The object of these notes is to give a self-contained and brief treatment of the important properties of Sobolev spaces

AN INTRODUCTION TO SOBOLEV SPACES

© Steve Taylor, Montana State University.

Preface: These notes were written to supplement the graduate level PDE course at Montana State University. Sobolev Spaces have become an indispensable tool in the theory of partial differential equations and all graduate-level courses on PDE's ought to devote some time to the study of the more important properties of these spaces. The object of these notes is to give a self-contained and brief treatment of the important properties of Sobolev spaces. The main aim is to give clear proofs of all of the main results without writing an entire book on the subject! Why did I write these notes? Much of the existing literature on the subject seems to fall into two categories, either long treatises on the subject with the most general assumptions possible (and thus unsuitable for part of a PDE course), or very sketchy discussions confined to a chapter of a PDE text.

CONTENTS:

1. The Spaces W^{j , p}(Ω) and W₀^{j , p}(Ω)... 1

2. Extension Theorems... 10

3. Sobolev Inequalities and Imbedding Theorems... 13

4. Compactness Theorems... 20

5. Interpolation Results... 25

6. The Spaces H^k(Ω) and H₀^k(Ω)... 27

7. Trace Theorems... 29

Appendix: Some Spaces of Continuous Functions... 34

References... 35 In these notes, Ω is a domain (i.e. an open, connected set) in Rⁿ.

1. The Spaces W^{j , p}(Ω) and W₀^{j , p}(Ω) Definitions: Suppose 1≤ p< ∞. Then

(i) L_loc^p (Ω)={u: u∈L^p(K) for every compact subset K of Ω} (ii) u is locally integrable in Ω if u∈L¹_loc(Ω) .

(iii) Let u and v be locally integrable functions defined in Ω. We say that v is the αth weak derivative of u if for every φ ∈C₀^∞(Ω)

uD^αφ dx

∫

∫

and we say that D^αu=v in the weak sense.

(iv) Let u and v be in L_loc^p (Ω) . We say that v is the αth strong derivative of u if for each compact subset K of Ω there exists a sequence {φj} in C^|α|(K) such that φ_j →u in L^p(K) and D^αφ_j →v in L^p(K).

THEOREM 1 If D^αu=v and D^βv =w in the weak sense then D^{α+ β}u=w in the weak sense.

PROOF Let ψ ∈C0

∞(Ω) and φ = D^βψ . Then

uD^{α +β}ψ dx

∫

∫

∫

∫

Definition (mollifiers): Let ρ ∈C₀^∞(Rⁿ) be such that

(i) Supp ρ ⊂B₁(0) , (recall that "supp" denotes the support of a function, and B_r(p) denotes an open ball of radius r and center p).

(ii)

∫

If ε > 0 then we set (provided that the integral exists) J_εu(x )= 1

εⁿ ρ(x−y

ε )u(y) dy

∫

J_εu is called a mollifier of u. Note that if u is locally integrable in Ω and if K is a compact subset of Ω then J_εu∈C^∞(K) provided that ε < dist(K,∂Ω). Suppose now that

u∈L_loc^p (Ω) . Clearly

J_εu(x )= ρ(y)u(x − εy) dy

B₁( 0)

∫

so for p>1 we have (if 1 / p+1 / q=1)

| J_εu(x)|≤ {ρ(y)}^{1/ q}{ρ(y)}^{1/ p}|u(x− εy)| dy

B₁( 0)

∫

≤( ({ρ(y)}^{1/ q})^qdx)^{1/ q}( ({ρ(y)}^{1/ p}| u(x− εy)|)^p dy)^{1/ p}

B₁( 0)

∫

B₁( 0)

∫

Hence | J_εu(x)|^p≤ ρ(y)|u(x− εy)|^p dy

B₁( 0)

∫

| J_εu(x)|^p

∫

∫

∫

≤ ρ(y) |u(x)|^pdx dy

K₀

∫

B₁( 0)

∫

= |u(x)|^p dx

K₀

∫

where K₀ is a compact subset of Ω, K⊂Interior(K₀) and ε <dist(K,∂K0) . i.e. we have

|| J_εu||_Lp( K)≤||u||_L^p_{( K}

0). (1)

LEMMA 2 If u∈L_loc^p (Ω) and K is a compact subset of Ω then || J_εu−u||_Lp( K )→0 as ε →0.

