Sobolev spaces

Sobolev spaces

Department of Mathematics, Aalto University 2021

1 SOBOLEV SPACES 1

1.1 Weak derivatives . . . 1

1.2 Sobolev spaces . . . 4

1.3 Properties of weak derivatives. . . 8

1.4 Completeness of Sobolev spaces . . . 9

1.5 Hilbert space structure . . . 11

1.6 Approximation by smooth functions . . . 12

1.7 Local approximation in Sobolev spaces. . . 16

1.8 Global approximation in Sobolev spaces . . . 17

1.9 Sobolev spaces with zero boundary values. . . 18

1.10 Chain rule. . . 21

1.11 Truncation. . . 23

1.12 Weak convergence methods for Sobolev spaces. . . 25

1.13 Difference quotients. . . 33

1.14 Absolute continuity on lines . . . 36

2 SOBOLEV INEQUALITIES 42 2.1 Gagliardo-Nirenberg-Sobolev inequality . . . 43

2.2 Sobolev-Poincaré inequalities . . . 49

2.3 Morrey’s inequality . . . 55

2.4 Lipschitz functions andW^1,∞. . . 59

2.5 Summary of the Sobolev embeddings. . . 62

2.6 Direct methods in the calculus of variations . . . 62

3 MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 70 3.1 Representation formulas and Riesz potentials . . . 71

3.2 Sobolev-Poincaré inequalities . . . 78

3.3 Sobolev inequalities on domains . . . 86

3.4 A maximal function characterization of Sobolev spaces. . . 89

3.5 Pointwise estimates . . . 92

3.6 Approximation by Lipschitz functions. . . 97

3.7 Maximal operator on Sobolev spaces . . . 102

4 POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 106

4.1 Sobolev capacity . . . .106

4.2 Capacity and measure . . . .109

4.3 Quasicontinuity . . . .116

4.4 Lebesgue points of Sobolev functions . . . 119

4.5 Sobolev spaces with zero boundary values. . . 124

Sobolev spaces 1

In this chapter we begin our study of Sobolev spaces. The Sobolev space is a vector space of functions that have weak derivatives. Motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces.

1.1 Weak derivatives

Notation. LetΩ⊂Rⁿbe open,f:Ω→Randk=1, 2, . . .. Then we use the following notations:

C(Ω⁾={f :f continuous inΩ^}

suppf={x∈Ω^{: f(x)}6=0}= the support off

C₀(Ω⁾={f∈C(Ω^{) : suppf} is a compact subset ofΩ^}

C^k(Ω⁾={f∈C(Ω^{) : f isk}times continuously diferentiable}

C₀^k(Ω⁾=C^k(Ω⁾∩C₀(Ω⁾ C^∞=

\∞ k=1

C^k(Ω)= smooth functions C^∞₀(Ω⁾=C^∞(Ω⁾∩C₀(Ω⁾

= compactly supported smooth functions

= test functions

■

W A R N I N G : In general, suppf*Ω^. Examples 1.1:

(1) Letu:B(0, 1)→R,u(x)=1− |x|. Then suppu=B(0, 1).

1

(2) Letf:R→Rbe

f(x)=







x², xÊ0,

−x², x<0.

Now f∈C¹(R) \C²(R) although the graph looks smooth.

(3) Let us defineϕ:Rⁿ→R, ϕ(x)=





 e

1

|x|2−1, x∈B(0, 1), 0, x∈Rⁿ\B(0, 1).

Nowϕ∈C^∞₀(Rⁿ) and suppϕ=B(0, 1) (exercise).

Let us start with a motivation for definition of weak derivatives. LetΩ⊂Rⁿ be open,u∈C¹(Ω^{) and}ϕ∈C^∞₀(Ω). Integration by parts gives

ˆ

Ω

uÇϕ Çxj

dx= − ˆ

Ω

Çu Çxjϕdx.

There is no boundary term, sinceϕhas a compact support inΩand thus vanishes nearÇΩ^.

Let thenu∈C^k(Ω^{), k}=1, 2, . . ., and letα=(α1,α2, . . . ,αn)∈Nⁿ (we use the convention that 0∈N) be a multi-index such that the order of multi-index|α| = α1+. . .+αnis at mostk. We denote

D^αu= Ç^|α|u Çx^α₁¹. . .Çx^αnⁿ

= Ç^α1 Çx₁^α1. . . Ç^αn

Çx^αnⁿ

u.

TH E M O R A L: A coordinate of a multi-index indicates how many times a function is differentiated with respect to the corresponding variable. The order of a multi-index tells the total number of differentiations.

Successive integration by parts gives ˆ

ΩuD^αϕdx=(−1)^|α|

ˆ

ΩD^αuϕdx.

Notice that the left-hand side makes sense even under the assumptionu∈L¹_loc(Ω).

Definition 1.2. Assume thatu∈L¹_loc(Ω^{) and let}α∈Nⁿbe a multi-index. Then v∈L¹_loc(Ω^{) is the}αth weak partial derivative ofu, writtenD^αu=v, if

ˆ

Ωu D^αϕdx=(−1)^|α|

ˆ

Ωvϕdx

for every test functionϕ∈C^∞₀ (Ω). We denoteD⁰u=D^(0,...,0)=u. If|α| =1, then Du=(D1u,D2u. . . ,Dnu)

is the weak gradient ofu. Here D_ju= Çu

Çx_j=D(0,...,1,...,0)u, j=1, . . . ,n, (the jth component is 1).

TH E M O R A L: Classical derivatives are defined as pointwise limits of differ- ence quotients, but the weak derivatives are defined as a functions satisfying the integration by parts formula. Observe, that changing the function on a set of measure zero does not affect its weak derivatives.

