Jost J. Partial Differential Equations

Подождите немного. Документ загружается.

8.2 The Sobolev Space W

1,2

187

Lemma 8.2.1: Let u ∈ L

loc

(Ω), and suppose v = D

u exists. If dist(x, ∂Ω)

>h, we have

(x)) = (D

(x).

Proof: By diﬀerentiating under the integral, we obtain

(x)) =



∂

∂x





x − y



u(y)dy

−1



∂

∂y





x − y



u(y)dy







x − y



u(y)dy by (8.2.1)

=(D

(x).



Lemmas A.3 and 8.2.1 and formula (8.2.1) imply the following theorem:

Theorem 8.2.1: Let u, v ∈ L

(Ω). Then

v = D

precisely if there exists a sequence (u

) ⊂ C

∞

(Ω) with

→ u,

∂

∂x

→ v in L

(Ω



) for any Ω



⊂⊂ Ω.



Deﬁnition 8.2.2: The Sobolev space W

1,2

(Ω) is deﬁned as the space of those

u ∈ L

(Ω) that possess a weak derivative of class L

(Ω) for each direction

(i =1,...,d).

In W

1,2

(Ω) we deﬁne a scalar product

(u, v)

1,2

(Ω)



uv +



i=1



u · D

and a norm

u

1,2

(Ω)

:= (u, u)

1,2

(Ω)

We also deﬁne H

1,2

(Ω) as the closure of C

∞

(Ω) ∩ W

1,2

(Ω) with respect to

the W

1,2

-norm, and H

1,2

(Ω) as the closure of C

∞

(Ω) with respect to this

norm.

Corollary 8.2.1: W

1,2

(Ω) is complete with respect to ·

1,2

, and is hence

a Hilbert space. W

1,2

(Ω)=H

1,2

(Ω).

188 8. Existence Techniques III

Proof: Let (u

)

n∈N

be a Cauchy sequence in W

1,2

(Ω). Then (u

)

n∈N

)

n∈N

(i =1,...,d) are Cauchy sequences in L

(Ω). Since L

(Ω)is

complete, there exist u, v

∈ L

(Ω)with

→ u, D

→ v

in L

(Ω)(i =1,...,d).

For φ ∈ C

(Ω), we have



· φ = −



φ,

and the left-hand side converges to



·φ, the right-hand side to −



u ·D

φ.

Therefore, D

u = v

, and thus u ∈ W

1,2

(Ω). This shows completeness.

In order to prove the equality H

1,2

(Ω)=W

1,2

(Ω), we need to verify that

the space C

∞

(Ω) ∩ W

1,2

(Ω) is dense in W

1,2

(Ω). For n ∈ N, we put



x ∈ Ω : x <n,dist(x, ∂Ω) >



with Ω

:= Ω

−1

:= ∅.Thus,

⊂⊂ Ω

n+1

and



n∈N

= Ω.

We let {ϕ

}

j∈N

be a partition of unity subordinate to the cover



n+1

n−1



of Ω.Letu ∈ W

1,2

(Ω). By Theorem 8.2.1, for every ε>0, we may ﬁnd a

positive number h

for any n ∈ N such that

≤ dist(Ω

,∂Ω

n+1

(ϕ

− ϕ

u

1,2

(Ω)

Since the ϕ

constitute a partition of unity, on any Ω



⊂⊂ Ω, at most ﬁnitely

many of the smooth functions (ϕ

are non-zero. Consequently,

˜u :=



(ϕ

∈ C

∞

(Ω).

We have

u − ˜u

1,2

(Ω)

≤



(ϕ

− ϕ

u <ε,

and we see that every u ∈ W

1,2

(Ω) can be approximated by C

∞

-functions.



8.2 The Sobolev Space W

1,2

189

Corollary 8.2.1 answers one of the questions raised in Section 8.1, namely

whether the function w considered there can be represented as the gradient

of an L

-function.

Examples: Ω =(−1, 1) ⊂ R.

(i) u(x):=|x|

In that case, u ∈ W

1,2

((−1, 1)), and

Du(x)=



1 for 0 <x<1,

−1for− 1 <x<0,

because for every φ ∈ C

((−1, 1)),



−1

−φ(x)dx +



φ(x)dx = −



−1



(x) ·|x|dx.

(ii)

u(x):=



1 for 0 ≤ x<1,

0for− 1 <x<0,

is not weakly diﬀerentiable, for if it were, necessarily Du(x)=0for

x = 0; hence as an L

loc

function Du ≡ 0, but we do not have, for every

φ ∈ C

((−1, 1)),



−1

φ(x) · 0 dx = −



−1



(x)u(x)dx = −





(x)dx = φ(0).