PROOF Let K₀ be a compact subset of Ω where K⊂Interior(K₀) and let ε <dist(K,∂K₀) . Let δ >0 and let w∈C^∞(K₀) be such that || u−w||_Lp( K₀)< δ. Then applying (1) to u−w , we obtain

|| J_εu−J_εw||_Lp( K )< δ. (2)

But J_εw(x)−w(x)= ρ(y){w(x− εy)−w(x)} dy

B₁( 0 )

∫

as ε →0. Hence, if ε is sufficiently small, we have

|| J_εw−w||_Lp( K )< δ. (3)

Hence, by (2) and (3)

|| J_εu−u||_Lp( K )≤||w−u||_Lp( K)+|| J_εu−J_εw||_Lp( K )+|| J_εw−w||_Lp( K)<3δ. (4) Since δ is arbitrary, || J_εu−u||_Lp( K )→0 as ε →0. 

The proof of the following theorem contains some other important approximating properties of mollifiers.

THEOREM 3 Suppose that u and v are in L_loc^p (Ω) . Then D^αu=v in the weak sense if and only if D^αu=v in the strong L^p sense.

PROOF Suppose that D^αu=v in the strong L^p sense. Let φ ∈C₀^∞(Ω) and let K = supp φ. Let ε >0 and take ψ ∈C^|α|(K) so that ||ψ −u||_Lp( K)< ε and

|| D^αψ −v||_Lp( K )< ε. Then

| uD^αφ dx

∫

∫

∫

∫

+ | (u− ψ)D^αφ dx|

∫

∫

≤||u− ψ||_Lp( K)|| D^αφ||_Lq( K )+||v−D^αψ||_Lp( K)||φ||_Lq( K )

≤ ε(||D^αφ||_Lq( K )+||φ||_Lq( K )),

where q is the conjugate exponent of p (if p=1 then q= ∞ and if p>1 then 1 / p+1 / q=1). But ε is arbitrary, so the LHS must be zero. So D^αu=v in the weak sense.

Conversely, suppose that D^αu=v in the weak sense and let K be a compact subset of Ω. Then J_εu∈C^∞(K) if ε < dist(K,∂Ω) and we have for all x in K

D^αJ_εu(x)= ε⁻ⁿ D_x^αρ(x−y

ε )u(y) dy

∫

= ε⁻ⁿ(−1)^|α| D_y^αρ(x−y

ε )u(y) dy

∫

= ε⁻ⁿ ρ(x−y

ε )v(y) dy

∫

= J_εv(x).

But by Lemma 2, || J_εu−u||_Lp( K )→0 and || D^αJ_εu−v||_Lp( K)=|| J_εv−v||_Lp( K)→0 as ε →0.

Thus D^αu=v in the strong sense. 

Definitions (i) |u|^Ω_{j, p}=( | D^αu(x)|^p dx

∫

|α|≤j

∑

(ii) C ˆ ^{j, p}(Ω)={u∈C^j(Ω): | u|j, p Ω < ∞} .

(iii) H^{j , p}(Ω) = completion of ˆ C ^{j, p}(Ω) with respect to the norm | |j, p Ω .

H^{j , p}(Ω) is called a Sobolev space. We will encounter other such spaces as well.

Recall that the completion of a normed linear space is a larger space in which all Cauchy sequences converge (i.e. it is a Banach space). It is constructed by first defining a space of equivalence classes of Cauchy sequences. Two Cauchy sequences {x_m} , {y_m} are said to be in the same equivalence class if lim

m→ ∞|| x_m −y_m||=0 . A member x of the old space is identified with the equivalence class of the sequence {x, x,x, . . .} of the new space and in this sense the new space contains the old space. Further, the old space is dense in its completion. Moreover, if a normed linear space X is dense in a Banach space Y, then Y is the completion of X.

Recall that for 1≤ p< ∞, L^p(Ω) is the completion of C₀^∞(Ω) with respect to the usual "p norm". This knowledge allows us to see what members of H^{j , p}(Ω) "look like".

Members of L^p(Ω) are equivalence classes of measurable functions with finite p norms, two functions being in the same equivalence class if they differ only on a set of measure zero.

