Jost J. Partial Differential Equations

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Exercises 217

8.2 What would be a natural deﬁnition of k-times weak diﬀerentiablity? (The

answer will be given in the next chapter, but you might wish to try

yourself at this point to deﬁne Sobolev spaces W

k,2

(Ω)ofk-times weakly

diﬀerentiably functions that are contained in L

(Ω) together with all

their weak derivatives and to prove results analogous to Theorem 8.2.1

and Corollary 8.2.1 for them.)

8.3 Consider a variational problem of the type

I(u)=



F (Du(x))dx

with a smooth function F : R

→ Ω satisfying an inequality of the form

|F (p)|≤c

|p|

+ c

for all p ∈ R

Derive the corresponding Euler–Lagrange equations for a minimizer (in

the weak sense; cf. (8.4.4)). Try more generally to ﬁnd conditions for

integrands of the type F (x, u(x),Du(x)) that allow one to derive weak

Euler–Lagrange equations for minimizers.

8.4 Following R. Courant, as a model problem for ﬁnite elements we consider

the Poisson equation

Δu = f in Ω,

u =0 on∂Ω

Figure 8.1.

in the unit square Ω =[0, 1] ×

[0, 1] ⊂ R

.Forh =

(n ∈

N), we subdivide

Ω into

) subsquares of side length h,

andeachsuchsquareinturnis

subdivided into two right-angled

symmetric triangles by the di-

agonal from the upper left to

the lower right vertex (see Fig-

ure 8.1). We thus obtain trian-

gles Δ

,i=1,...,2

2n+1

.What

is the number of interior vertices

of this triangulation?

We consider the space of continuous triangular ﬁnite elements

:= {ϕ ∈ C

(Ω):ϕ

|Δ

linear for all i, ϕ =0on∂Ω}.

The triangular elements ϕ

with

)=δ

218 8. Existence Techniques III

constitute a basis of S

(proof?).

Compute



Dϕ

· Dϕ

for all pairs i, j

and establish the system of linear equations for the approximating solu-

tion of the Poisson equation in S

, i.e., for the minimizer ϕ



|Dϕ|



fϕ

for ϕ ∈ S

, with respect to the above basis ϕ

of S

(for that purpose,

you have just computed the coeﬃcients a

!).

9. Sobolev Spaces and L

Regularity Theory

9.1 General Sobolev Spaces. Embedding Theorems of

Sobolev, Morrey, and John–Nirenberg

Deﬁnition 9.1.1: Let u : Ω → R be integrable, α := (α

,...,α

ϕ :=



∂

∂x



···



∂

∂x



ϕ for ϕ ∈ C

|α|

(Ω).

An integrable function v : Ω → R is called an αth weak derivative of u,in

symbols v = D

u,if



ϕv dx =(−1)

|α|



ϕdx for all ϕ ∈ C

|α|

(Ω). (9.1.1)

For k ∈ N, 1 ≤ p<∞, we deﬁne the Sobolev space

k,p

(Ω):={u ∈ L

(Ω):D

u exists and is contained in L

(Ω) for

all |α|≤k},

u

k,p

(Ω)

⎛

⎝



|α|≤k



⎞

⎠

The spaces H

k,p

(Ω) and H

k,p

(Ω) are deﬁned to be the closures of C

∞

(Ω)

and C

∞

(Ω), respectively, with respect to ·

k,p

(Ω)

. Occasionally, we shall

employ the abbreviation ·

= ·

(Ω)

Concerning notation: The multi-index notation will be used in the present

section only. Later on, for u ∈ W

1,p

(Ω), ﬁrst weak derivatives will be denoted

by D

u, i =1,...,d, as in Deﬁnition 8.2.1, and we shall denote the vector

u,...,D

u)byDu. Likewise, for u ∈ W

2,p

(Ω), second weak derivatives

will be written D

u, i, j =1,...,d, and the matrix of second weak derivatives

will be denoted by D

As in Section 8.2, one proves the following lemma:

Lemma 9.1.1: W

k,p

(Ω)=H

k,p

(Ω). The space W

k,p

(Ω) is complete with

respect to ·

k,p

(Ω)

, i.e., it is a Banach space. 

