Page 47 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

34 Preliminaries

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Definition 1.40 (i) Let Ω ⊂ R be open and bounded. We say that Ω is a
k
bounded open set with C , k ≥ 1, boundary if for every x ∈ ∂Ω,there exist a
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neighborhood U ⊂ R of x and a one-to-one and onto map H : Q → U,where
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Q = {x ∈ R : |x j | < 1,j =1, 2, ..., n}
Q ,H
H ∈ C k ¡ ¢ −1 ∈ C k ¡ ¢
U ,H (Q + )= U ∩ Ω,H (Q 0 )= U ∩ ∂Ω
with Q + = {x ∈ Q : x n > 0} and Q 0 = {x ∈ Q : x n =0}.
(ii) If H is in C k,α , 0 <α ≤ 1,we will say that Ω is a boundedopenset
with C k,α boundary.
(iii) If H is only in C 0,1 , we will say that Ω is a bounded open set with
Lipschitz boundary.

Figure 1.1: regular boundary

Remark 1.41 Every polyhedron has Lipschitz boundary, while the unit ball in
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R has a C ∞ boundary.
In the next two theorems (see for references Theorems 5.4 and 6.2 in Adams
[1]) we will write some inclusions between spaces; they have to be understood
up to a choice of a representative.

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Theorem 1.42 (Sobolev imbedding theorem). Let Ω ⊂ R be a bounded
open set with Lipschitz boundary.