Suppose that {u_m} is a Cauchy sequence in ˆ C ^{j, p}(Ω). Then for |α|≤ j , {D^αu_m} is a Cauchy sequence in L^p(Ω). Hence, there are members u^α of L^p(Ω) such that D^αu_m →u^α in L^p(Ω). Hence, according to our definition of strong derivatives, u⁰ is in L^p(Ω) and u^α is the α strong derivative of u⁰. Hence we see that

H^{j , p}(Ω)={u∈L^p(Ω): u has strong L^p derivatives of order ≤ j in L^p(Ω) and there exists a sequence {u_m} in ˆ C ^{j, p}(Ω) such that D^αu_m→D^αu in L^p(Ω)}.

Definition W^{j , p}(Ω)={u∈L^p(Ω): the weak derivatives of order ≤ j of u are in L^p(Ω)}

Note that by Theorem 3, an equivalent definition of W^{j , p}(Ω) is obtained by writing "strong derivatives" instead of "weak derivatives". Because of this, we see easily that

H^{j , p}(Ω)⊂W^{j , p}(Ω) . In fact, H^{j , p}(Ω)=W^{j, p}(Ω). This is not obvious because for

members of H^{j , p}(Ω) we can find sequences of nice functions such that D^αu_m→D^αu in the topology of L^p(Ω), while according to our definition of strong derivatives, such limits exist only in the topology of L_loc^p (Ω) for members of W^{j , p}(Ω) . Before proving that

H^{j , p}(Ω)=W^{j, p}(Ω), we need the concept of a partition of unity.

LEMMA 4 Let E⊂Rⁿ and let G be a collection of open sets U such that E⊂{∪U: U∈G} . Then there exists a family F of non-negative functions f ∈C₀^∞(Rⁿ) such that 0≤ f (x)≤1 and

(i) for each f ∈F, there exists U∈G such that supp f ⊂U ,

(ii) if K⊂E is compact then supp f ∩K is non-empty for only finitely many f ∈F ,

(iii)

f (x)

f∈F

∑

(iv) if G ={Ω1,Ω2, . ..} where each Ωi is bounded and Ω i ⊂E then the family F of such functions can be constructed so that F ={f₁, f₂, ..} and supp f_j ⊂ Ω_j.

The family of functions F is called a partition of unity subordinate to the cover G.

PROOF Suppose first that E is compact, so there exists a positive integer N such that E⊂ ∪i=1

N U_i, where each U_i ∈G. Pick compact sets E_i ⊂U_i such that E⊂ ∪i=1

N E_i. Let g_i = J_εiχE_i , where εi is chosen to be so small that supp g_i ⊂U_i. Then g_i ∈C0

∞(U_i) and g_i >0 on a neighborhood of E_i. Let g=

∑

If E is open, let

E_i =E∩B _i(0)∩{x: dist(x,∂E)≥1 i}.

Thus E_i is compact and E= ∪i=1

N E_i. Let G_i be the collection of all open sets of the form U∩[Interior(E_i+1)−E_i−2], where U∈G and E₀ = E₋₁= ∅. The members of G_i provide an open cover for the compact set E_i−Interior(E_i−1), so they possess a partition of unity F_i with finitely many elements. We let

s(x)= _g∈F g(x)

∑

i=1

∑

and observe that only finitely many terms are represented and that s>0 on E. Now we let F be the collection of all functions of the form

f (x)= g(x)

s(x), x∈E 0, x∉E

 

 This F does the job.

If E is not open, note that any partition of unity for ∪U is a partition of unity for E.

For the proof of (iv), let H be the partition of unity obtained above and let f_i = sum of functions h in H such that supp h⊂ Ω_i, but supp h⊄ Ω_j, j<i . Note that each h is represented in one and only one of these sums and that the sums are finite since each Ω i is a compact subset of E. Thus the functions f_i provide the required partition of unity.  THEOREM 5 (Meyers and Serrin, 1964) H^{j , p}(Ω)=W^{j, p}(Ω).

PROOF We already know that H^{j , p}(Ω)⊂W^{j , p}(Ω) . The opposite inclusion follows if we can show that for every u∈W^{j ,p} and for every ε >0 we can find w∈C ˆ ^{j, p} such that for |α|≤ j , || D^αw−D^αu||_Lp(Ω)< ε.