Remark: Any u ∈ L

loc

(Ω) deﬁnes a distribution (cf. Section 1.1) l

[ϕ]:=



u(x)ϕ(x)dx for ϕ ∈ C

∞

(Ω).

Every distribution l possesses distributional derivatives D

l, i =1,...,d,

deﬁned by

l[ϕ]:=−l

∂ϕ

∂x

If v = D

u ∈ L

loc

(Ω) is the weak derivative of u,then

= l

because

[ϕ]=



u(x)ϕ(x)dx = −



u(x)

∂ϕ

∂x

(x)dx = D

[ϕ]

for all ϕ ∈ C

∞

(Ω).

190 8. Existence Techniques III

Whereas the distributional derivative D

always exists, the weak deriva-

tive need not exist. Thus, in general, the distributional derivative is not of

the form l

for some v ∈ L

loc

(Ω), i.e., not represented by a locally integrable

function. This is what happens in Example 2. Here, Dl

= δ

, the delta

distribution at 0, because

[ϕ]=−l

[ϕ



]=−



−1



(x)dx = −





(x)dx = ϕ(0).

The delta distribution cannot be represented by some locally integrable func-

tion v, because, as one easily veriﬁes, there is no function v ∈ L

loc

((−1, 1))

with



−1

v(x)ϕ(x)dx = ϕ(0) for all ϕ ∈ C

∞

(Ω).

This explains why u from Example 2 is not weakly diﬀerentiable.

We now prove a replacement lemma exhibiting a characteristic property

of Sobolev functions:

Lemma 8.2.2: Let Ω

⊂⊂ Ω, g ∈ W

1,2

(Ω), u ∈ W

1,2

(Ω

), u − g ∈

1,2

(Ω

). Then

v(x):=



u(x) for x ∈ Ω

g(x) for x ∈ Ω \ Ω

is contained in W

1,2

(Ω),and

v(x)=



u(x) for x ∈ Ω

g(x) for x ∈ Ω \ Ω

Proof: By Corollary 8.2.1, there exist g

∈ C

∞

(Ω), u

∈ C

∞

(Ω

)with

→ g in W

1,2

(Ω),

→ u in W

1,2

(Ω

− g

=0 on∂Ω

. (8.2.2)

We put

(x):=



(x)forx ∈ Ω

(x)forx ∈ Ω \ Ω

(x):=



(x)forx ∈ Ω

(x)forx ∈ Ω \ Ω

(x):=



u(x)forx ∈ Ω

g(x)forx ∈ Ω \ Ω

8.2 The Sobolev Space W

1,2

191

We then have for ϕ ∈ C

(Ω),



ϕw



ϕw



Ω\Ω

ϕw



ϕD



Ω\Ω

ϕD

= −



ϕ −



Ω\Ω

since the two boundary terms resulting from integrating the two

integrals by parts have opposite signs and thus cancel because

of g

= u

on ∂Ω

= −



by (8.2.2). Now for n →∞,



ϕw

→



ϕD

u +



Ω\Ω

ϕD



ϕ →



ϕ,

and the claim follows. 

The next lemma is a chain rule for Sobolev functions:

Lemma 8.2.3: For u ∈ W

1,2

(Ω), f ∈ C

(R), suppose

sup

y∈R



(y)| < ∞.

Then f ◦u ∈ W

1,2

(Ω), and the weak derivative satisﬁes D(f ◦u)=f



(u)Du.

Proof: Let u

∈ C

∞

(Ω), u

→ u in W

1,2

(Ω)forn →∞.Then



|f(u

) − f(u)|

dx ≤ sup |f





− u|

dx → 0

and





)Du

− f



(u)Du|

dx ≤ 2sup|f





|Du

− Du|





) − f



(u)|

|Du|

dx.

By a well-known result about L

-functions, after selection of a subsequence,

converges to u pointwise almost everywhere in Ω.

Since f



is continuous,



) then also converges pointwise almost everywhere to f



(u), and since

See J. Jost, Postmodern Analysis, p. 240 [12].

192 8. Existence Techniques III



is also bounded, the last integral converges to 0 for n →∞by Lebesgue’s

theorem on dominated convergence.

Thus

f(u

) → f (u)inL

(Ω)

and

D(f(u

)) = f



)Du

→ f



(u)Du in L

(Ω),

and hence f ◦ u ∈ W

1,2

(Ω)andD(f ◦ u)=f



(u)Du. 

Corollary 8.2.2: If u ∈ W

1,2

(Ω), then also |u|∈W

1,2

(Ω),andD|u| =

sign u · Du.

Proof: We consider f

(u):=(u

+ε

)

−ε, apply Lemma 8.2.3, and let ε → 0,

using once more Lebesgue’s theorem on dominated convergence to justify the

limit as before. 

We next prove the Poincar´e inequality (see also Corollary 9.5.1 below).

