Chapter 37

Basic Theory Of Sobolev SpacesDefinition 37.0.1 Let U be an open set of Rn. Define Xm,p (U) as the set of all functions inLp (U) whose weak partial derivatives up to order m are also in Lp (U) where 1 ≤ p. Thenorm1 in this space is given by

||u||m,p ≡

(∫U

∑|α|≤m

|Dα u|p dx

)1/p

.

where α = (α1, · · · ,αn)∈Nn and |α| ≡∑α i. Here D0u≡ u.C∞(U)

is defined to be the setof functions which are restrictions to U of a function in C∞

c (Rn). Thus C∞(U)⊆W m,p (U) .

The Sobolev space, W m,p (U) is defined to be the closure of C∞(U)

in Xm,p (U) with respectto the above norm. Denote this norm by ||u||W m,p(U), ||u||Xm,p(U) , or ||u||m,p,U when it isimportant to identify the open set, U.

Also the following notation will be used pretty consistently.

Definition 37.0.2 Let u be a function defined on U. Define

ũ(x)≡{

u(x) if x ∈U0 if x /∈U .

Theorem 37.0.3 Both Xm,p (U) and W m,p (U) are separable reflexive Banach spaces pro-vided p > 1.

Proof: Define Λ : Xm,p (U)→ Lp (U)w where w equals the number of multi indices, α,such that |α| ≤ m as follows. Letting {α i}w

i=1 be the set of all multi indices with α1 = 0,

Λ(u)≡ (Dα1u,Dα2u, · · · ,Dαw u) = (u,Dα2u, · · · ,Dαw u) .

Then Λ is one to one because one of the multi indices is 0. Also

Λ(Xm,p (U))

is a closed subspace of Lp (U)w . To see this, suppose

(uk,Dα2uk, · · · ,Dαwuk)→ ( f1, f2, · · · , fw)

in Lp (U)w . Then uk → f1 in Lp (U) and Dα j uk → f j in Lp (U) . Therefore, letting φ ∈C∞

c (U) and letting k→ ∞,∫U (Dα j uk)φdx = (−1)|α|

∫U ukDα j φdx

↓ ↓∫U f jφdx (−1)|α|

∫U f1Dα j φdx ≡ Dα j ( f1)(φ)

1You could also let the norm be given by ||u||m,p ≡ ∑|α|≤m ||Dα u||p or ||u||m,p ≡ max{||Dα u||p : |α| ≤ m

}because all norms are equivalent on Rp where p is the number of multi indices no larger than m. This is usedwhenever convenient.

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