For m ≥1 let

Ω_m={x∈Ω: || x||<m, dist(x,∂Ω)> 1 m}

and let Ω₀= Ω₋₁ = ∅. Let {ψ_m} be the partition of unity of part (iv), Theorem 4, subordinate to the cover {Ω_m+2− Ω _m} . Each uψ_m is j times weakly differentiable and has support in Ω_m+2 − Ω _m. As in the "conversely" part of the proof of Theorem 3, we can pick

εm >0 so small that w_m =J_εm(uψm) has support in Ωm+3− Ω m−1 and |w_m −uψm|_{j, p}< ε 2^m. Let w = Σm=1

∞ w_m. This is a C^∞ function because on each set Ωm+2− Ω m we have w =w_m−2+w_m−1+w_m+w_m+1+w_m+2. Further,

|| D^αw−D^αu||_Lp(Ω)=||Σm=1

∞ D^α(w_m−uψm)||_Lp(Ω)

≤ Σm=1

∞ || D^α(w_m−uψm)||_Lp(Ω)

≤ Σm=1

∞ ε / 2^m = ε.



Remarks

(i) The proof shows that in fact C^∞(Ω)∩C ˆ ^{j, p}(Ω) is dense in W^{j , p}(Ω) .

(ii) Clearly members of C^∞(Ω)∩C ˆ ^{j, p}(Ω) are not necessarily continuous on ∂Ω or even bounded near ∂Ω. It would be very useful to have the knowledge that C^∞(Ω )∩C ˆ ^{j , p}(Ω) or C^j(Ω )∩C ˆ ^{j, p}(Ω) is also dense in W^{j , p}(Ω) . But the following example shows that this cannot always be expected.

Problem 1 Let Ω ={(x, y) : 1<x²+y² <2, y≠0 if x>0}, i.e. an annulus minus the positive x-axis. Let w(x, y)= θ, the angular polar coordinate of (x,y). Clearly w is in W^1,1(Ω) because it is a bounded continuously differentiable function. Show that we cannot find a φ ∈C¹(Ω ) such that |u− ϕ|_1,1<2π. (Note that Ω is the whole annulus).

The reason for the failure of the domain in Problem 1 is that the domain is on each side of part of its boundary. The following definition expresses the idea of a domain lying on only one side of its boundary.

Definition A domain Ω has the segment property if for each x∈∂Ω there exists an open ball U centered at x and a vector y such that if z∈Ω ∩U then z+ty∈Ω for

0<t<1.

We will not need the following theorem, so we don't prove it. For a proof, see Adam's book. However, see Lemma 9 for the simpler version of the result that we will need.

THEOREM 6 If Ω has the segment property then the set of restrictions to Ω of functions in C₀^∞(Rⁿ) is dense in W^{m, p}(Ω) .

THEOREM 7 Change of Variables and the Chain Rule. Let V, Ω be domains in Rⁿ and let T: V → Ω be invertible. Suppose that T and T⁻¹ have continuous, bounded derivatives of order ≤ j . Then if u∈W^{j ,p}(Ω) we have v =u^oT∈W^{j , p}(V) and the derivatives of v are given by the chain rule.

PROOF Let y denote coordinates in Ω and let x denote coordinates in V ( y=T(x) ). If f ∈L^p(Ω) then f^oT∈L^p(V ) because

| f^oT|^p dx

∫

∫

∫

(Here J is the Jacobian of T⁻¹).

If u∈W^{j ,p}(Ω), let {um} be a sequence in ˆ C ^{j, p}(Ω) converging to u in W^{j , p}(Ω) and set v^m=u_{m o}T . By the chain rule, if |α|≤ j

D_x^αv_m =

∑

where the R_α,β are bounded terms involving T and its derivatives. But for |β|≤ j D_y^βu∈L^p(Ω) ⇒(D_y^βu)^oT ∈L^p(V)⇒(D_y^βu)^oTR_α,β ∈L^p(V) since the R_α,β are bounded.