220 9. Sobolev Spaces and L

Regularity Theory

We now state the Sobolev embedding theorem:

Theorem 9.1.1:

1,p

(Ω) ⊂



d−p

(Ω) for p<d,

(

Ω) for p>d.

Moreover, for u ∈ H

1,p

(Ω) ,

u

d−p

≤ c Du

for p<d, (9.1.2)

sup

|u|≤c |Ω|

−

·Du

for p>d, (9.1.3)

where the constant c depends on p and d only.

In order to better understand the content of the Sobolev embedding the-

orem, we ﬁrst consider the scaling behavior of the expressions involved: Let

f ∈ H

1,p

) ∩ L

). We look at the scaling y = λx (with λ>0) and

(y):=f





= f(x).

Then, with y = λx,





|Df

(y)|



= λ

d−p





|Df(x)|



(note that on the left, the derivative is taken with respect to y, and on the

right with respect to x; this explains the −p in the exponent) and





(y)|



= λ





|f(x)|



Thus in the limit λ → 0, f



is controlled by Df



≤ λ

d−p

for λ<1

holds, i.e.,

≥

d − p

i.e.,

q ≤

d − p

if p<d.

(We have implicitly assumed Df

> 0 here, but you will easily convince

yourself that this is the essential case of the embedding theorem.) We treat

only the limit λ → 0 here, since only for λ ≤ 1(forf ∈ H

1,p

)) do we have

9.1 General Sobolev Spaces. Embedding Theorems 221

supp f

⊂ supp f,

and the Sobolev embedding theorem covers only the case where the functions

have their support contained in a ﬁxed bounded set Ω. Looking at the scaling

properties for λ →∞, one observes that this assumption on the support is

necessary for the theorem. The scaling properties for p>dwill be examined

after Corollary 9.1.5.

Proof of Theorem 9.1.1: We shall ﬁrst prove the inequalities (9.1.2) and

(9.1.3) for u ∈ C

(Ω). We put u =0onR

\ Ω again. As in the proof

of Theorem 8.2.2,

|u(x)|≤



−∞

u(x

,...,x

i−1

,ξ,x

i+1

,...,x

)|dξ with x =(x

,...,x

)

for 1 ≤ i ≤ d, and hence

|u(x)|

≤

i=1



∞

−∞

u|dx

and

|u(x)|

d−1

≤



i=1



∞

−∞

u|dx



d−1

It follows that



∞

−∞

|u(x)|

d−1

≤





∞

−∞

u|dx



d−1



i=1



∞

−∞



∞

−∞

u|dx



d−1

wherewehaveused(A.6)forp

= ···= p

d−1

= d − 1. Iteratively, we obtain



|u(x)|

d−1

dx ≤



i=1



u|dx



d−1

and hence

u

d−1

≤



i=1



u|dx



≤





i=1

u|dx,

since the geometric mean is not larger than the arithmetic one, and conse-

quently

u

d−1

≤

Du

, (9.1.4)

222 9. Sobolev Spaces and L

Regularity Theory

which is (9.1.2) for p =1.

Applying (9.1.4) to |u|

(γ>1) (|u|

is not necessarily contained in

(Ω), even if u is, but as will be explained at the end of the present proof,

by an approximation argument, if shown for C

(Ω), (9.1.4) continues to hold

for H

1,1

, and we shall choose γ such that for u ∈ H

1,p

(Ω), we have |u|

∈

1,1

(Ω)), we obtain

|u|



d−1

≤



|u|

γ−1

|Du|dx ≤

)

|u|

γ−1

)

·Du

for

(9.1.5)

applying H¨older’s inequality (A.4). For p<d, γ =

(d−1)p

d−p

satisﬁes

γd

d − 1

(γ −1)p

p − 1

and (9.1.5) yields, taking q =

p−1

into account,

u

γd

d−1

≤

u

γ−1

γd

d−1

·Du

i.e.,

u

γd

d−1

≤

Du

which is (9.1.2). In order to establish (9.1.3), we need the following general-

ization of Lemma 8.2.4:

Lemma 9.1.2: For μ ∈ (0, 1], f ∈ L

(Ω) let

f)(x):=



|x − y|

d(μ−1)

f(y)dy.

