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.
In general, we denote by x points in
and write x = (x1, · · · , xn) in terms
of its coordinates. For any x ∈
we denote by |x| the standard Euclidean
norm, unless otherwise stated. Namely, for any x = (x1, · · · , xn), we have
Sometimes, we need to distinguish one particular direction as the time di-
rection and write points in
as (x, t) for x ∈
and t ∈ R. In this
case, we call x = (x1, · · · , xn) ∈
the space variable and t ∈ R the time
we also denote points by (x, y).
Let Ω be a domain in
that is, an open and connected subset in
We denote by C(Ω) the collection of all continuous functions in Ω, by
the collection of all functions with continuous derivatives up to order
m, for any integer m ≥ 1, and by
the collection of all functions with
continuous derivatives of arbitrary order. For any u ∈
we denote by