WA R N I N G We use the same notation for the weak and classical derivatives. It should be clear from the context which interpretation is used.

Remarks 1.3:

(1) Ifu∈C^k(Ω), then the classical partial derivatives up to orderkare also the corresponding weak derivatives ofu. In this sense, weak derivatives generalize classical derivatives.

(2) Ifu=0 almost everywhere in an open set, thenD^αu=0 almost everywhere in the same set.

Lemma 1.4. A weakαth partial derivative ofu, if it exists, is uniquely defined up to a set of measure zero.

Proof. Assume thatv,ev∈L¹_loc(Ω) are both weakαth partial derivatives ofu, that

is, ˆ

ΩuD^αϕdx=(−1)^|α|

ˆ

Ωvϕdx=(−1)^|α|

ˆ

Ωevϕdx for everyϕ∈C^∞₀ (Ω). This implies that

ˆ

Ω

(v−v)eϕdx=0 for every ϕ∈C₀^∞(Ω^{). (1.1)} Claim:v=evalmost everywhere inΩ^.

Reason. Let Ω⁰bΩ (i.e. Ω⁰ is open and Ω⁰ is a compact subset of Ω). The spaceC^∞₀(Ω⁰) is dense inL^p(Ω⁰) (we shall return to this later). There exists a sequence of functions ϕi∈C₀^∞(Ω⁰) such that|ϕi| É2 inΩ⁰ andϕi→sgn(v−ev) almost everywhere inΩ^0asi→ ∞. Here sgn is the signum function.

Identity (1.1) and the dominated convergence theorem, with the majorant

|(v−v)eϕi| É2(|v| + |ev|)∈L¹(Ω^{0), give} 0=lim

i→∞

ˆ

Ω⁰

(v−v)eϕidx= ˆ

Ω⁰

lim

i→∞(v−ev)ϕidx

= ˆ

Ω⁰(v−v) sgn(ve −v)dxe = ˆ

Ω⁰|v−ev|dx

This implies thatv=evalmost everywhere inΩ^{0for every}Ω⁰bΩ^{. Thus v}=ve

almost everywhere inΩ. _■

From the proof we obtain a very useful corollary.

Corollary 1.5 (Fundamental lemma of the calculus of variations). Iff∈L¹_loc(Ω⁾

satisfies ˆ

Ωfϕdx=0

for everyϕ∈C^∞₀ (Ω^{), thenf}=0 almost everywhere inΩ^.

TH E M O R A L: This is an integral way to say that a function is zero almost everywhere.

Example 1.6. Letn=1 andΩ=(0, 2). Consider

u(x)=







x, 0<x<1, 1, 1Éx<2, and

v(x)=







1, 0<x<1, 0, 1Éx<2.

We claim thatu⁰=vin the weak sense. To see this, we show that ˆ ₂

0

uϕ⁰dx= − ˆ ₂

0

vϕdx for everyϕ∈C^∞₀ ((0, 2)).

Reason. An integration by parts and the fundamental theorem of calculus give ˆ ₂

0

u(x)ϕ⁰(x)dx= ˆ ₁

0

xϕ⁰(x)dx+ ˆ ₂

1

ϕ⁰(x)dx

=xϕ(x)

¯

¯

¯

¯

1 0

| {z }

=ϕ(1)

− ˆ ₁

0

ϕ(x)dx+ϕ(2)

| {z }

=0

−ϕ(1)

= − ˆ ₁

0

ϕ(x)dx= − ˆ ₂

0

vϕ(x)dx

for everyϕ∈C^∞₀ ((0, 2)). _■

1.2 Sobolev spaces

Definition 1.7. Assume that Ω is an open subset of Rⁿ. The Sobolev space W^k,p(Ω) consists of functionsu∈L^p(Ω) such that for every multi-indexαwith

|α| Ék, the weak derivativeD^αuexists andD^αu∈L^p(Ω^{). Thus} W^k,p(Ω⁾={u∈L^p(Ω^{) :Dαu}∈L^p(Ω^),|α| Ék}.

Ifu∈W^k,p(Ω), we define its norm kukW^k,p(Ω)=

à X

|α|Ék

ˆ

Ω|D^αu|^pdx

!¹_p

, 1Ép< ∞, and

kukW^k,∞(Ω)= X

|α|Ék

ess sup

Ω |D^αu|.

Notice thatD⁰u=D^(0,...,0)u=u. Assume thatΩ⁰is an open subset ofΩ^{. We say} thatΩ⁰is compactly contained inΩ^{, denoted}Ω⁰bΩ^{, if}Ω⁰is a compact subset of Ω. A functionu∈W^k,p

loc (Ω), ifu∈W^k,p(Ω⁰) for everyΩ⁰bΩ.

TH E M O R A L: Thus Sobolev spaceW^k,p(Ω) consists of functions inL^p(Ω^{) that} have weak partial derivatives up to orderkand they belong toL^p(Ω).

Remarks 1.8:

(1) As inL^pspaces we identifyW^k,pfunctions which are equal almost every- where.

(2) There are several ways to define a norm onW^k,p(Ω). The normk · kW^k,p(Ω)

is equivalent, for example, with the norm X

|α|ÉkkD^αukL^p(Ω), 1ÉpÉ ∞. andk · kW^k,∞(Ω)is also equivalent with

max|α|ÉkkD^αukL^∞(Ω). (3) Fork=1 we use the norm

kukW^1,p(Ω)=³

kuk_L^pp(Ω)+ kDuk^p_Lp(Ω)

´¹_p

= µˆ

Ω|u|^pdx+ ˆ

Ω|Du|^pdx

¶¹_p

, 1Ép< ∞, and

kukW^1,∞(Ω)=ess sup

Ω |u| +ess sup

Ω |Du|. We may also consider equivalent norms

kukW^1,p(Ω)= Ã

kuk_L^pp(Ω)+

n

X

j=1

°°D_ju°

°

p L^p(Ω)

!¹_p ,

kukW^1,p(Ω)= kukL^p(Ω)+

n

X

j=1

°°D_ju°

°L^p(Ω), and

kukW^1,p(Ω)= kukL^p(Ω)+ kDukL^p(Ω)

when 1Ép< ∞and kukW^1,∞(Ω)=max©

kukL^∞(Ω),kD1ukL^∞(Ω), . . . ,kDnukL^∞(Ω)

ª.

