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
1
http://dx.doi.org/10.1090/gsm/120/01
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