**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 W_{0}^{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 H

_{0}

*(Ω)... 27*

^{k}**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 R** ^{n}*.

**1.** **The Spaces W**^{j , p}**(**Ω) and W_{0}^{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*∈

*L*

^{1}

*(Ω) .*

_{loc}(iii) *Let u and v be locally integrable functions defined in *Ω*. We say that v is *
the α*th weak derivative of u if for every φ ∈C*_{0}^{∞}(Ω)

*uD*^{α}φ *dx*

### ∫

Ω^{=}

^{(−1)}

^{|}

^{α}

^{|}

### ∫

Ω

^{vφ}

^{dx}^{,}

*and we say that D*^{α}*u*=*v in the weak sense.*

(iv) *Let u and v be in L*_{loc}* ^{p}* (Ω) . We say that v is the α

*th strong derivative of u if*

*for each compact subset K of*Ω there exists a sequence {φ

*j*

*} in C*

^{|α|}

*(K)*such that φ

*→*

_{j}*u in L*

^{p}*(K) and D*

^{α}φ

*→*

_{j}*v in L*

^{p}*(K).*

**T****HEOREM ****1** *If D*^{α}*u*=*v and D*^{β}*v* =*w in the weak sense then D*^{α+ β}*u*=*w in the weak*
*sense.*

**P****ROOF** Let ψ ∈C0

∞(Ω) and φ = *D*^{β}ψ . Then

*uD*^{α +β}ψ *dx*

### ∫

Ω^{=}

^{(−1)}

^{|α|}

### ∫

^{Ω}

^{φ}

^{v dx}^{=}

^{(−1)}

^{|α}

^{|}

### ∫

^{Ω}

^{vD}^{β}

^{ψ}

^{dx}^{=}

^{(}

^{−1)}

^{|α}

^{|+|β|}

### ∫

^{Ω}

^{ψ}

^{w dx}^{.}

^{}

**Definition (mollifiers):** Let ρ ∈C_{0}^{∞}*(R** ^{n}*) be such that

(i) Supp ρ ⊂*B*_{1}(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)

### ∫

ρ(x) dx=1^{,}(iii) ρ(x)≥0 .

If ε > 0 then we set (provided that the integral exists)
*J*_{ε}*u(x )*= 1

ε* ^{n}* ρ(

*x*−

*y*

ε *)u(y) dy*

### ∫

Ω^{.}

*J*_{ε}*u 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*_{ε}*u*∈C^{∞}*(K) provided that ε <* *dist(K,∂Ω). Suppose now that*

*u*∈*L*_{loc}* ^{p}* (Ω) . Clearly

*J*_{ε}*u(x )*= ρ(y)u(x − εy) dy

*B*_{1}( 0)

### ∫

^{,}

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

*| J*_{ε}*u(x)|≤* {ρ(y)}* ^{1/ q}*{ρ(y)}

^{1/ p}*|u(x*− ε

*y)| dy*

*B*_{1}( 0)

### ∫

≤( ({ρ(y)}* ^{1/ q}*)

^{q}*dx)*

*( ({ρ*

^{1/ q}*(y)}*

^{1/ p}*| u(x*− ε

*y)|)*

^{p}*dy)*

^{1/ p}*B*_{1}( 0)

### ∫

*B*_{1}( 0)

### ∫

^{.}

*Hence | J*_{ε}*u(x)|** ^{p}*≤ ρ(y)|u(x− εy)|

^{p}*dy*

*B*_{1}( 0)

### ∫

*, and this trivially holds if p*=1 too. Integrating this, we see that

*| J*_{ε}*u(x)|*^{p}

### ∫

*K*

^{dx}^{≤}

### ∫

*1( 0)*

^{B}^{ρ(y)}

### ∫

^{K}

^{|u(x}^{− εy)|}

^{p}

^{dx dy}≤ ρ*(y)* *|u(x)|*^{p}*dx dy*

*K*_{0}

### ∫

*B*_{1}( 0)

### ∫

= *|u(x)|*^{p}*dx*

*K*_{0}

### ∫

^{,}

*where K*_{0} is a compact subset of Ω*, K*⊂*Interior(K*_{0}) and ε <*dist(K,*∂K0) . i.e. we have

*|| J*_{ε}*u||*_{L}*p**( K)*≤||u||_{L}^{p}_{( K}

0). (1)

