Elements of Advanced Mathematical Analysis for Physics and Engineering

Filippo Gazzola, Alberto Ferrero, Maurizio Zanotti

• Publisher: Società Editrice Esculapio
• Year: 2015
• Print ISBN: 9788874886456
• eBook ISBN: 9788874888955
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Deep comprehension of applied sciences requires a solid knowledge of Mathematical Analysis. For most of high level scientific research, the good understanding of Functional Analysis and weak solutions to differential equations is essential. This book aims to deal with the main topics that are necessary to achieve such a knowledge. Still, this is the goal of many other texts in advanced analysis; and then, what would be a good reason to read or to consult this book? In order to answer this question, let us introduce the three Authors. Alberto Ferrero got his degree in Mathematics in 2000 and presently he is researcher in Mathematical Analysis at the Università del Piemonte Orientale. Filippo Gazzola got his degree in Mathematics in 1987 and he is now full professor in Mathematical Analysis at the Politecnico di Milano. Maurizio Zanotti got his degree in Mechanical Engineering in 2004 and presently he is structural and machine designer and lecturer professor in Mathematical Analysis at the Politecnico di Milano.
The three Authors, for the variety of their skills, decided to join their expertises to write this book. One of the reasons that should encourage its reading is that the presentation turns out to be a reasonable compromise among the essential mathematical rigor, the importance of the applications and the clearness, which is necessary to make the reference work pleasant to the readers, even to the inexperienced ones. The range of treated topics is quite wide and covers the main basic notions of the scientific research which is based upon mathematical models. We start from vector spaces and Lebesgue integral to reach the frontier of theoretical research such as the study of critical exponents for semilinear elliptic equations and recent problems in fluid dynamics. This long route passes through the theory of Banach and Hilbert spaces, Sobolev spaces, differential equations, Fourier and Laplace transforms, before which we recall some appropriate tools of Complex Analysis. We give all the proofs that have some didactic or applicative interest, while we omit the ones which are too technical or require too high level knowledge. This book has the ambitious purpose to be useful to a broad variety of readers. The first possible beneficiaries are of course the second or third year students of a scientific course of degree: in what follows they will find the topics that are necessary to approach more advanced studies in Mathematics and in other fields, especially Physics and Engineering. This text could be also useful to graduate students who want to start a Ph.D. course: indeed it contains the matter of a multidisciplinary Ph.D. course given by Filippo Gazzola for several years at Politecnico di Milano. Finally, this book could be addressed also to the ones who have already left education far-back but occasionally need to use mathematical tools: we refer both to university professors and their research, and to professionals and designers who want to model a certain phenomenon, but also to the nostalgics of the good old days when they were students. It is precisely for this last type of reader that we have also reported some elementary topics, such as the properties of numerical sets and of the integrals; moreover, every chapter is provided with examples and specific exercises aimed at the involvement of the reader.

Chapter 1

Banach and Hilbert spaces

In this chapter we present the basic notions of Functional Analysis that will be useful in the following chapters. Abstract functional spaces and their properties are an essential tool in the study of integral transforms and partial differential equations.

The first part of the chapter is devoted to the definitions and the properties of Banach spaces. Such spaces are nothing else than vector spaces in which is defined a positive function, called norm. The norm allows to measure the distance between two vectors of the space. In this way, thanks to the acquired structure of metric space, on a Banach space it will be possible to introduce the notions of convergence of a sequence, of limit and of continuity of a functional. Banach spaces are endowed with a further fundamental property, the completeness, whose definition can be found in this chapter.

In the remaining part of the chapter we analyze more in detail Hilbert spaces and the linear mappings between them. Hilbert spaces are Banach spaces on which is possible to define a scalar product. Particular emphasis is also given to the spectral properties of the linear mappings from a Hilbert space into itself.

1.1Vector spaces and norms

Let us begin with a list of basic notions of Linear Algebra.

Definition 1.1.1. Let

be the real field

or the complex field

. The elements of the field

will be called scalars. A vector space (or linear space) over the field

is a non empty set X such that:

(i) for every pair of elements x, y ∈ X there exists a third element of X, denoted by x + y and called sum of x plus y,

(ii) for every x ∈ X and λ ∈

there exists an element of X, denoted by λx and called product of λ and x.

Such operations must satisfy the following properties:

For every x, y, z ∈ X there holds (x + y) + z = x + (y + z).

There exists an element 0 ∈ X, called zero vector or zero, such that for every x ∈ X there holds 0 + x = x + 0 = x.

For every x ∈ X there exists an element −x, called the opposite of x, such that x + (−x) = (−x) + x = 0.

For every x, y ∈ X there holds x + y = y + x.

For every x, y ∈ X and for every λ ∈

there holds λ(x + y) = λx + λy.

For every x ...

Table of contents

• Cover
• Title Page
• Copyright
• Contents
• Preface
• 1. Banach and Hilbert spaces
• 2. Lebesgue integral and Lp spaces
• 3. Classical solutions of linear partial differential equations
• 4. Complex Analysis
• 5. Integral transforms
• 6. Hilbertian Sobolev spaces
• 7. Weak solutions of linear partial differential equations
• 8. Some nonlinear partial differential equations
• Glossary of the prerequisites
• Bibliography
• Index