This is an introduction to advanced analysis that supports a modern presentation with concrete examples and applications, in particular in the areas of calculus of variations and partial differential equations. The book aims 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 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 expanded third edition contains all-new material on cover theorems, and added material on properties of various classes of weakly differential functions.

A very streamlined text on mathematical analysis at the "advanced calculus" level Particularly adapted to the needs of those going on to applications in physics Includes supplementary material: sn.pub/extras

Autorentext

Jürgen Jost studierte von 1975 bis 1980 Mathematik, Physik, Volkswirtschaftslehre und Philosophie an der Universität Bonn. Er promovierte 1980 in der Mathematik und wurde nach verschiedenen internationalen Forschungsaufenthalten 1984 als Professor für Mathematik an die Ruhruniversität Bochum und 1996 als Direktor an das neu zu gründende Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig berufen. Er ist auch Honorarprofessor an der Universität Leipzig und externes Fakultätsmitglied des Santa Fe Institute for the Sciences of Complexity in den USA. 1993 erhielt er den Gottfried-Wilhelm-Leibniz-Preis der DFG und 2010 einen Advanced Grant des European Research Council. Er ist Autor von mehr als 20 Forschungsmonographien und Lehrbüchern und von über 400 wissenschaftlichen Fachpublikationen. In seinen Forschungen verbindet er eine Vielzahl von mathematischen Disziplinen und Methoden mit einer allgemeinen Theorie komplexer Systeme und vielfältigen Anwendungen in der mathematischen und theoretischen Biologie und Neurobiologie.

Inhalt
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.- Topological Concepts.- Metric Spaces: Continuity, Topological Notions, Compact Sets.- Calculus in Euclidean and Banach Spaces.- Differentiation in Banach Spaces.- Differential Calculus in $$\mathbb{R}$$ d.- The Implicit Function Theorem. Applications.- Curves in $$\mathbb{R}$$ d. Systems of ODEs.- The Lebesgue Integral.- Preparations. Semicontinuous Functions.- The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets.- Lebesgue Integrable Functions and 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.- and Sobolev Spaces.- The Lp-Spaces.- Integration by Parts. Weak Derivatives. Sobolev Spaces.- 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.