**L****EMMA ****2** *If u*∈*L*_{loc}* ^{p}* (Ω) and K is a compact subset of Ω

*then || J*

_{ε}

*u*−

*u||*

_{L}*p*

*( K )*→

*0 as*ε →

*0.*

**P****ROOF** *Let K*_{0} be a compact subset of Ω* where K*⊂*Interior(K*_{0}) and let
ε <*dist(K,*∂K_{0}) . Let δ >*0 and let w*∈C^{∞}*(K*_{0}*) be such that || u*−*w||*_{L}*p**( K*_{0})< δ. Then
*applying (1) to u*−*w , we obtain*

*|| J*_{ε}*u*−*J*_{ε}*w||*_{L}*p**( K )*< δ. (2)

*But J*_{ε}*w(x)*−*w(x)*= ρ(y){w(x− ε*y)*−*w(x)} dy*

*B*_{1}( 0 )

### ∫

*, and this goes to zero uniformly on K*

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

*|| J*_{ε}*w*−*w||*_{L}*p**( K )*< δ. (3)

Hence, by (2) and (3)

*|| J*_{ε}*u*−*u||*_{L}*p**( K )*≤||w−*u||*_{L}*p**( K)*+|| J_{ε}*u*−*J*_{ε}*w||*_{L}*p**( K )*+|| J_{ε}*w*−*w||*_{L}*p**( K)*<3δ. (4)
Since δ* is arbitrary, || J*_{ε}*u*−*u||*_{L}*p**( K )*→0 as ε →0.

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

**T****HEOREM ****3** *Suppose that u and v are in L*_{loc}* ^{p}* (Ω) . Then D

^{α}

*u*=

*v in the weak sense if and*

*only if D*

^{α}

*u*=

*v in the strong L*

^{p}*sense.*

**P****ROOF** *Suppose that D*^{α}*u*=*v in the strong L** ^{p}* sense. Let φ ∈C

_{0}

^{∞}(Ω) and let

*K = supp*φ. Let ε >0 and take ψ ∈C

^{|}

^{α}

^{|}

*(K) so that ||*ψ −

*u||*

_{L}*p*

*( K)*< ε and

*|| D*^{α}ψ −*v||*_{L}*p**( K )*< ε. Then

| *uD*^{α}φ *dx*

### ∫

*K*

^{−}

^{(−1)}

^{|}

^{α}

^{|}

### ∫

^{K}

^{vφ}

^{dx}^{|≤|}

### ∫

^{K}^{ψD}

^{α}

^{φ}

^{dx}^{−}

^{(−1)}

^{|}

^{α}

^{|}

### ∫

^{K}^{φD}

^{α}

^{ψ}

^{dx}^{|}

+ | *(u*− ψ*)D*^{α}φ *dx|*

### ∫

*K*

^{+|}

### ∫

^{K}

^{(v}^{−}

^{D}^{α}

^{ψ)φ}

^{dx}^{|}

≤||u− ψ||_{L}*p**( K)**|| D*^{α}φ||_{L}*q**( K )*+||v−*D*^{α}ψ||_{L}*p**( K)*||φ||_{L}*q**( K )*

≤ ε(||D^{α}φ||_{L}*q**( K )*+||φ||_{L}*q**( 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*^{α}*u*=*v in the weak*
sense.

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

*D*^{α}*J*_{ε}*u(x)*= ε^{−n} *D*_{x}^{α}ρ(*x*−*y*

ε *)u(y) dy*

### ∫

Ω= ε^{−}* ^{n}*(−1)

^{|}

^{α}

^{|}

*D*

_{y}^{α}ρ(

*x*−

*y*

ε *)u(y) dy*

### ∫

Ω= ε^{−n} ρ(*x*−*y*

ε *)v(y) dy*

### ∫

Ω= *J*_{ε}*v(x).*

*But by Lemma 2, || J*_{ε}*u*−*u||*_{L}*p**( K )*→*0 and || D*^{α}*J*_{ε}*u*−*v||*_{L}*p**( K)*=|| J_{ε}*v*−*v||*_{L}*p**( K)*→0 as ε →0.

*Thus D*^{α}*u*=*v in the strong sense.*

**Definitions (i)** *|u|*^{Ω}* _{j, p}*=(

*| D*

^{α}

*u(x)|*

^{p}*dx*

### ∫

Ω|α|≤*j*

### ∑

^{)}

^{1/ p}^{.}

(ii) *C *ˆ * ^{j, p}*(Ω)=

*{u*∈C

*(Ω): | u|*

^{j}*j, p*Ω < ∞} .

