Mathematical Analysis

Foundations and Advanced Techniques for Functions of Several Variables

verfasst von: Mariano Giaquinta, Giuseppe Modica

Verlag: Birkhäuser Boston

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Mathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables builds upon the basic ideas and techniques of differential and integral calculus for functions of several variables, as outlined in an earlier introductory volume. The presentation is largely focused on the foundations of measure and integration theory.

The book begins with a discussion of the geometry of Hilbert spaces, convex functions and domains, and differential forms, particularly k-forms. The exposition continues with an introduction to the calculus of variations with applications to geometric optics and mechanics. The authors conclude with the study of measure and integration theory – Borel, Radon, and Hausdorff measures and the derivation of measures. An appendix highlights important mathematicians and other scientists whose contributions have made a great impact on the development of theories in analysis.

This work may be used as a supplementary text in the classroom or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering. One of the key strengths of this presentation, along with the other four books on analysis published by the authors, is the motivation for understanding the subject through examples, observations, exercises, and illustrations.

Inhaltsverzeichnis

Frontmatter

1. Spaces of Summable Functions and Partial Differential Equations

Abstract

This chapter aims at substantiating the abstract theory of Hilbert spaces developed in [GM3]. After introducing the Laplace, heat and wave equations we present the classical method of separation of variables in the study of partial differential equations. Then we introduce Lebesgue’s spaces of psummable functions and we continue with some elements of the theory of Sobolev spaces. Finally, we present some basic facts concerning the notion of weak solution, the Dirichlet principle and the alternative theorem.