Page 48 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Sobolev spaces 35

Case 1. If 1 ≤ p< n then
q
W 1,p (Ω) ⊂ L (Ω)
∗
for every q ∈ [1,p ] where
1 1 1 np
∗
= − , i.e. p = .
p ∗ p n n − p
More precisely, for every q ∈ [1,p ] there exists c = c (Ω,p,q) so that
∗
kuk L q ≤ c kuk W 1,p .
Case 2. If p = n then
q
W 1,n (Ω) ⊂ L (Ω) , for every q ∈ [1, ∞) .
More precisely, for every q ∈ [1, ∞) there exists c = c (Ω,p,q) so that

kuk L q ≤ c kuk W 1,n .
Case 3. If p> n then

W 1,p (Ω) ⊂ C 0,α ¡ ¢
Ω , for every α ∈ [0, 1 − n/p] .
In particular, there exists a constant c = c (Ω,p) so that

kuk L ∞ ≤ c kuk W 1,p .
The above theorem gives, not only imbeddings, but also compactness of these
imbeddings under further restrictions.
n
Theorem 1.43 (Rellich-Kondrachov Theorem). Let Ω ⊂ R be a bounded
open set with Lipschitz boundary.
q
Case 1. If 1 ≤ p<n then the imbedding of W 1,p in L is compact, for every
q ∈ [1,p ). This means that any bounded set of W 1,p is precompact (i.e., its
∗
q
closure is compact) in L for every 1 ≤ q< p (the result is false if q = p ).
∗
∗
q
Case 2. If p = n then the imbedding of W 1,n in L is compact, for every
q ∈ [1, ∞).
Ω is compact, for
Case 3. If p> n then the imbedding of W 1,p in C 0,α ¡ ¢
every 0 ≤ α< 1 − n/p.
In particular in all cases (i.e., 1 ≤ p ≤∞) the imbedding of W 1,p (Ω) in
p
L (Ω) is compact.