(iii) *H** ^{j , p}*(Ω) = completion of ˆ

*C*

*(Ω) with respect to the norm | |*

^{j, p}*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 C

_{0}

^{∞}(Ω) with respect to the

*usual "p norm". This knowledge allows us to see what members of H*

*(Ω) "look like".*

^{j , p}*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*

*(Ω). Then for |α|≤*

^{j, p}*j , {D*

^{α}

*u*

*} is a*

_{m}*Cauchy sequence in L*

*(Ω). Hence, there are members u*

^{p}^{α}

*of L*

*(Ω) such that D*

^{p}^{α}

*u*

*→*

_{m}*u*

^{α}

*in L*

*(Ω). Hence, according to our definition of strong derivatives, u*

^{p}^{0}

*is in L*

*(Ω) and u*

^{p}^{α}is the α

*strong derivative of u*

^{0}. Hence we see that

*H** ^{j , p}*(Ω)=

*{u*∈

*L*

*(Ω): u has strong L*

^{p}*derivatives of order ≤ j in L*

^{p}*(Ω) and there exists a*

^{p}*sequence {u*

*} in ˆ*

_{m}*C*

*(Ω) such that D*

^{j, p}^{α}

*u*

*→*

_{m}*D*

^{α}

*u in L*

*(Ω)}.*

^{p}**Definition** *W** ^{j , p}*(Ω)=

*{u*∈

*L*

*(Ω): the weak derivatives of order ≤ j of u are in*

^{p}*L*

*(Ω)}*

^{p}*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*

*(Ω) . In fact, H*

^{j , p}*(Ω)=*

^{j , p}*W*

*(Ω). This is not obvious because for*

^{j, p}*members of H** ^{j , p}*(Ω) we can find sequences of nice functions such that D

^{α}

*u*

*→*

_{m}*D*

^{α}

*u in*

*the topology of L*

*(Ω), while according to our definition of strong derivatives, such limits*

^{p}*exist only in the topology of L*

_{loc}*(Ω) for members of W*

^{p}*(Ω) . Before proving that*

^{j , p}*H** ^{j , p}*(Ω)=

*W*

*(Ω), we need the concept of a partition of unity.*

^{j, p}**L****EMMA**** 4** *Let E*⊂*R*^{n}* 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 ∈C_{0}^{∞}*(R*^{n}*) 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

### ∑

^{=}

*1 for each x*∈

*E (because of (ii), this sum is finite),*

(iv) *if * G ={Ω1,Ω2*, . ..} where each *Ω*i** is bounded and *Ω *i* ⊂*E then the family*
*F of such functions can be constructed so that * F =*{f*_{1}*, f*_{2}*, ..} and*
*supp f** _{j}* ⊂ Ω

*.*

_{j}*The family of functions F is called a partition of unity subordinate to the cover G.*

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

*N* *U** _{i}*, where each

*U*

*∈G*

_{i}*. Pick compact sets E*

*⊂*

_{i}*U*

_{i}*such that E*⊂ ∪

*i*=1

*N* *E** _{i}*. Let

*g*

*=*

_{i}*J*

_{ε}

*χ*

_{i}*E*

*, where ε*

_{i}*i*

*is chosen to be so small that supp g*

*⊂*

_{i}*U*

_{i}*. Then g*

*∈C0*

_{i}∞*(U** _{i}*) and

*g*

*>*

_{i}*0 on a neighborhood of E*

_{i}*. Let g*=

### ∑

_{i=1}

^{N}*g*

_{i}*, and let S*=

*supp g*⊂ ∪

_{i}

^{N}_{=1}

*U*

*. If ε <*

_{i}*dist(E,∂S) then k*=

*J*

_{ε}χ

_{S}*is zero on E and h*=

*g*+

*k*∈

*C*

^{∞}

*(R*

^{n}*). Further, h*>

*0 on R*

^{n}*and h*=

*g on E. Thus*F =

*{f*

_{i}*: f*

*=*

_{i}*g*

_{i}*/ h} does the job.*

*If E is open, let*

*E** _{i}* =

*E*∩

*B*

*(0)∩*

_{i}*{x: dist(x,∂E)*≥1

*i*}.