Further,

|| D_x^αv_m − Σ_{β ≤α}(D_y^βu)^oTR_α,β||_Lp( V )=||Σ_{β ≤α}(D_y^βu_m−D_y^βu)^oTR_α,β||_Lp( V )

≤ Σ_{β ≤α}||(D_y^βu_m −D_y^βu)^oTR_α,β||_Lp( V )

≤const.Σ_{β ≤α}||(D_y^βu_m−D_y^βu)^oT||_Lp( V )

≤const.Σ_{β ≤α}|| D_y^βu_m−D_y^βu||_Lp(Ω)

by (5). So (α =0 case), v^m→v=u^oT in L^p(V ) and

D_x^αv_m→ Σ_{β ≤α}(D_y^βu)^oT R_α,β in L^p(V ). This shows that v∈W^{j, p}(V) and

D_x^αv = Σ_{β ≤ α}(D_y^βu)^oT R_α,β.  Definition W₀^{j , p}(Ω) =completion of C0

∞(Ω) with respect to the norm | |j, p Ω .

Remarks (i) Clearly W₀^{j , p}(Ω)⊂W^{j , p}(Ω) because C0

∞(Ω)⊂C ˆ ^{j , p}(Ω).

(ii) Saying that f ∈W0

j, p(Ω) is a generalized way of saying that f and its derivatives of order ≤ j−1 vanish on ∂Ω. e.g. W₀^{1, p}(Ω)∩W^{2, p}(Ω) is a useful space for studying solutions of the Dirichlet problem for second order elliptic PDE's.

(iii) C₀^j(Ω)⊂W₀^{j, p}(Ω) because if f ∈C0

j(Ω), we know that if ε is sufficiently small then J_ε f ∈C0

∞(Ω) and J_ε f → f in | |^Ω_{j, p} norm.

Problem 2 Show that W^{j , p}(Rⁿ)=W₀^{j, p}(Rⁿ) . Hint: Why is it enough to show that C ˆ ^{j, p}(Rⁿ)⊂W₀^{j , p}(Rⁿ) ?

Problem 3 Show that if Ω is a domain in Rⁿ, f ∈W0

j, p(Ω) and if f is extended to be zero outside Ω then the new function is in W^{j , p}(Rⁿ).

Problem 4 Show that if y∈C¹[0,1] and y(0)=y(1)=0 then y∈W0

1, p(0,1) . Use this fact to show that for any f ∈L^p(0,1) there is a unique y∈W0

1, p(0,1)∩W^{2, p}(0,1) such that y"−y= f . Hint: Solve the problem first with f ∈C0

∞(0,1) and then take limits.

2. Extension Theorems

Most of the important Sobolev inequalities and imbedding theorems that we will derive in the next section are most easily derived for the space W₀^{j , p}(Ω) which (see Problem 3) can be viewed as being a subspace of W^{j , p}(Rⁿ). Direct derivations of these results for the spaces W^{j , p}(Ω) are tedious and difficult because of the boundary behavior of the functions (Adams uses the direct derivation approach in his book). In this section we investigate the existence of extension operators that allow us to extend functions in

W^{j , p}(Ω) to be functions in W^{j , p}(Rⁿ). This will allow us to easily deduce the Sobolev

imbedding theorems for the spaces W^{j , p}(Ω) from the corresponding results for W^{j , p}(Rⁿ).

LEMMA 8 Let u∈Rⁿ and f ∈L^p(Rⁿ). Set f_δ(x)= f (x +δu) . Then lim

δ →0 f_δ = f in L^p(Rⁿ).

PROOF Given ε >0, let φ ∈C₀^∞(Rⁿ) be such that || f − φ||_Lp< ε. Since φ_δ → φ uniformly on a sufficiently large ball containing the supports of all φ_δ (say, for δ ≤1), we can pick δ so small that ||φ − φ_δ||_Lp< ε. Then

|| f − f_δ||_Lp≤|| f − φ||_Lp+||φ − φ_δ||_Lp+||φ_δ − f_δ||_Lp<3ε.  LEMMA 9 Let R₊ⁿ ={x ∈Rⁿ: x_n >0} . C^∞(R ₊ⁿ)∩C ˆ ^{j, p}(R₊ⁿ) is dense in W^{j , p}(R₊ⁿ).