Example 1.9. Letu:B(0, 1)→[0,∞],u(x)= |x|^−α,α>0. Clearlyu∈C^∞(B(0, 1) \ {0}), butuis unbounded in any neighbourhood of the origin.

We start by showing thatuhas a weak derivative in the entire unit ball. When x6=0 , we have

Çu

Çx_j(x)= −α|x|^−α−1xj

|x|= −α xj

|x|^α+2, j=1, . . . ,n.

Thus

Du(x)= −α x

|x|^α+2. Gauss’ theorem gives

ˆ

B(0,1)\B(0,ε)

Dj(uϕ)dx= ˆ

Ç(B(0,1)\B(0,ε))

uϕνjdS,

whereν=(ν1, . . . ,νn) is the outward pointing unit (|ν| =1) normal of the boundary andϕ∈C^∞₀(B(0, 1)). Asϕ=0 onÇB(0, 1), this can be written as

ˆ

B(0,1)\B(0,ε)

D_juϕdx+ ˆ

B(0,1)\B(0,ε)

uD_jϕdx= ˆ

ÇB(0,ε)

uϕνjdS.

By rearranging terms, we obtain ˆ

B(0,1)\B(0,ε)

uD_jϕdx= − ˆ

B(0,1)\B(0,ε)

D_juϕdx+ ˆ

ÇB(0,ε)uϕνjdS. (1.2) Let us estimate the last term on the right-hand side. Sinceν(x)= −_|^x_x|, we have νj(x)= −_|x|^xj, whenx∈ÇB(0,ε). Thus

¯

¯

¯

¯ ˆ

ÇB(0,ε)

uϕνjdS

¯

¯

¯

¯É kϕkL^∞(B(0,1))

ˆ

ÇB(0,ε)

ε^−αdS

= kϕkL^∞(B(0,1))ωn−1ε^n−1−α→0 as ε→0,

ifn−1−α>0. Hereωn−1=Hⁿ⁻¹(ÇB(0, 1)) is the (n−1)-dimensional measure of the sphereÇB(0, 1).

Next we study integrability ofD_ju. We need this information in order to be able to use the dominated convergence theorem. A straightforward computation gives

ˆ

B(0,1)

¯¯D_ju¯

¯dxÉ ˆ

B(0,1)|Du|dx=α ˆ

B(0,1)|x|^−α−1dx

=α ˆ 1

0

ˆ

ÇB(0,r)|x|^−α−1dS dr=αω_n−1 ˆ 1

0

r^{−α−1+n−1}dr

=αωn−1

ˆ ₁

0

r^n−α−2dr= αωn−1

n−α−1r^n−α−1

¯

¯

¯

¯

1 0

< ∞, ifn−1−α>0.

The following argument shows thatD_ju is a weak derivative ofualso in a neighbourhood of the origin. By the dominated convergence theorem

ˆ

B(0,1)

uD_jϕdx= ˆ

B(0,1)

limε→0

³

uD_jϕχB(0,1)\B(0,ε)

´ dx

=lim

ε→0

ˆ

B(0,1)\B(0,ε)

uD_jϕdx

= −lim

ε→0

ˆ

B(0,1)\B(0,ε)

Djuϕdx+lim

ε→0

ˆ

ÇB(0,ε)

uϕνjdS

= − ˆ

B(0,1)

limε→0D_juϕχB(0,1)\B(0,ε)dx

= − ˆ

B(0,1)

D_juϕdx.

Here we used the dominated convergence theorem twice: First to the function uD_jϕχB(0,1)\B(0,ε),

which is dominated by|u|kDϕk_∞∈L¹(B(0, 1)), and then to the function D_juϕχB(0,1)\B(0,ε),

which is dominated by|Du|kϕk_∞∈L¹(B(0, 1)). We also used (1.2) and the fact that the last term there converges to zero asε→0.

Now we have proved thatuhas a weak derivative in the unit ball. We note that u∈L^p(B(0, 1)) if and only if−pα+n>0, or equivalently, α< ⁿ_p. On the other hand,|Du| ∈L^p(B(0, 1), if−p(α+1)+n>0, or equivalently,α<^n−p_p . Thus u∈W^1,p(B(0, 1)) if and only ifα<ⁿ⁻_p^p.

Let (r_i) be a countable and dense subset ofB(0, 1) and defineu:B(0, 1)→[0,∞], u(x)=

X∞ i=1

1

2ⁱ|x−ri|^−α. Thenu∈W^1,p(B(0, 1)) ifα<^n−p_p .

Reason.

kukW^1,p(B(0,1))É X∞ i=1

1 2ⁱ

°

°|x−r_i|^−α°

°W^1,p(B(0,1))

É X∞ i=1

1 2ⁱ

°

°|x|^−α°

°W^1,p(B(0,1))

=°

°|x|^−α°

°W^1,p(B(0,1))< ∞. _■

Note that ifα>0, thenuis unbounded in every open subset ofB(0, 1) and not differentiable in the classical sense in a dense subset.

TH E M O R A L: Functions inW^1,p, 1Ép<n,nÊ2, may be unbounded in every open subset.