*Thus E*_{i}* is compact and E*= ∪*i*=1

*N* *E** _{i}*. Let G

*be the collection of all open sets of the form*

_{i}*U*∩

*[Interior(E*

_{i}_{+}

_{1})−

*E*

_{i}_{−}

_{2}], where

*U*∈G

*and E*

_{0}=

*E*

_{−}

_{1}= ∅. The members of G

*provide*

_{i}*an open cover for the compact set E*

*−*

_{i}*Interior(E*

_{i}_{−}

_{1}), so they possess a partition of unity F

*with finitely many elements. We let*

_{i} *s(x)*= _{g}_{∈F} *g(x)*

### ∑

*i*

*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*

*provide the required partition of unity. *

_{i}**T**

**HEOREM**

**5**(Meyers and Serrin, 1964)

*H*

*(Ω)=*

^{j , p}*W*

*(Ω).*

^{j, p}**P****ROOF** *We already know that H** ^{j , p}*(Ω)⊂

*W*

*(Ω) . The opposite inclusion follows*

^{j , p}*if we can show that for every u*∈W

*and for every ε >*

^{j ,p}*0 we can find w*∈

*C*ˆ

*such that for |α|≤*

^{j, p}*j , || D*

^{α}

*w*−

*D*

^{α}

*u||*

_{L}*p*(Ω)< ε.

*For m* ≥1 let

Ω* _{m}*=

*{x*∈Ω

*: || x||<m, dist(x,*∂Ω)> 1

*m*}

and let Ω_{0}= 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

*− Ω*

_{m+2}*. As in the "conversely" part of the proof of Theorem 3, we can pick*

_{m}ε*m* >*0 so small that w** _{m}* =

*J*

_{ε}

_{m}*(uψ*

*m*) has support in Ω

*m*+3− Ω

*m*−1

*and |w*

*−*

_{m}*uψ*

*m*|

*< ε 2*

_{j, p}*.*

^{m}*Let w*= Σ

*m*=1

∞ *w*_{m}*. This is a C*^{∞} function because on each set Ω*m*+2− Ω *m* we have
*w* =*w*_{m}_{−}_{2}+*w*_{m}_{−}_{1}+*w** _{m}*+

*w*

_{m}_{+}

_{1}+

*w*

_{m}_{+}

_{2}. Further,

*|| D*^{α}*w*−*D*^{α}*u||*_{L}*p*(Ω)=||Σ*m*=1

∞ *D*^{α}*(w** _{m}*−

*uψ*

*m*)||

_{L}*p*(Ω)

≤ Σ*m*=1

∞ *|| D*^{α}*(w** _{m}*−

*uψ*

*m*)||

_{L}*p*(Ω)

≤ Σ*m*=1

∞ ε / 2* ^{m}* = ε.

**Remarks**

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

*(Ω) .*

^{j , p}(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

*(Ω) or C*

^{j , p}*(Ω )∩*

^{j}*C*ˆ

*(Ω) is also dense in W*

^{j, p}*(Ω) . But the following example shows that this cannot always be expected.*

^{j , p}**Problem 1** Let Ω =*{(x, y) : 1*<*x*^{2}+*y*^{2} <*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 ∈C^{1}(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.

**T****HEOREM**** 6** *If *Ω* has the segment property then the set of restrictions to *Ω* of functions*
*in C*_{0}^{∞}*(R*^{n}*) is dense in W** ^{m, p}*(Ω) .

**T****HEOREM**** 7** **Change of Variables and the Chain Rule.** *Let V, Ω be domains*
*in R*^{n}* 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*

^{o}

*T*∈W

^{j , p}*(V) and the*

*derivatives of v are given by the chain rule.*

**P****ROOF** *Let y denote coordinates in Ω and let x denote coordinates in V*
*( y*=*T(x) ). If f* ∈*L** ^{p}*(Ω

*) then f*

^{o}

*T*∈

*L*

^{p}*(V ) because*

*| f*^{o}*T|*^{p}*dx*

### ∫

*V*

^{=}

### ∫

Ω

^{| f |}

^{p}

^{J dy}^{≤}

^{const.}### ∫

Ω

^{| f |}

^{p}

^{dy}^{(5)}

*(Here J is the Jacobian of T*^{−}^{1}).