Theorem 8.2.2: For u ∈ H

1,2

(Ω), we have

u

(Ω)

≤



|Ω|



Du

(Ω)

, (8.2.3)

where |Ω| denotes the (Lebesgue) measure of Ω,andω

is the measure of

the unit ball in R

. In particular, for any u ∈ H

1,2

(Ω),itsW

1,2

-norm is

controlled by the L

-norm of Du:

u

1,2

(Ω)

≤





|Ω|





Du

(Ω)

Proof: Suppose ﬁrst u ∈ C

(Ω); we put u(x)=0forx ∈ R

\Ω.Forω ∈ R

with |ω| = 1, by the fundamental theorem of calculus we obtain by integrating

along the ray {rω :0≤ r<∞} that

u(x)=−



∞

∂

∂r

u(x + rω)dr.

Integrating with respect to ω then yields, as in the proof of Theorem 1.2.1,

u(x)=−

dω



∞



|ω|=1

∂

∂r

u(x + rω) dωdr

= −

dω



∞



∂B(x,r)

d−1

∂u

∂ν

(z)dσ(z)dr

= −

dω



|x − y|

d−1



i=1

∂

∂y

u(y)

− y

|x − y|

dy,

(8.2.4)

8.2 The Sobolev Space W

1,2

193

and thus with the Schwarz inequality,

|u(x)|≤

dω



|x − y|

d−1

·|Du(y)|dy. (8.2.5)

We now need a lemma:

Lemma 8.2.4: For f ∈ L

(Ω), 0 <μ≤ 1,let

f)(x):=



|x − y|

d(μ−1)

f(y)dy.

Then

V

f

(Ω)

≤

1−μ

|Ω|

f

(Ω)

Proof: B(x, R):={y ∈ R

: |x −y|≤R}.LetR be chosen such that |Ω| =

|B(x, R)| = ω

. Since in that case

|Ω \(Ω ∩ B(x, R))| = |B(x, R) \ (Ω ∩ B(x, R))|

and

|x − y|

d(μ−1)

≤ R

d(μ−1)

for |x − y|≥R,

|x − y|

d(μ−1)

≥ R

d(μ−1)

for |x − y|≤R,

it follows that



|x − y|

d(μ−1)

dy ≤



B(x,R)

|x − y|

d(μ−1)

dy =

dμ

1−μ

|Ω|

(8.2.6)

We now write

|x − y|

d(μ−1)

|f(y)| =



|x − y|

(μ−1)



|x − y|

(μ−1)

|f(y)|



and obtain, applying the Cauchy Schwarz inequality,

|(V

f)(x)|≤



|x − y|

d(μ−1)

|f(y)|dy

≤





|x − y|

d(μ−1)







|x − y|

d(μ−1)

|f(y)|



and hence



f(x)|

dx ≤

1−μ

|Ω|



|x − y|

d(μ−1)

|f(y)|

dy dx

by estimating the ﬁrst integral of the preceding inequality

with (8.2.6)

≤



1−μ

|Ω|





|f(y)|

194 8. Existence Techniques III

by interchanging the integrations with respect to x and y and applying (8.2.6)

once more, whence the claim. 

We may now complete the proof of Theorem 8.2.2: Applying Lemma 8.2.4

with μ =

and f = |Du| to the right-hand side of (8.2.5), we obtain

(8.2.3) for u ∈ C

(Ω). Since by deﬁnition of H

1,2

(Ω), it contains C

(Ω)

as a dense subspace, we may approximate u in the H

1,2

-norm by some se-

quence (u

)

n∈N

⊂ C

(Ω). Thus, u

converges to u in L

,andDu

to u.

Thus, the inequality (8.2.3) that has been proved for u

extends to u. 

Remark: The assumption that u is contained in H

1,2

(Ω), and not only in

1,2

(Ω), is necessary for Theorem 8.2.2, since otherwise the nonzero con-

stants would constitute counterexamples. However, the assumption u ∈

1,2

(Ω) may be replaced by other assumptions that exclude nonzero con-

stants, for example by



u(x)dx =0.

For our treatment of eigenvalues of the Laplace operator in Section 9.5,

the fundamental tool will be the compactness theorem of Rellich:

Theorem 8.2.3: Let Ω ∈ R

be open and bounded. Then H

1,2

(Ω) is com-

pactly embedded in L

(Ω); i.e., any sequence (u

)

n∈N

⊂ H

1,2

(Ω) with

u



1,2

(Ω)

≤ c

(8.2.7)

contains a subsequence that converges in L

(Ω).

