Mathematical Principles of Theoretical Physics

124 CHAPTER 3. MATHEMATICAL FOUNDATIONS

In this case,φis called thek-th weak derivative ofuj, denoted by

(3.2.3) φ=∂kuj.

The spaceC 0 ∞(M)consists of infinitely differentiable functions, and

(3.2.4) C 0 ∞(M) =

{

C∞(M) ifMis compact and∂M=/0,

{u∈C∞(M)|u 6 =0 in a compact set ofM}.

The definition (3.2.2)-(3.2.3) for weak derivatives is very abstract. In the following, we
discuss the distinction between continuity and weak differentiability in an intuitive fashion.
Letube a function defined onRn. It is known that ifuis differentiable atx=0, thenu
can be Taylor expanded as

(3.2.5) u(x) =ax+o(|x|),

wherea=∇u( 0 )is the first order derivative ofuatx=0, i.e. the gradient ofuatx=0.
However, ifuis weakly but not continuously differentiable atx=0, then in a neighbor-
hood ofx= 0 ,umust contain at least a term as|xi|α(α≤ 1 ). Without loss of generality, we
expressuin the form

(3.2.6) u=a|x|α+continuously differentiable terms,

wherea 6 =0 is a constant, and 1−n<α≤1. The indexαin (3.2.6) determines the regularity
ofuas follows:

(3.2.7)

u=Lipschitz ifα= 1 ,

u=H ̈older if 0<α< 1 ,

u=singularity ifα< 1.

Expressions (3.2.5) and (3.2.6) exhibit the essential difference between continuity and
weak differentiability.

Remark 3.6.For a weakly differentiable function as (3.2.6), its indexαhas to satisfy 1−n<
α≤1, which is crucial in the Sobolev embedding theorems in the next subsection.

3.Wk,pspaces (Sobolev spaces).Letu:M→M⊗pEN. Then,u={uj| 1 ≤j≤N},
and each componentujofuis a function onM. We denote

∂αuj=

∂kuj

(∂x^1 )α^1 ···(∂xn)αn

forα= (α 1 ,···,αn),

andk=|α|=

n

∑

i= 1

αi(αi≥ 0 ). Then, we defineWk,pspaces as

(3.2.8) Wk,p(M⊗pEN) ={u∈Lp(M⊗pEN)|∂βuj∈Lp(M), 1 ≤j≤n, 0 ≤β≤k},

and∂βujin (3.2.8) are weak derivatives ofuj.