*If u*∈W* ^{j ,p}*(Ω), let {u

*m*} be a sequence in ˆ

*C*

*(Ω) converging to u in W*

^{j, p}*(Ω) and*

^{j , p}*set v*

*=*

^{m}*u*

_{m o}*T . By the chain rule, if |α|≤*

*j*

*D*_{x}^{α}*v** _{m}* =

### ∑

_{β ≤ α}

*(D*

_{y}^{β}

*u*

*)*

_{m}^{o}

*T R*

_{α}

_{,}

_{β}

*where the R*_{α}_{,}_{β}* are bounded terms involving T and its derivatives. But for |β|≤* *j*
*D*_{y}^{β}*u*∈*L** ^{p}*(Ω) ⇒

*(D*

_{y}^{β}

*u)*

^{o}

*T*∈L

^{p}*(V)*⇒

*(D*

_{y}^{β}

*u)*

^{o}

*TR*

_{α}

_{,β}∈

*L*

^{p}*(V) since the R*

_{α}

_{,β}are bounded.

Further,

*|| D*_{x}^{α}*v** _{m}* − Σ

_{β ≤α}

*(D*

_{y}^{β}

*u)*

^{o}

*TR*

_{α}

_{,}

_{β}||

_{L}*p*

*( V )*=||Σ

_{β ≤α}

*(D*

_{y}^{β}

*u*

*−*

_{m}*D*

_{y}^{β}

*u)*

^{o}

*TR*

_{α}

_{,}

_{β}||

_{L}*p*

*( V )*

≤ Σ_{β ≤α}*||(D*_{y}^{β}*u** _{m}* −

*D*

_{y}^{β}

*u)*

^{o}

*TR*

_{α}

_{,}

_{β}||

_{L}*p*

*( V )*

≤*const.Σ*_{β ≤α}*||(D*_{y}^{β}*u** _{m}*−

*D*

_{y}^{β}

*u)*

^{o}

*T||*

_{L}*p*

*( V )*

≤*const.Σ*_{β ≤α}*|| D*_{y}^{β}*u** _{m}*−

*D*

_{y}^{β}

*u||*

_{L}*p*(Ω)

by (5). So (α =*0 case), v** ^{m}*→

*v*=

*u*

^{o}

*T in L*

^{p}*(V ) and*

*D*_{x}^{α}*v** _{m}*→ Σ

_{β ≤α}

*(D*

_{y}^{β}

*u)*

^{o}

*T R*

_{α}

_{,}

_{β}in

*L*

^{p}*(V ). This shows that v*∈W

^{j, p}*(V) and*

*D*_{x}^{α}*v* = Σ_{β ≤ α}*(D*_{y}^{β}*u)*^{o}*T R*_{α}_{,β}.
**Definition** *W*_{0}* ^{j , p}*(Ω) =completion of C0

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

**Remarks** (i) *Clearly W*_{0}* ^{j , p}*(Ω)⊂

*W*

*(Ω) because C0*

^{j , p}∞(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. W*_{0}* ^{1, p}*(Ω)∩

*W*

*(Ω) is a useful space for studying solutions of the Dirichlet problem for second order elliptic PDE's.*

^{2, p}(iii) *C*_{0}* ^{j}*(Ω)⊂

*W*

_{0}

*(Ω) because if f ∈C0*

^{j, p}*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}*(R** ^{n}*)=

*W*

_{0}

^{j, p}*(R*

*) . Hint: Why is it enough to show that*

^{n}*C*ˆ

^{j, p}*(R*

*)⊂*

^{n}*W*

_{0}

^{j , p}*(R*

*) ?*

^{n}**Problem 3** Show that if Ω* is a domain in R*^{n}*, f* ∈W0

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

**Problem 4** *Show that if y*∈C^{1}*[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 W*_{0}* ^{j , p}*(Ω) which (see

*Problem 3) can be viewed as being a subspace of W*

^{j , p}*(R*

*). Direct derivations of these*

^{n}*results for the spaces W*

*(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*

^{j , p}*W** ^{j , p}*(Ω) to be functions in W

^{j , p}*(R*

*). This will allow us to easily deduce the Sobolev*

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

^{j , p}*(R*

*).*

^{n}**L****EMMA**** 8** *Let u*∈*R*^{n}* and f* ∈*L*^{p}*(R*^{n}*). Set f*_{δ}*(x)*= *f (x* +δu) . Then lim

