Page 24 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 1

Preliminaries

1.1 Introduction

In this chapter we will introduce several notions that will be used throughout
the book. Most of them are concerned with different spaces of functions. We
recommend for the first reading to omit this chapter and to refer to it only when
needed in the next chapters.
In Section 1.2, we just fix the notations concerning spaces of k-times, k ≥ 0
k
an integer, continuously differentiable functions, C (Ω). We next introduce the
spaces of Hölder continuous functions, C k,α (Ω),where k ≥ 0 is an integer and
0 <α ≤ 1.
p
In Section 1.3 we consider the Lebesgue spaces L (Ω), 1 ≤ p ≤∞.We
will assume that the reader is familiar with Lebesgue integration and we will
not recall theorems such as, Fatou lemma, Lebesgue dominated convergence
theorem or Fubini theorem. We will however state, mostly without proofs, some
other important facts such as, Hölder inequality, Riesz theorem and some density
p
results. We will also discuss the notion of weak convergence in L and the
Riemann-Lebesgue theorem. We will conclude with the fundamental lemma of
the calculus of variations that will be used throughout the book, in particular
for deriving the Euler-Lagrange equations. There are many excellent books on
this subject and we refer, for example to Adams [1], Brézis [14], De Barra [37].
In Section 1.4 we define the Sobolev spaces W k,p (Ω),where 1 ≤ p ≤∞
and k ≥ 1 is an integer. We will recall several important results concerning
these spaces, notably the Sobolev imbedding theorem and Rellich-Kondrachov
theorem. We will, in some instances, give some proofs for the one dimensional
case in order to help the reader to get more familiar with these spaces. We
recommend the books of Brézis [14] and Evans [43] for a very clear introduction

11