Example 1.10. Observe, thatu(x)= |x|^−α,α>0, does not belong toW^1,n(B(0, 1).

However, there are unbounded functions inW^1,n,nÊ2. Letu:B(0, 1)→R,

u(x)=





 log³

log³ 1+_|x|¹

´´

, x6=0, 0, x=0.

Then u∈W^1,n(B(0, 1)) when nÊ2, but u∉L^∞(B(0, 1)). This can be used to construct a function inW^1,n(B(0, 1) that is unbounded in every open subset of B(0, 1) (exercise).

TH E M O R A L: Functions inW^1,p, 1ÉpÉn,nÊ2, are not continuous. Later we shall see, that everyW^1,p function with p>n coincides with a continuous function almost everywhere.

Example 1.11. The functionu:B(0, 1)→R,

u(x)=u(x₁, . . . ,x_n)=







1, x_n>0, 0, x_n<0, does not belong toW^1,p(B(0, 1) for any 1ÉpÉ ∞(exercise).

1.3 Properties of weak derivatives

The following general properties of weak derivatives follow rather directly from the definition.

Lemma 1.12. Assume thatu,v∈W^k,p(Ω^{) and}|α| Ék. Then (1) D^αu∈W^k−|α|,p(Ω^),

(2) D^β(D^αu)=D^α(D^βu) for all multi-indicesα,βwith|α| + |β| Ék, (3) for everyλ,µ∈R,λu+µv∈W^k,p(Ω^{) and}

D^α(λu+µv)=λD^αu+µD^αv, (4) ifΩ⁰⊂Ωis open, thenu∈W^k,p(Ω^0),

(5) (Leibniz’s formula) ifη∈C^∞₀ (Ω^{), then}ηu∈W^k,p(Ω^{) and}

D^α(ηu)=X

βÉα

Ãα β

!

D^βηD^α−βu,

where

Ãα β

!

= α!

β!(α−β)!, α!=α1!. . .αn! andβÉαmeans thatβjÉαjfor everyj=1, . . . ,n.

TH E M O R A L: Weak derivatives have the same properties as classical deriva- tives of smooth functions.

Proof. (1) Follows directly from the definition of weak derivatives. See also (2).

(2) Letϕ∈C^∞₀(Ω^{). ThenDβ}ϕ∈C^∞₀(Ω). Therefore (−1)^|β|

ˆ

ΩD^β(D^αu)ϕdx= ˆ

ΩD^αuD^βϕdx

=(−1)^|α|

ˆ

Ω

uD^α+βϕdx

=(−1)^|α|(−1)^|α+β|

ˆ

Ω

D^α+βuϕdx for all test functionsϕ∈C^∞₀ (Ω). Notice that

|α| + |α+β| =α1+. . .+αn+(α1+β1)+. . .+(αn+βn)

=2(α1+. . .+αn)+β1+. . .+βn

=2|α| + |β|.

As 2|α|is an even number, the estimate above, together with the uniqueness results Lemma1.4and Corollary1.5, implies thatD^β(D^αu)=D^α+βu.

(3) and (4) Clear.

(5) First we consider the case|α| =1. Letϕ∈C^∞₀ (Ω). By Leibniz’s rule for differentiable functions and the definition of weak derivative

ˆ

ΩηuD^αϕdx= ˆ

Ω

(uD^α(ηϕ)−u(D^αη)ϕ)dx

= − ˆ

Ω(ηD^αu+uD^αη)ϕdx

for allϕ∈C^∞₀(Ω). The case|α| >1 follows by induction (exercise). ä

1.4 Completeness of Sobolev spaces

One of the most useful properties of Sobolev spaces is that they are complete.

Thus Sobolev spaces are closed under limits of Cauchy sequences.

A sequence (ui) of functionsui∈W^k,p(Ω),i=1, 2, . . ., converges inW^k,p(Ω) to a functionu∈W^k,p(Ω), if for everyε>0 there existsi_εsuch that

kui−ukW^k,p(Ω)<ε when iÊi_ε. Equivalently,

ilim→∞ku_i−ukW^k,p(Ω)=0.

A sequence (ui) is a Cauchy sequence inW^k,p(Ω), if for everyε>0 there exists i_εsuch that

kui−ujkW^k,p(Ω)<ε when i,jÊi_ε.

WA R N I N G: This is not the same condition as

ku_i+1−u_ikW^k,p(Ω)<ε when iÊi_ε.

Indeed, the Cauchy sequence condition implies this, but the converse is not true (exercise).

Theorem 1.13 (Completeness). The Sobolev space W^k,p(Ω), 1ÉpÉ ∞, k= 1, 2, . . ., is a Banach space.

TH E M O R A L: The spacesC^k(Ω^),k=1, 2, . . . , are not complete with respect to the Sobolev norm, but Sobolev spaces are. This is important in existence arguments for PDEs.

Proof. Step 1: k · kW^k,p(Ω)is a norm.

Reason. (1) kukW^k,p(Ω)=0⇐⇒u=0 almost everywhere inΩ^.

=⇒ kukW^k,p(Ω)=0 implieskukL^p(Ω)=0, which implies thatu=0 almost every- where inΩ^.

⇐= u=0 almost everywhere inΩ^implies ˆ

Ω

D^αuϕdx=(−1)^|α|

ˆ

Ω

uD^αϕdx=0

for allϕ∈C^∞₀(Ω). This together with Corollary1.5implies thatD^αu=0 almost everywhere inΩfor allα,|α| Ék.

(2) kλukW^k,p(Ω)= |λ|kukW^k,p(Ω),λ∈R. Clear.