PROOF Suppose f is in W^{j , p}(R₊ⁿ) let ε > 0 and pick φ ∈C^∞(R₊ⁿ)∩C ˆ ^{j, p}(R₊ⁿ) so that

|| D^αφ − D^αf ||_Lp( R₊ⁿ)< ε for all |α|≤ j . We take the vector of Lemma 8 to be u=(0,0,0, . . ,1) and define functions ψ^α ∈L^p(Rⁿ) as

ψ^α(x )= D^αφ(x) , x_n>0 0 , x_n≤0

 



Observe that for each δ >0, φ_δ ∈C^∞(R ₊ⁿ)∩C ˆ ^{j, p}(R₊ⁿ) . By Lemma 8, we can pick δ >0 so that, for all |α|≤ j , ||ψ_δ^α− ψ^α||_Lp( Rⁿ)< ε. But this implies that || D^αφ_δ −D^αφ||_L^p_{( R}

+n)< ε. Hence

|| D^αφ_δ −D^αf ||_Lp( R₊ⁿ)≤|| D^αφ_δ −D^αφ||_Lp( R₊ⁿ)+||D^αφ − D^αf ||_Lp( R₊ⁿ)<2ε. 

LEMMA 10 There exists a linear mapping E₀: W^{j, p}(R₊ⁿ)→W^{j, p}(Rⁿ) such that E₀f = f in R₊ⁿ and |E₀f |_{j , p}^Rn ≤C| f |^R_{j, p}⁺ⁿ, where C depends on only n and p.

PROOF If f ∈C^∞(R ₊ⁿ), define

E₀f (x)= f (x) , x_n ≥0

c_kf (x₁, x₂, . . , x_n−1,−kxn)

k=1

∑

 



where the constants c_k are chosen so that E₀f (x)∈C^j(Rⁿ) , i.e.

(−k)^mc_k

k=1 j+1

∑

It is easy to check that there is a constant C depending on only n and p such that

|| D^αE₀f ||_Lp( Rⁿ)≤C|| D^αf ||_Lp( R₊ⁿ). (6) If now f ∈W^{j, p}(R₊ⁿ) , take a sequence f_m ∈C^∞(R ₊ⁿ)∩C ˆ ^{j, p}(R₊ⁿ) converging to f in

W^{j , p}(R₊ⁿ) (we can do this by Lemma 9). Then f_m is a Cauchy sequence and (6) implies

that E₀f_m is a Cauchy sequence in W^{j , p}(Rⁿ). We denote the limit by E₀f . Since

|| D^αE₀f_m||_Lp( Rⁿ)≤C|| D^αf_m||_Lp( R₊ⁿ), taking limits shows that f satisfies (6). 

Definition A domain Ω is of class C^m if ∂Ω can be covered by bounded open sets Ωj

such that there are mappings ψ_j:Ω _j →B , where B is the unit ball centered at the origin and

(i) ψj(Ωj∩ Ω)=B∩R₊ⁿ (ii) ψ_j(Ω_j∩ ∂Ω)=B∩ ∂R₊ⁿ (iii) ψj ∈C^m(Ω j) and ψj

−1 ∈C^m(B ).

(Because of (iii), all derivatives of order ≤m of ψj and its inverse are bounded).

THEOREM 11 If Ω is a bounded domain of class C^m then there exists a bounded linear extension operator E:W^{m , p}(Ω)→W^{m , p}(Rⁿ).

PROOF Since ∂Ω is compact (boundaries are always closed), we might as well assume that the number of sets Ω_j covering ∂Ω is a finite number N. Let U= ∪^N_j=1Ω_j and let d =dist(∂Ω,∂U ). Setting Ω0 ={x ∈Ω: dist(x,∂Ω)>d / 2} , we see that Ω0,Ω1,Ω2, . .,ΩN cover Ω. These sets also cover Ω , which is compact, so by the first part of the proof of Lemma 4, there exists a finite partition of unity θ0,θ1,θ2, . . ,θN for Ω and supp θ_j ⊂ Ω_j. Recall that the support of a function is the closure of the set on which that function is non-zero. Hence, supp θ_j is even bounded away from ∂Ω_j.

