An Introduction to Second Order Partial Differential Equations

Classical and Variational Solutions

Doina Cioranescu, Patrizia Donato;Marian P Roque

• Publisher: WSPC
• Year: 2017
• eBook ISBN: 9789813229198
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

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The book extensively introduces classical and variational partial differential equations (PDEs) to graduate and post-graduate students in Mathematics. The topics, even the most delicate, are presented in a detailed way. The book consists of two parts which focus on second order linear PDEs. Part I gives an overview of classical PDEs, that is, equations which admit strong solutions, verifying the equations pointwise. Classical solutions of the Laplace, heat, and wave equations are provided. Part II deals with variational PDEs, where weak (variational) solutions are considered. They are defined by variational formulations of the equations, based on Sobolev spaces. A comprehensive and detailed presentation of these spaces is given. Examples of variational elliptic, parabolic, and hyperbolic problems with different boundary conditions are discussed.

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--> Contents:

• Preface
• List of Symbols
• Classical Partial Differential Equations:
  • What is a Partial Differential Equation?
  • Classification of Partial Differential Equations
  • Elliptic Equations
  • Parabolic Equations
  • Hyperbolic Equations
• Variational Partial Differential Equations:
  • L p -spaces
  • The Sobolev Spaces W 1,p
  • Sobolev Embedding Theorems
  • Variational Elliptic Problems
  • Variational Evolution Problems
• Bibliography
• Index

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--> Readership: Graduate and post-graduate students as well as researchers who are interested in PDEs in both classical and variational approaches. -->
Keywords:Partial Differential Equations;Strong and Weak Solutions;Sobolev Spaces;Variational MethodsReview: Key Features:

• Consists of two parts, each self-contained, often discussed in separated textbooks in the literature. The foundation of the theories is presented with the properties and main results, most of them given with full proofs. Some theorems and proofs presented are not included in many PDE books but are vital foundational concepts
• Can be used as a textbook for one to three courses. It is detailed and yet, easy to read, thanks to many examples and remarks, and it is mathematically rigorous
• The detailed presentation of the book allows graduate and post-graduate students in Mathematics to be introduced to the world of second order PDEs. It can also be used by young researchers as a reference for variational second order PDEs
• This book is based on the long experience of the authors as researchers and teachers in the field of PDEs, teaching both in their home universities and in master or research schools in many countries abroad

Part II

Variational Partial Differential Equations

Chapter 6

L^p-spaces

In the previous chapters, we have solved well-posed PDEs which admit classical solutions, that is, solutions that are smooth functions possessing continuous derivatives. We have also seen in Section 3.6 that some PDEs do not have solutions. In particular, Example 3.21 shows that the problem −Δu = f does not always have a solution when N ≥ 2 and f is only continuous. Such equations model physical phenomena, hence a solution is expected to exist, but maybe not in the classical sense. In this case, the idea is to consider suitable functional spaces where a solution can exist in a “ weaker” sense. Sobolev spaces are the appropriate ones in this setting. We introduce them in the next chapter.

Here, we give some preliminary notions. We start with a short presentation of Banach, Hilbert, and in particular, L^p-spaces. We also give a review of the notions of weak and weak^* convergence which will be needed in our discussion. For details on L^p-spaces, we refer the reader to [3], [4], [33], and [34] and for proofs and more details concerning Banach and Hilbert spaces, to [14], [33], [34], [43], and [44].

6.1Some properties of Banach spaces

Let us begin by recalling the definition of Banach spaces which will be the functional spaces used here.

Definition 6.1. Let X be a vector space. A mapping ||⋅|| : X

R₀⁺ is called a norm on X if for every x, y ∈ X, the following are satisfied:

(1) ||x|| = 0

x = 0,

(2) ||λx|| = |λ| ||x|| for any λ ∈ R ,

(3) ||x + y|| ≤ ||x|| + ||y||.

The space X together with its norm denoted by ||⋅||_X is called a normed space. If the space X is a complete metric space with respect to the associated metric ρ given by

then X is called a Banach space.

Remark 6.2. Let X be a normed vector space with...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Preface
• Contents
• List of Symbols
• Classical Partial Differential Equations
• Variational Partial Differential Equations
• Bibliography
• Index