(3) The triangle inequality for 1Ép< ∞follows from the elementary inequal- ity (a+b)^αÉa^α+b^α,a,bÊ0, 0<αÉ1, and Minkowski’s inequality, since

ku+vkW^k,p(Ω)= Ã

X

|α|Ék

kD^αu+D^αvk_L^pp(Ω)

!¹_p

É Ã

X

|α|Ék

¡kD^αukL^p(Ω)+ kD^αvkL^p(Ω)

¢p

!¹_p

É Ã

X

|α|Ék

kD^αuk_L^pp(Ω)

!¹_p +

à X

|α|Ék

kD^αvk_L^pp(Ω)

!¹_p

= kukW^k,p(Ω)+ kvkW^k,p(Ω). _■ Step 2:Let (ui) be a Cauchy sequence inW^k,p(Ω^{). As}

kD^αui−D^αujkL^p(Ω)É kui−ujkW^k,p(Ω), |α| Ék,

it follows that (D^αui) is a Cauchy sequence inL^p(Ω),|α| Ék. The completeness ofL^p(Ω) implies that there existsu_α∈L^p(Ω) such thatD^αu_i→u_αinL^p(Ω^{). In} particular,ui→u_(0,...,0)=uinL^p(Ω).

Step 3:We show thatD^αu=u_α,|α| Ék. We would like to argue ˆ

ΩuD^αϕdx=lim

i→∞

ˆ

Ωu_iD^αϕdx

=lim

i→∞(−1)^|α|

ˆ

ΩD^αuiϕdx

=(−1)^|α|

ˆ

Ω

u_αϕdx

for every ϕ∈C^∞₀ (Ω). On the second line we used the definition of the weak derivative. Next we show how to conclude the first and last equalities above.

1<p< ∞ Letϕ∈C^∞₀ (Ω). By Hölder’s inequality we have

¯

¯

¯

¯ ˆ

Ωu_iD^αϕdx− ˆ

ΩuD^αϕdx

¯

¯

¯

¯=

¯

¯

¯

¯ ˆ

Ω(u_i−u)D^αϕdx

¯

¯

¯

¯

É ku_i−ukL^p(Ω)kD^αϕk_Lp0(Ω)→0

and consequently we obtain the first inequality above. The last inequality follows in the same way, since

¯

¯

¯

¯ ˆ

Ω

D^αu_iϕdx− ˆ

Ω

u_αϕdx

¯

¯

¯

¯É kD^αu_i−u_αkL^p(Ω)kϕk_L^p0_(Ω)→0.

p=1,p= ∞ A similar argument as above (exercise).

This means that the weak derivativesD^αuexist andD^αu=u_α,|α| Ék. As we also know thatD^αui→u_α=D^αu,|α| Ék, we conclude thatkui−ukW^k,p(Ω)→0.

Thusu_i→uinW^k,p(Ω^). ä

Remark 1.14. W^k,p(Ω), 1Ép< ∞is separable. In the case k=1 consider the mappingu7→(u,Du) fromW^1,p(Ω^{) toLp(}Ω⁾×L^p(Ω⁾ⁿand recall that a subset of a separable space is separable. However,W^1,∞(Ω) is not separable (exercise).

1.5 Hilbert space structure

The spaceW^k,2(Ω) is a Hilbert space with the inner product

〈u,v〉W^k,2(Ω)= X

|α|Ék〈D^αu,D^αv〉L²(Ω), where

〈D^αu,D^αv〉L²(Ω)= ˆ

Ω

D^αuD^αv dx.

Observe that

kukW^k,2(Ω)= 〈u,u〉

1 2 W^k,2(Ω).

1.6 Approximation by smooth functions

This section deals with the question whether every function in a Sobolev space can be approximated by a smooth function.

Defineφ∈C₀^∞(Rⁿ) by

φ(x)=





 c e

1

|x|2−1, |x| <1, 0, |x| Ê1, wherec>0 is chosen so that

ˆ

Rⁿφ(x)dx=1.

Forε>0, set

φε(x)= 1 εⁿφ³x

ε

´ .

The functionφis called the standard mollifier or Friedrich’s mollifier. Observe thatφεÊ0, suppφε=B(0,ε) and

ˆ

Rⁿφε(x)dx= 1 εⁿ

ˆ

Rⁿφ³x ε

´ dx= 1

εⁿ ˆ

Rⁿφ(y)εⁿd y= ˆ

Rⁿφ(x)dx=1 for allε>0. Here we used the change of variabley=^x_ε,dx=εⁿd y.

Notation. IfΩ⊂Rⁿis open withÇΩ6= ;, we write

Ωε={x∈Ω^{: dist(x,}ÇΩ⁾>ε}, ε>0.

If f∈L¹_loc(Ω), we obtain its standard convolution mollification f_ε:Ωε→[−∞,∞], f_ε(x)=(f∗φε)(x)=

ˆ

Ωf(y)φε(x−y)d y.

■

TH E M O R A L: Since the convolution is a weighted integral average off over the ballB(x,ε) for everyx, instead ofΩit is well defined only inΩε. IfΩ=Rⁿ, we do not have this problem.

Remarks 1.15:

(1) For everyx∈Ωε, we have f_ε(x)=

ˆ

Ωf(y)φ_ε(x−y)d y= ˆ

B(x,ε)

f(y)φ_ε(x−y)d y.

(2) By a change of variablesz=x−ywe have ˆ

Ωf(y)φε(x−y)d y= ˆ

Ωf(x−z)φε(z)d z

(3) For everyx∈Ωε, we have

|f_ε(x)| É

¯

¯

¯

¯ ˆ

B(x,ε)

f(y)φ_ε(x−y)d y

¯

¯

¯

¯É kφ_εk∞

ˆ

B(x,ε)|f(y)|d y< ∞. (4) Iff∈C₀(Ω), thenf_ε∈C₀(Ωε), whenever

0<ε<ε0=1

2dist(suppf,ÇΩ^).