δ →0 *f*_{δ} = *f in*
*L*^{p}*(R*^{n}*).*

**P****ROOF** Given ε >0, let φ ∈C_{0}^{∞}*(R*^{n}*) be such that || f* − φ||_{L}*p*< 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}||_{L}*p*< ε. Then

*|| f* − *f*_{δ}||_{L}*p*≤|| f − φ||_{L}*p*+||φ − φ_{δ}||_{L}*p*+||φ_{δ} − *f*_{δ}||_{L}*p*<3ε.
**L****EMMA**** 9** *Let R*_{+}* ^{n}* =

*{x*∈R

^{n}*: x*

*>*

_{n}*0} . C*

^{∞}

*(R*

_{+}

*)∩*

^{n}*C*ˆ

^{j, p}*(R*

_{+}

^{n}*) is dense in W*

^{j , p}*(R*

_{+}

*).*

^{n}**P****ROOF** *Suppose f is in W*^{j , p}*(R*_{+}* ^{n}*) let ε > 0 and pick φ ∈C

^{∞}

*(R*

_{+}

*)∩*

^{n}*C*ˆ

^{j, p}*(R*

_{+}

*) so that*

^{n}*|| D*^{α}φ − *D*^{α}*f ||*_{L}*p**( 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}*(R*

*) as*

^{n}ψ^{α}*(x )*= *D*^{α}φ*(x)* *, x** _{n}*>0
0

*, x*

*≤0*

_{n}

Observe that for each δ >0, φ_{δ} ∈C^{∞}*(R *_{+}* ^{n}*)∩

*C*ˆ

^{j, p}*(R*

_{+}

*) . By Lemma 8, we can pick δ >0 so that, for all |α|≤*

^{n}*j , ||*ψ

_{δ}

^{α}− ψ

^{α}||

_{L}*p*

*( R*

*)< ε*

^{n}*. But this implies that || D*

^{α}φ

_{δ}−

*D*

^{α}φ||

_{L}

^{p}

_{( R}+*n*)< ε.
Hence

*|| D*^{α}φ_{δ} −*D*^{α}*f ||*_{L}*p**( R*_{+}* ^{n}*)≤|| D

^{α}φ

_{δ}−

*D*

^{α}φ||

_{L}*p*

*( R*

_{+}

*)+||D*

^{n}^{α}φ −

*D*

^{α}

*f ||*

_{L}*p*

*( R*

_{+}

*)<2ε. *

^{n}**L****EMMA**** 10** *There exists a linear mapping E*_{0}*: W*^{j, p}*(R*_{+}* ^{n}*)→

*W*

^{j, p}*(R*

^{n}*) such that E*

_{0}

*f*=

*f*

*in R*

_{+}

^{n}*and |E*

_{0}

*f |*

_{j , p}

^{R}*≤*

^{n}*C| f |*

^{R}

_{j, p}^{+}

^{n}*, where C depends on only n and p.*

**P****ROOF** *If f* ∈C^{∞}*(R *_{+}* ^{n}*), define

*E*_{0}*f (x)*= *f (x)* *, x** _{n}* ≥0

*c*_{k}*f (x*_{1}*, x*_{2}*, . . , x*_{n}_{−}_{1},−kx*n*)

*k*=1

### ∑

*j+1*

^{, x}

^{n}^{<}

^{0}

*where the constants c*_{k}* are chosen so that E*_{0}*f (x)*∈C^{j}*(R** ^{n}*) , i.e.

(−k)^{m}*c*_{k}

*k=1*
*j*+1

### ∑

^{=}

^{1,}

^{m}^{=}

*0,1,2, . . , j .*

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

*|| D*^{α}*E*_{0}*f ||*_{L}*p**( R** ^{n}*)≤

*C|| D*

^{α}

*f ||*

_{L}*p*

*( R*

_{+}

*). (6)*

^{n}*If now f*∈W

^{j, p}*(R*

_{+}

^{n}*) , take a sequence f*

*∈C*

_{m}^{∞}

*(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 E*_{0}*f*_{m}* is a Cauchy sequence in W*^{j , p}*(R*^{n}*). We denote the limit by E*_{0}*f . Since*

