functions. Differential Equations
The Banach Fixed Point Theorem. The Concept of Banach Space
Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli
Integrals and Ordinary Differential Equations

Chapter II. Topological Concepts

Metric Spaces: Continuity, Topological Notions, Compact

Chapter III. Calculus in Euclidean and Banach Spaces

Differentiation in Banach Spaces
Differential Calculus in R(d)
The Implicit Function Theorem. Applications
Curves in R(d). Systems of ODEs

Chapter IV. The Lebesgue Integral

Preparations. Semicontinuous Functions
The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets
Null Functions and Null Sets. The Theorem of Fubini
The Convergence Theorems of Lebesgue Integration Theory
Measurable Functions and Sets. Jensen's Inequality. The Theorem of Egorov
The Transformation Formula

Chapter V. L(p) and Sobolev Spaces

The L(p)-Spaces
Integration by Parts. Weak
Derivatives. Sobolev Spaces Weak derivatives. Sobolev Spaces

Chapter VI. Introduction to the Calculus of Variations and Elliptic Partial Differential Equations

Hilbert Spaces. Weak Convergence
Variational Principles and Partial Differential Equations
Regularity of Weak Solutions
The Maximum Principle
The Eigenvalue Problem for the Laplace Operator

L'auteur - Jürgen Jost

Honorary Professor, Department of Mathematics, University of Leipzig
Member, Academy of Sciences and Literature, Mainz, the Saxonian Academy of Sciences, Leipzig, and the German Academy of the Natural Scientists - Leopoldina
External Faculty Member, Santa Fe Institute for the Sciences of Complexity, 1399 Hyde Park Road, Santa Fe, NM 87501, USA

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Jürgen Jost
Parution	12/11/2002
Édition	2^eme édition
Nb. de pages	384
Format	15,5 x 23,5
Couverture	Broché
Poids	597g
Intérieur	Noir et Blanc
EAN13	9783540438731