Reason. If x∈Ωεs.t. dist(x, suppf)>ε0(in particular, for everyx∈Ωε\ Ωε0) thenB(x,ε)∩suppf= ;, which implies thatf_ε(x)=0. _■ Lemma 1.16 (Properties of mollifiers).

(1) f_ε∈C^∞(Ωε).

(2) f_ε→f almost everywhere asε→0.

(3) Iff∈C(Ω), thenf_ε→f uniformly in everyΩ⁰bΩ.

(4) Iff∈L^p_loc(Ω^{), 1}Ép< ∞, then f_ε→f inL^p(Ω⁰) for everyΩ⁰bΩ^.

W A R N I N G : Assertion (4) does not hold forp= ∞, since there are functions inL^∞(Ω) that are not continuous.

Proof. (1) Letx∈Ωε,j=1, . . . ,n,e_j=(0, . . . , 1, . . . , 0) (the jth component is 1). Let h₀>0 such thatB(x,h₀)⊂Ωεand leth∈R,|h| <h₀. Then

f_ε(x+he_j)−f_ε(x)

h = 1

εⁿ ˆ

B(x+he_j,ε)∪B(x,ε)

1 h

· φ

µx+he_j−y ε

¶

−φ³x−y ε

´¸

f(y)d y.

LetΩ⁰=B(x,h₀+ε). ThenΩ⁰bΩ^andB(x+he_j,ε)∪B(x,ε)⊂Ω^0. Claim:

1 h

· φ

µx+he_j−y ε

¶

−φ³x−y ε

´¸

→1 ε

Çφ Çx_j

³x−y ε

´

for everyy∈Ω^0ash→0.

Reason. Letψ(x)=φ¡_x−y

ε

¢. Then Çψ Çxj

(x)=1 ε

Çφ Çxj

³x−y ε

´

, j=1, . . . ,n and

ψ(x+he_j)−ψ(x)= ˆ _h

0

Ç

Çt(ψ(x+te_j))dt= ˆ _h

0

Dψ(x+te_j)·e_jdt.

■

Thus

|ψ(x+he_j)−ψ(x)| É ˆ _|h|

0 |Dψ(x+te_j)·e_j|dt É1

ε ˆ _|h|

0

¯

¯

¯

¯Dφ

µx+te_j−y ε

¶¯

¯

¯

¯ dt É|h|

ε kDφkL^∞(Rⁿ).

This estimate shows that we can use the Lebesgue dominated convergence theorem (on the third row) to obtain

Çf_ε Çxj

(x)=lim

h→0

f_ε(x+he_j)−f_ε(x) h

=lim

h→0

1 εⁿ

ˆ

Ω⁰

1 h

· φ

µx+he_j−y ε

¶

−φ³x−y ε

´¸ f(y)d y

= 1 εⁿ

ˆ

Ω⁰

1 ε

Çφ Çxj

³x−y ε

´ f(y)d y

= ˆ

Ω⁰

Çφε

Çxj

(x−y)f(y)d y

= µÇφε

Çxj ∗f

¶ (x).

A similar argument shows thatD^αf_εexists and D^αf_ε=D^αφε∗f inΩε

for every multi-indexα. (2) Recall that´

B(x,ε)φ_ε(x−y)d y=1. Therefore we have

|f_ε(x)−f(x)| =

¯

¯

¯

¯ ˆ

B(x,ε)φε(x−y)f(y)d y−f(x) ˆ

B(x,ε)φε(x−y)d y

¯

¯

¯

¯

=

¯

¯

¯

¯ ˆ

B(x,ε)φε(x−y)(f(y)−f(x))d y

¯

¯

¯

¯ É 1

εⁿ ˆ

B(x,ε)φ³x−y ε

´

|f(y)−f(x)|d y ÉΩnkφkL^∞(Rⁿ)

1

|B(x,ε)| ˆ

B(x,ε)|f(y)−f(x)|d y→0

for almost every x∈Ω ^asε→0. Here Ωn= |B(0, 1)|and the last convergence follows from the Lebesgue’s differentiation theorem.

(3) LetΩ⁰bΩ⁰⁰bΩ^{, 0}<ε<dist(Ω^0,ÇΩ^{00), andx}∈Ω^{0. Because}Ω^{00is compact} andf∈C(Ω^),f is uniformly continuous inΩ⁰⁰, that is, for everyε⁰>0 there exists δ>0 such that

|f(x)−f(y)| <ε⁰for everyx,y∈Ω^00with|x−y| <δ.

By combining this with an estimate from the proof of (ii), we conclude that

|f_ε(x)−f(x)| ÉΩnkφkL^∞(Rⁿ)

1

|B(x,ε)| ˆ

B(x,ε)|f(y)−f(x)|d y<ΩnkφkL^∞(Rⁿ)ε⁰ for everyx∈Ω^0ifε<δ.

(4) LetΩ⁰bΩ⁰⁰bΩ^.

Claim: ˆ

Ω⁰|f_ε|^pdxÉ ˆ

Ω⁰⁰|f|^pdx whenever 0<ε<dist(Ω⁰,ÇΩ⁰⁰) and 0<ε<dist(Ω⁰⁰,ÇΩ).