Proof: The strategy is to ﬁnd functions w

n,ε

∈ C

(Ω), for every ε>0, with

u

− w

n,ε



1,2

(Ω)

(8.2.8)

and

w

n,ε



1,2

(Ω)

≤ c

(8.2.9)

(the constant c

will depend on ε, but not on n). By the Aszela–Ascoli theo-

rem, (w

n,ε

)

n∈N

then contains a subsequence that converges uniformly, hence

also in L

. Since this holds for every ε>0, one may appeal to a general

theorem about compact subsets of metric spaces to conclude that the closure

of (u

)

n∈N

is compact in L

(Ω) and thus contains a convergent subsequence.

That theorem

states that a subset of a metric space is compact precisely if

it is complete and totally bounded, i.e., if for any ε>0, it is contained in

the union of a ﬁnite number of balls of radius ε.

Applying this result to the (closure of the) sequence (w

n,ε

)

n∈N

, we infer

that there exist ﬁnitely many z

, ν =1,...,N,inL

(Ω) such that for every

n ∈ N,

see, e.g., J. Jost, Postmodern Analysis, Springer, 1998, Theorem 7.38.

8.2 The Sobolev Space W

1,2

195

w

n,ε

− z



(Ω)

for some ν ∈{1,...,N}. (8.2.10)

Hence, from (8.2.8) and (8.2.10), for every n ∈ N,

u

− z



(Ω)

<ε for some ν.

Since this holds for every ε>0, the sequence (u

)

n∈N

is totally bounded,

and so its closure is compact in L

(Ω), and we get the desired convergent

subsequence in L

(Ω).

It remains to construct the w

n,ε

. First of all, by deﬁnition of H

1,2

(Ω),

there exists w

∈ C

(Ω)with

u

− w



1,2

(Ω)

. (8.2.11)

By (8.2.7), then also

w



1,2

(Ω)

≤ c



for some constant c



. (8.2.12)

We then deﬁne w

n,ε

as the molliﬁcation of w

with a parameter h = h(ε)to

be determined subsequently:

n,ε

(x)=







x − y



(y)dy.

The crucial step now is to control the L

-norm of the diﬀerence w

− w

n,ε

with the help of the W

1,2

-bound on the original u

.Thisgoesasfollows:



(x) − w

n,ε

(x)|

dx =







|y|≤1

(y)(w

(x) − w

(x − hy))dy



≤







|y|≤1

(y)



h|y|



∂

∂r

(x − rω)



dr dy



dx with ω =

|y|







|y|≤1

(y)



h|y|



∂

∂r

(x − rω)



dr dy



≤





|y|≤1

(y)dy





|y|≤1

(y)h

|y|



|Dw

(x)|

dx dy



by H¨older’s inequality ((A.4) of the Appendix) and Fubini’s theorem. Since



|y|≤1

(y)dy = 1, we obtain the estimate

w

− w

n,ε



(Ω)

≤ h Dw



(Ω)

Because of (8.2.12), we may then choose h such that

w

− w

n,ε



(Ω)

. (8.2.13)

Then (8.2.11) and (8.2.13) yield the desired estimate (8.2.8). 

196 8. Existence Techniques III

8.3 Weak Solutions of the Poisson Equation

As before, let Ω be an open and bounded subset of R

, g ∈ H

1,2

(Ω). With the

concepts introduced in the previous section, we now consider the following

version of the Dirichlet principle. We seek a solution of

Δu =0 inΩ,

u = g for ∂Ω



meaning u − g ∈ H

1,2

(Ω)



by minimizing the Dirichlet integral



|Dv|

(here, Dv =(D

v,...,D

v))

among all v ∈ H

1,2

(Ω)withv −g ∈ H

1,2

(Ω). We want to convince ourselves

that this approach indeed works. Let

κ := inf





|Dv|

: v ∈ H

1,2

(Ω),v− g ∈ H

1,2

(Ω)



and let (u

)

n∈N

be a minimizing sequence, meaning that u

− g ∈ H

1,2

(Ω),

and



|Du

→ κ.

We have already argued in Section 8.1 that for a minimizing sequence

)

n∈N

, the sequence of (weak) derivatives (Du

) is a Cauchy sequence

in L

(Ω). Theorem 8.2.2 implies

u

− u



(Ω)

≤ const Du

− Du



(Ω)

Thus, (u

) also is a Cauchy sequence in L

(Ω). We conclude that (u

)

n∈N

converges in W

1,2

(Ω) to some u.Thisu satisﬁes



|Du|

= κ

as well as

u − g ∈ H

1,2

(Ω),

because H

1,2

(Ω) is a closed subspace of W

1,2

(Ω). Furthermore, for every

v ∈ H

1,2

(Ω), t ∈ R, putting Du · Dv :=



i=1

u · D

v,wehave

κ ≤



|D(u + tv)|



|Du|

+2t



Du · Dv + t



|Dv|

and diﬀerentiating with respect to t at t = 0 yields



|D(u + tv)|

t=0



Du · Dv for all v ∈ H

1,2

(Ω).