*|| D*^{α}*E*_{0}*f** _{m}*||

_{L}*p*

*( R*

*)≤*

^{n}*C|| D*

^{α}

*f*

*||*

_{m}

_{L}*p*

*( 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 Ω

*j*

such that there are mappings ψ* _{j}*:Ω

*→*

_{j}*B , where B is the unit ball centered at the origin*and

(i) ψ*j*(Ω*j*∩ Ω)=*B*∩*R*_{+}* ^{n}*
(ii) ψ

*(Ω*

_{j}*∩ ∂Ω)=*

_{j}*B*∩ ∂R

_{+}

*(iii) ψ*

^{n}*j*∈C

*(Ω*

^{m}*j*) and ψ

*j*

−1 ∈C^{m}*(B ).*

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

**T****HEOREM*** 11 If *Ω

*is a bounded domain of class C*

^{m}*then there exists a bounded linear*

*extension operator E:W*

*(Ω)→*

^{m , p}*W*

^{m , p}*(R*

^{n}*).*

**P****ROOF** 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}*Ω*

_{j=1}*and*

_{j}*let d*=

*dist(∂Ω,∂U ). Setting*Ω0 =

*{x*∈Ω

*: dist(x,∂Ω)*>

*d / 2} , we see that*Ω0,Ω1,Ω2, . .,Ω

*N*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, . . ,θ

*N*for Ω and supp θ

*⊂ 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}*.*

_{j}*Let f* ∈W* ^{m, p}*(Ω) . Then fθ

*j*∈W

*(Ω ∩ Ω*

^{m, p}*j*), so by our chain rule theorem (Theorem 7)

*w** _{j}* =

*( fθ*

*)*

_{j}^{o}ψ

_{j}^{−}

^{1}∈

*W*

^{m, p}*(R*

_{+}

*∩*

^{n}*B). Clearly supp w*

*is bounded away from*

_{j}∂B , so we can extend w*j** 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 R*

^{n}*by use of the extension operator E*

_{0}of Lemma 10. Let

*w *˜ * _{j}* =

*E*

_{0}

*w . If*ρ <1 is chosen so that supp θ

*j o*ψ

*j*

−1 ⊂*B *_{ρ}(0) , then we observe from the way
*that E*_{0} was constructed that supp ˜ *w ** _{j}* ⊂

*B*

_{ρ}(0). Consequently, supp

*w *˜ * _{j o}*ψ

*j*⊂ ψ

*j*

*(B*

_{ρ}(0)) is bounded away from ∂Ω

_{j}*. Further, again by Theorem 7, this function is in W*

*(Ω*

^{m, p}*). We*

_{j}*extend it to be in W*

^{m, p}*(R*

*) by defining it to be zero outside Ω*

^{n}*j*. If we call the extended

*function g*

_{j}*, it is clear from our construction that g*

*=*

_{j}*fθ*

*j*on Ω ∩ Ω

*j*and that

*|g** _{j}*|

_{m, p}

^{R}*≤*

^{n}*C| f |*

_{m , p}^{Ω}

*, where C is independent of f. Finally, we let g*

_{0}denote the function

*obtained by extending fθ*

_{0}to be zero outside Ω

*and define Ef*= Σ

_{j}

^{N}_{=0}

*g*

*. *

_{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

*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*

_{j}*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**

**T****HEOREM *** 12 If *Ω ⊂

*R*

^{n}*satisfies the cone condition (with height h and opening α) and if*

*p*>

*1, mp*>

*n then W*

*(Ω)⊂*

^{m, p}*C*

*(Ω) and there is a constant C depending on only α*

_{B}*, h,*

*n and p such that for all u*∈W

*(Ω), sup|u|≤*

^{m , p}*C|u|*

*.*

_{m , p}Note: Ω does not have to be bounded as Friedman suggests in his Theorem 9.1!

**P****ROOF** *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, . . ,θ

*n*−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

### ∫

*h*

^{u(r,θ)} dr} = (−1)^{m}

*(m*−1)! *r*^{m}^{−}^{1} ∂^{m}

∂r^{m}*{g(r / h)*

0

### ∫

*h*

^{u(r,}^{θ}

^{)} dr ,}*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

### ∫

*h*

^{u(r,θ}^{)} drdS}^{θ}

### ∫

*A*

=*c* *r*^{m}^{−}* ^{n}* ∂

^{m}∂r^{m}*{g(r / h)*

0

### ∫

*h*

^{u(r,}^{θ}

^{)} r}

^{n}^{−}

^{1}

^{drdS}^{θ}

### ∫

*A*

=*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}*||

_{L}*q*

*( V*

*)||*

_{x}_{∂r}

^{∂}

^{m}*m*

*{g(r / h)u(r,*θ)}||

_{L}

^{p}

_{( V}*x*)