Reason. Takex∈Ω⁰. Hölder’s inequality implies

|f_ε(x)| =

¯

¯

¯

¯ ˆ

B(x,ε)φ_ε(x−y)f(y)d y

¯

¯

¯

¯ É

ˆ

B(x,ε)φε(x−y)^1−1pφε(x−y)^1p|f(y)|d y É

µˆ

B(x,ε)φε(x−y)d y

¶_p0¹ µˆ

B(x,ε)φε(x−y)|f(y)|^pd y

¶¹_p

By raising the previous estimate to powerpand by integrating overΩ⁰, we obtain ˆ

Ω⁰|f_ε(x)|^pdxÉ ˆ

Ω⁰

ˆ

B(x,ε)φ_ε(x−y)|f(y)|^pd y dx

= ˆ

Ω⁰⁰

ˆ

Ω⁰φε(x−y)|f(y)|^pdx d y

= ˆ

Ω⁰⁰|f(y)|^p ˆ

Ω⁰φε(x−y)dx d y

= ˆ

Ω⁰⁰|f(y)|^pd y.

Here we used Fubini’s theorem and once more the fact that the integral ofφεis

one. _■

SinceC(Ω⁰⁰) is dense inL^p(Ω⁰⁰). Therefore for everyε⁰>0 there existsg∈C(Ω⁰⁰⁾ such that

µˆ

Ω⁰⁰|f−g|^pdx

¶¹_p Éε⁰

3. By (3), we haveg_ε→guniformly inΩ⁰asε→0. Thus

µˆ

Ω⁰⁰|g_ε−g|^pdx

¶¹_p Ésup

Ω⁰ |g_ε−g|¯

¯Ω^0¯¯1p<ε⁰ 3,

whenε>0 is small enough. Now we use Minkowski’s inequality and the previous claim to conclude that

µˆ

Ω⁰|f_ε−f|^pdx

¶¹_p É

µˆ

Ω⁰|f_ε−g_ε|^pdx

¶¹_p

+ µˆ

Ω⁰|g_ε−g|^pdx

¶¹_p +

µˆ

Ω⁰|g−f|^pdx

¶¹_p

É2 µˆ

Ω⁰⁰|g−f|^pdx

¶¹_p +

µˆ

Ω⁰|g_ε−g|^pdx

¶¹_p

É2ε⁰ 3+ε⁰

3 =ε⁰.

Thusf_ε→f inL^p(Ω⁰) asε→0. ä

1.7 Local approximation in Sobolev spaces

Next we show that the convolution approximation converges locally in Sobolev spaces.

Theorem 1.17. Letu∈W^k,p(Ω^{), 1}Ép< ∞. then (1) D^αu_ε=D^αu∗φεinΩεand

(2) u_ε→uinW^k,p(Ω⁰) for everyΩ⁰bΩ^.

TH E M O R A L: Smooth functions are dense in local Sobolev spaces. Thus every Sobolev function can be locally approximated with a smooth function in the Sobolev norm.

Proof. (1) Fixx∈Ωε. Then

D^αu_ε(x)=D^α(u∗φε)(x)=(u∗D^αφε)(x)

= ˆ

ΩD^α_xφ_ε(x−y)u(y)d y

=(−1)^|α|

ˆ

ΩD^α_y(φ_ε(x−y))u(y)d y.

Here we first used the proof of Lemma1.16(1) and then the fact that Ç

Çx_j

³φ³x−y ε

´´

= − Ç Çx_j

³φ³y−x ε

´´

= − Ç Çy_j

³φ³x−y ε

´´

. For everyx∈Ωε, the functionϕ(y)=φε(x−y) belongs toC^∞₀(Ω). Therefore

ˆ

ΩD^α_y(φε(x−y))u(y)d y=(−1)^|α|

ˆ

ΩD^αu(y)φε(x−y)d y.

By combining the above facts, we see that D^αu_ε(x)=(−1)^|α|+|α|

ˆ

Ω

D^αu(y)φε(x−y)d y=(D^αu∗φε)(x).

Notice that (−1)^|α|+|α|=1.

(2) LetΩ⁰bΩ, and chooseε>0 s.t. Ω⁰⊂Ωε. By (i) we know thatD^αu_ε= D^αu∗φ_εinΩ^0,|α| Ék. By Lemma1.16, we haveD^αu_ε→D^αuinL^p(Ω^{0) as}ε→0,

|α| Ék. Consequently

ku_ε−ukW^k,p(Ω⁰)= Ã

X

|α|Ék

kD^αu_ε−D^αuk_L^pp(Ω⁰)

!¹_p

→0.

ä

1.8 Global approximation in Sobolev spaces

The next result shows that the convolution approximation converges also globally in Sobolev spaces.

Theorem 1.18 (Meyers-Serrin). If u∈W^k,p(Ω), 1É p< ∞, then there exist functionsu_i∈C^∞(Ω⁾∩W^k,p(Ω) such thatu_i→uinW^k,p(Ω^).

TH E M O R A L: Smooth functions are dense in Sobolev spaces. Thus every Sobolev function can be approximated with a smooth function in the Sobolev norm.

In particular, this holds true for the function with a dense infinity set in Example 1.9.

Proof. LetΩ0= ;and Ωi=

½

x∈Ω: dist(x,ÇΩ⁾>1 i

¾

∩B(0,i), i=1, 2, . . . . Then

Ω= [∞ i=1

Ωi and Ω1bΩ2b^{. . .}bΩ^.

Claim:There existηi∈C^∞₀ (Ωi+2\Ωi−1),i=1, 2, . . ., such that 0ÉηiÉ1 and X∞

i=1

ηi(x)=1 for every x∈Ω^. This is a partition of unity subordinate to the covering {Ωi}.

Reason. By using the distance function and convolution approximation we can constructeηi∈C^∞₀ (Ωi+2\Ωi−1) such that 0ÉeηiÉ1 andeηi=1 inΩi+1\Ωi(exercise).

Then we define

ηi(x)= ηei(x) P_∞

j=1eηj(x), i=1, 2, . . . .