Let f ∈W^{m, p}(Ω) . Then fθj ∈W^{m, p}(Ω ∩ Ωj), so by our chain rule theorem (Theorem 7)

w_j =( fθ_j)^oψ_j⁻¹∈W^{m, p}(R₊ⁿ∩B). Clearly supp w_j is bounded away from

∂B , so we can extend wj to be a member of W^{m, p}(R₊ⁿ) by letting it be zero in R₊ⁿ−B . We can further extend w_j to all of Rⁿ by use of the extension operator E₀ of Lemma 10. Let

w ˜ _j =E₀w . If ρ <1 is chosen so that supp θj oψj

−1 ⊂B _ρ(0) , then we observe from the way that E₀ was constructed that supp ˜ w _j ⊂B _ρ(0). Consequently, supp

w ˜ _{j o}ψj ⊂ ψj(B _ρ(0)) is bounded away from ∂Ω_j. Further, again by Theorem 7, this function is in W^{m, p}(Ω_j). We extend it to be in W^{m, p}(Rⁿ) by defining it to be zero outside Ωj. If we call the extended function g_j, it is clear from our construction that g_j = fθj on Ω ∩ Ωj and that

|g_j|_{m, p}^Rn ≤C| f |_{m , p}^Ω , where C is independent of f. Finally, we let g₀ denote the function obtained by extending fθ₀ to be zero outside Ω and define Ef = Σ_j^N₌₀g_j. 

Remarks The theorem can be improved in a number of ways:

(i) We can allow Ω to be unbounded if ∂Ω is bounded (e.g. Ω is the exterior of a bounded domain).

(ii) We can allow Ω to be of class C^m−1,1 instead of C^m (i.e. the derivatives of order m−1 of the functions ψ_j are Lipschitz continuous. The proof of this requires a better version of Theorem 7 which we don't have time to prove here. Note that for the case m =1, the boundary could have corners.

(iii) Calderón has proved an extension theorem for domains satisfying the cone property (see the definition below) and a few other minor assumptions. The proof is much too time-consuming for us and it relies on the Calderón-Zygmund inequality, which also has a very lengthy proof. (See [Ad] for this).

Definition A domain Ω is said to satisfy the cone property if there exist positive constants α, h such that for each x∈Ω there exists a right spherical cone V_x⊂ Ω with height h and opening α.

3. Sobolev Inequalities and Imbedding Theorems

THEOREM 12 If Ω ⊂Rⁿ satisfies the cone condition (with height h and opening α) and if p>1, mp>n then W^{m, p}(Ω)⊂C_B(Ω) and there is a constant C depending on only α, h, n and p such that for all u∈W^{m , p}(Ω), sup|u|≤C|u|_{m , p}.

Note: Ω does not have to be bounded as Friedman suggests in his Theorem 9.1!

PROOF Initially, suppose that u is in ˆ C ^{m, p}(Ω). Let g∈C^∞(R) be such that g(t)=1 if t≤1 / 2 and g(t)=0 if t≥1. Let x∈Ω and let (r, θ) denote polar coordinates centered at x. Here, θ =(θ1,θ2, . . ,θn−1) denotes the angular coordinates and we can describe the cone with vertex x in polar coordinates as V_x ={(r,θ) : 0≤r≤h,θ ∈A}. Clearly, we have

u(x)= − ∂

∂r{g(r / h)

0

∫

= (−1)^m

(m−1)! r^m−1 ∂^m

∂r^m{g(r / h)

0

∫

after m-1 integrations by parts. Next, we integrate with respect to the angular measure dS_θ, noting that the left-hand-side becomes a constant times u(x).

u(x)=c r^m−1 ∂^m

∂r^m{g(r / h)

0

∫

∫

=c r^m−n ∂^m

∂r^m{g(r / h)

0

∫

∫

=c r^m−n ∂^m

∂r^m{g(r / h)u(r,θ)} dV

V_x

∫

Applying Hölder's inequality to this, we obtain

|u(x)|≤const.||r^m−n||_Lq( V_x)||_∂r^∂mm{g(r / h)u(r,θ)}||_L^p_{( V}

x)

≤const.|| r^m−n||_Lq( V_x)|u|_{m, p}^Ω .