≤*const.|| r*^{m}^{−}* ^{n}*||

_{L}*q*

*( 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

*(Ω),*

^{m , p}*take a sequence {u*

*} of functions in ˆ*

_{k}*C*

*(Ω) converging to u in the | |*

^{m, p}

_{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*

*(Ω).*

_{B}*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 W*_{0}* ^{m, p}*(Ω) and the cone condition is

*not required. Note that here we can even conclude that W*

_{0}

*(Ω)⊂*

^{m, p}*{u*∈

*C*

*(Ω*

_{B}*) : u*=0 on

∂Ω} .

**C****OROLLARY**** 13** *If *Ω ⊂*R*^{n}* 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}*(Ω) and there is a constant C depending*

^{k}*on only*α

*, h, n, k and p such that for all u*∈W

*(Ω) sup*

^{m , p}|α|≤*k**| D*^{α}*u|*≤*C|u|** _{m , p}*.

**P****ROOF** *Apply the previous theorem to the derivatives D*^{α}*u 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 W*_{0}* ^{m, p}*(Ω) ? See Problem 6.

**T****HEOREM *** 14 If *Ω ⊂

*R*

^{n}*is any domain and p*>

*n then W*

_{0}

*(Ω)⊂*

^{1, p}*C*

^{0,α}(Ω

*), where*α =1−

^{n}*p*

*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*_{i}*u||*_{L}*p*(Ω)
*i=1*

### ∑

*n*

^{.}

**P****ROOF** *Let u*∈C_{0}^{∞}(Ω) . We might as well assume that u∈C_{0}^{∞}*(R** ^{n}*) . Let

*d*=|| x −

*y|| , S*

*=*

_{x}*B*

_{d}*(x), S*

*=*

_{y}*B*

_{d}*(y) and S*=

*S*

*∩*

_{x}*S*

*. Then*

_{y}*|u(x)*−*u(y)| vol(S)*= *|u(x)*−*u(y)| dz*

### ∫

*S*

≤ *|u(x)*−*u(z)|+|u(z)*−*u(y)| dz*

### ∫

*S*

≤ *|u(x)*−*u(z)| dz*

*S*_{x}

### ∫

^{+}

### ∫

^{S}*y*

^{|u(z)}^{−}

^{u(y)| dz}*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}

### ∫

^{≤}

### ∫ ∫

^{0}

^{d}### ∫

^{0}

^{r}^{|}

_{∂ρ}

^{∂u}

^{| dρ}^{r}

^{n}^{−}

^{1}

^{drdS}^{θ}

≤ |∂u

∂ρ*| dρr*^{n}^{−}^{1}*drdS*_{θ}

0

### ∫

*d*0

### ∫

*d*

### ∫

= *d*^{n}

*n* |∂u

∂ρ*| dρ*

0

### ∫

*d*

^{dS}^{θ}

### ∫

= *d*^{n}

*n* ρ^{1}^{−}* ^{n}*|∂u

∂ρ|ρ^{n}^{−}^{1}*dρ*

0

### ∫

*d*

^{dS}^{θ}

### ∫

= *d*^{n}

*n* ρ^{1−}* ^{n}*|∂u

∂ρ*| dz*

*S*_{x}

### ∫

≤ *d*^{n}

*n* ||ρ^{1}^{−}* ^{n}*||

_{L}*q*

*( S*

*)||∂u*

_{x}∂ρ||_{L}*p**( S** _{x}*)

*where q*= *p*

*p*−1. A simple calculation shows that

||ρ^{1}^{−}* ^{n}*||

_{L}*q*

*( S*

*)=*

_{x}*const. d*

1−*n*
*p*

and it is easy to see that

||∂u

∂ρ||_{L}*p**( S** _{x}*)≤

*const.*

*|| D*

_{i}*u||*

_{L}*p*(Ω)

*i*=1

### ∑

*n*

^{.}

*Also, vol(S)*=*const. d*^{n}* 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*_{i}*u||*_{L}*p*(Ω)
*i=1*

### ∑

*n*

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

*for u*∈C0

∞(Ω) we have || u||* _{C}*0,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 C*_{0}^{∞}(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.