Observe that the sum is only over four indices in a neighbourhood of a given

point. _■

Now by Lemma1.12(5),ηiu∈W^k,p(Ω^{) and} supp(ηiu)⊂Ωi+2\Ωi−1. Letε>0. Chooseεi>0 so small that

supp(φεi∗(ηiu))⊂Ωi+2\Ωi−1

(see Remark1.15(4)) and

kφεi∗(ηiu)−ηiukW^k,p(Ω)< ε

2ⁱ, i=1, 2, . . . .

By Theorem1.17(2), this is possible. Define v=

X∞ i=1

φεi∗(ηiu).

This function belongs toC^∞(Ω), since in a neighbourhood of any pointx∈Ω^{, there} are at most finitely many nonzero terms in the sum. Moreover,

kv−ukW^k,p(Ω)=

°

°

°

°

° X∞ i=1

φεi∗(ηiu)− X∞ i=1

ηiu

°

°

°

°

°_Wk,p(Ω) É

X∞ i=1

°

°φεi∗(ηiu)−ηiu°

°W^k,p(Ω)

É X∞ i=1

ε

2ⁱ=ε. ä

Remarks 1.19:

(1) The Meyers-Serrin theorem1.18gives the following characterization for the Sobolev spacesW^k,p(Ω^{), 1}Ép< ∞: u∈W^k,p(Ω) if and only if there exist functions u_i∈C^∞(Ω⁾∩W^k,p(Ω^{), i}=1, 2, . . . , such that u_i→u in W^k,p(Ω^{) asi}→ ∞. In other words,W^k,p(Ω) is the completion ofC^∞(Ω^{) in} the Sobolev norm.

Reason. =⇒ Theorem1.18.

⇐= Theorem1.13. _■

(2) The Meyers-Serrin theorem1.18is false forp= ∞. Indeed, ifu_i∈C^∞(Ω⁾∩ W^1,∞(Ω) such that u_i→u inW^1,∞(Ω^{), thenu}∈C¹(Ω) (exercise). Thus special care is required when we consider approximations inW^1,∞(Ω^).

(3) LetΩ⁰bΩ. The proof of Theorem1.17and Theorem1.18shows that for everyε>0 there existsv∈C^∞₀(Ω) such thatkv−ukW^1,p(Ω⁰)<ε.

(4) The proof of Theorem1.18shows that not onlyC^∞(Ω^{) but alsoC}₀^∞(Ω^{) is} dense inL^p(Ω^{), 1}Ép< ∞.

1.9 Sobolev spaces with zero boundary values

In this section we study definitions and properties of first order Sobolev spaces with zero boundary values in an open subset ofRⁿ. A similar theory can be developed for higher order Sobolev spaces as well. Recall that, by Theorem1.18, the Sobolev spaceW^1,p(Ω) can be characterized as the completion ofC^∞(Ω^{) with} respect to the Sobolev norm when 1Ép< ∞.

Definition 1.20. Let 1Ép< ∞. The Sobolev space with zero boundary values W₀^1,p(Ω) is the completion of C^∞₀ (Ω) with respect to the Sobolev norm. Thus u∈W₀^1,p(Ω) if and only if there exist functionsu_i∈C^∞₀ (Ω^),i=1, 2, . . . , such that u_i→u inW^1,p(Ω^{) as i}→ ∞. The spaceW₀^1,p(Ω) is endowed with the norm of W^1,p(Ω^).

TH E M O R A L: The only difference compared toW^1,p(Ω) is that functions in W₀^1,p(Ω) can be approximated byC^∞₀(Ω) functions instead ofC^∞(Ω) functions, that is,

W^1,p(Ω⁾=C^∞(Ω^{) and W}₀^1,p(Ω⁾=C^∞₀ (Ω^),

where the completions are taken with respect to the Sobolev norm. A function inW₀^1,p(Ω) has zero boundary values in Sobolev’s sense. We may say thatu,v∈ W^1,p(Ω) have the same boundary values in Sobolev’s sense, ifu−v∈W₀^1,p(Ω). This is useful, for example, in Dirichlet problems for PDEs.

WA R N I N G: Roughly speaking a function inW^1,p(Ω) belongs toW₀^1,p(Ω), if it vanishes on the boundary. This is a delicate issue, since the function does not have to be zero pointwise on the boundary. We shall return to this question later.

Remark 1.21. W₀^1,p(Ω) is a closed subspace ofW^1,p(Ω) and thus complete (exer- cise).

Remarks 1.22:

(1) ClearlyC^∞₀(Ω⁾⊂W₀^1,p(Ω⁾⊂W^1,p(Ω⁾⊂L^p(Ω^).

(2) Ifu∈W₀^1,p(Ω), then the zero extensionue:Rⁿ→[−∞,∞],

u(x)e =







u(x), x∈Ω^, 0, x∈Rⁿ\Ω, belongs toW^1,p(Rⁿ) (exercise).

Lemma 1.23. If u∈W^1,p(Ω^{) and suppu} is a compact subset of Ω^{, then u} ∈ W₀^1,p(Ω^).

Proof. Letη∈C^∞₀(Ω) be a cutoff function such thatη=1 on the support ofu.

Claim:Ifu_i∈C^∞(Ω^),i=1, 2, . . . , such thatu_i→uinW^1,p(Ω^{), then}ηu_i∈C^∞₀ (Ω⁾ converges toηu=uinW^1,p(Ω^).

Reason. We observe that kηui−ηukW^1,p(Ω)=

³

kηui−ηuk_L^pp_(Ω)+ kD(ηui−ηu)k_L^pp_(Ω)

´¹_p

É kηu_i−ηukL^p(Ω)+ kD(ηu_i−ηu)kL^p(Ω),