But r^m−n is in L^q(V_x) if n−1+(m−n)q> −1, which is the case because q= p−1^p and mp>n . Thus, we obtain sup|u|≤C|u|_{m , p}. To extend this result to arbitrary u∈W^{m , p}(Ω), take a sequence {u_k} of functions in ˆ C ^{m, p}(Ω) converging to u in the | |_{m, p}^Ω norm. Then sup|u_j −u_k|≤C|u_j −u_k|_{m , p}, showing that the sequence is a Cauchy sequence in C_B(Ω).

Thus u is in C_B(Ω) and taking the limit of sup|u_j|≤C|u_j|_{m , p} shows that u satisfies the same

inequality. 

Problem 5. Modify the proof to show that the theorem also applies to the case p=1, m=n.

Problem 6. Show that a similar theorem holds for W₀^{m, p}(Ω) and the cone condition is not required. Note that here we can even conclude that W₀^{m, p}(Ω)⊂{u∈C_B(Ω ) : u=0 on

∂Ω} .

COROLLARY 13 If Ω ⊂Rⁿ satisfies the cone condition (with height h and opening α) and if p>1, (m−k)p>n then W^{m, p}(Ω)⊂C_B^k(Ω) and there is a constant C depending on only α, h, n, k and p such that for all u∈W^{m , p}(Ω) sup

|α|≤k| D^αu|≤C|u|_{m , p}.

PROOF Apply the previous theorem to the derivatives D^αu for |α|≤k .  Problem 7 What can you conclude if p=1 and m−k=n ? See Problem 5.

Problem 8 What is the corresponding theorem for W₀^{m, p}(Ω) ? See Problem 6.

THEOREM 14 If Ω ⊂Rⁿ is any domain and p>n then W₀^{1, p}(Ω)⊂C^0,α(Ω ), where α =1−ⁿp and there exists a constant C depending on only p and n such that for all u∈W0

1, p(Ω)

|u(x)−u(y)|

|| x−y||^α ≤C || D_iu||_Lp(Ω) i=1

∑

PROOF Let u∈C₀^∞(Ω) . We might as well assume that u∈C₀^∞(Rⁿ) . Let d =|| x −y|| , S_x =B_d(x), S_y =B_d(y) and S =S_x∩S_y. Then

|u(x)−u(y)| vol(S)= |u(x)−u(y)| dz

∫

≤ |u(x)−u(z)|+|u(z)−u(y)| dz

∫

≤ |u(x)−u(z)| dz

S_x

∫

∫

But if (r,θ) are the polar coordinates of z in a coordinate system centered at x, we get

|u(x)−u(z)|≤ ∫₀^r |_∂ρ^∂u| dρ, which implies

|u(x)−u(z)| dz

S_x

∫

∫ ∫

∫

≤ |∂u

∂ρ| dρrⁿ⁻¹drdS_θ

0

∫

∫

∫

= dⁿ

n |∂u

∂ρ| dρ

0

∫

∫

= dⁿ

n ρ¹⁻ⁿ|∂u

∂ρ|ρⁿ⁻¹dρ

0

∫

∫

= dⁿ

n ρ¹⁻ⁿ|∂u

∂ρ| dz

S_x

∫

≤ dⁿ

n ||ρ¹⁻ⁿ||_Lq( S_x)||∂u

∂ρ||_Lp( S_x)

where q= p

p−1. A simple calculation shows that

||ρ¹⁻ⁿ||_Lq( S_x)=const. d

1−n p

and it is easy to see that

||∂u

∂ρ||_Lp( S_x)≤const. || D_iu||_Lp(Ω) i=1

∑

Also, vol(S)=const. dⁿ and the integral over S_y can be estimated in a similar fashion.

Putting this together yields

|u(x)−u(y)|≤Cd

1−n

p || D_iu||_Lp(Ω) i=1

∑

which is precisely the inequality that we wanted. Further, we know from Theorem 12 applied to Rⁿ that sup|u|≤C|u|_{1, p}. Combining this with the previous inequality shows that for u∈C0

∞(Ω) we have || u||_C0,α(Ω )≤C|u|_{1, p}. Thus, if we now let u∈W0

1, p(Ω) and take a sequence {u_m} of functions in C₀^∞(Ω) converging to u in | |1,p norm, it follows that {u_m} converges in C^0,α(Ω ) . Thus u∈C^0,α(Ω ) , and taking limits shows that u satisfies the

inequality in the statement of the theorem. 