Mathematical Principles of Theoretical Physics

128 CHAPTER 3. MATHEMATICAL FOUNDATIONS

for 0<R<1 andβp>1. It follows from (3.2.16) that
∫

BR

|∇u|pdx<∞⇔k=n+ (α− 1 )p− 1 ≥ − 1.

Hence we obtain that

(3.2.17) u=

|x|α

|ln|x||β

∈W^1 ,p(BR)⇔α≥ 1 −

n

p

.

On the other hand, in the same fashion we see that

(3.2.18) u=

|x|α

|ln|x||β

∈Lq(BR)⇔α≥ −

n

q

.

Hence, by (3.2.17) we can see that asp>n,

u∈C^0 ,α(BR), α= 1 −

n

p

,

and asp≤n, then from (3.2.17) and (3.2.18), at the critical embedding exponentq∗we have

α= 1 −

n

p

, α=−

n

q∗

.

It follows that

q∗







=

np

n−p

forn>p

<∞ forn=p.

Thus, we deduce that

(3.2.19) u∈W^1 ,p(BR)⇒















u∈Lq(BR) for 1≤q≤

np

n−p

ifn>p,

u∈Lq(BR) for 1≤q<∞ ifn=p,

u∈C^0 ,α(BR) forα= 1 −

n

p

ifp>n.

The relations (3.2.19) are the embeddings given by (3.2.11).

3.2.3 Differential operators

A differential operator defined on a manifoldMis a mapping given by

G:Wk,p(M⊗pEN 11 )→Lp(M⊗pE 2 N^2 ) for somek≥ 1 ,

and is called ak-th order differential operator.
The most important operators in physics are:

1) the gradient operator:∇,

2) the divergent operator: div,