Chapter 1

Introduction

This chapter serves as an introduction of the entire book.

In Section 1.1, we first list several notations we will use throughout this

book. Then, we introduce the concept of partial differential equations.

In Section 1.2, we discuss briefly well-posed problems for partial differ-

ential equations. We also introduce several function spaces whose associated

norms are used frequently in this book.

In Section 1.3, we present an overview of this book.

1.1. Notation

In general, we denote by x points in

Rn

and write x = (x1, · · · , xn) in terms

of its coordinates. For any x ∈

Rn,

we denote by |x| the standard Euclidean

norm, unless otherwise stated. Namely, for any x = (x1, · · · , xn), we have

|x| =

n

i=1

xi

2

1

2

.

Sometimes, we need to distinguish one particular direction as the time di-

rection and write points in

Rn+1

as (x, t) for x ∈

Rn

and t ∈ R. In this

case, we call x = (x1, · · · , xn) ∈

Rn

the space variable and t ∈ R the time

variable. In

R2,

we also denote points by (x, y).

Let Ω be a domain in

Rn,

that is, an open and connected subset in

Rn.

We denote by C(Ω) the collection of all continuous functions in Ω, by

Cm(Ω)

the collection of all functions with continuous derivatives up to order

m, for any integer m ≥ 1, and by

C∞(Ω)

the collection of all functions with

continuous derivatives of arbitrary order. For any u ∈

Cm(Ω),

we denote by

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http://dx.doi.org/10.1090/gsm/120/01