Let 1 ≤ p ≤ q ≤∞,

0 ≤ δ =

−

<μ.

Then V

maps L

(Ω) continuously to L

(Ω),andforf ∈ L

(Ω), we have

V

f

≤



1 − δ

μ − δ



1−δ

1−μ

|Ω|

μ−δ

f

. (9.1.6)

Proof: Let

:= 1 +

−

=1−δ.

9.1 General Sobolev Spaces. Embedding Theorems 223

Then

(x − y):=|x − y|

d(μ−1)

∈ L

(Ω),

andasintheproofofLemma8.2.4,wechooseR such that |Ω| = |B(x, R)| =

, and we estimate as follows:







|x − y|

d(μ−1)

1−δ



1−δ

≤





B(x,R)

|x − y|

d(μ−1)

1−δ



1−δ



1 − δ

μ − δ



1−δ

d(μ−δ)



1 − δ

μ − δ



1−δ

1−μ

|Ω|

μ−δ

We write

 |f | = 

r(1−1/p)

(

|f|

)

|f|

pδ

and the generalized H¨older inequality (A.6) yields

f(x)|

≤







(x − y) |f(y)|









(x − y)dy



1−





|f(y)|



;

hence, integrating with respect to x and interchanging the integrations in the

ﬁrst integral, we obtain

V

f

≤ sup







(x − y)dy



f

≤



1 − δ

μ − δ



1−δ

1−μ

|Ω|

μ−δ

f

by the above estimate for 

. 

In order to complete the proof of Theorem 9.1.1, we use (8.2.4), assuming

ﬁrst u ∈ C

(Ω) as before, i.e.,

u(x)=

dω





i=1

− y

)

|x − y|

u(y)dy (9.1.7)

for x ∈ Ω. This implies

|u|≤

dω

(|D|). (9.1.8)

224 9. Sobolev Spaces and L

Regularity Theory

Inequality (9.1.6) for q = ∞, μ =1/d then yields (9.1.3), again at this

moment for u ∈ C

(Ω)only.

If now u ∈ H

1,p

(Ω), we approximate u in the W

1,p

-norm by C

∞

functions

, and apply (9.1.2) and (9.1.3) to the diﬀerence u

− u

. It follows that

) is a Cauchy sequence in L

dp/(d−p)

(Ω)(forp<d)orC

(

Ω)(forp>d),

respectively. Thus u itself is contained in the same space and satisﬁes (9.1.2)

or (9.1.3), respectively 

Corollary 9.1.1:

k,p

(Ω) ⊂



d−kp

(Ω) for kp < d,

(Ω) for 0 ≤ m<k−

Proof: The ﬁrst embedding iteratively follows from Theorem 9.1.1, and the

second one then from the ﬁrst and the case p>din Theorem 9.1.1. 

Corollary 9.1.2: If u ∈ H

k,p

(Ω) for some p and all k ∈ N, then u ∈

∞

(Ω). 

The embedding theorems to follow will be used in Chapter 12 only. First

we shall present another variant of the Sobolev embedding theorem. For a

function v ∈ L

(Ω), we deﬁne the mean of v on Ω as

−



v(x)dx :=

|Ω|



v(x)dx,

|Ω| denoting the Lebesgue measure of Ω. We then have the following result:

Corollary 9.1.3: Let 1 ≤ p<dand u ∈ H

1,p

(B(x

,R)). Then



−



B(x

,R)

|u|

d−p



d−p

≤ c



−



B(x

,R)

|Du|

+ −



B(x

,R)

|u|



, (9.1.9)

where c

depends on p and q only.

Proof: Without loss of generality, x

= 0. Likewise, we may assume R =

1, since we may consider the functions ˜u(x)=u(Rx) and check that the

expressions in (9.1.9) scale in the right way. Thus, let u ∈ H

1,p

(B(0, 1)). We

extend u to the ball B(0, 2), by putting

u(x)=u



|x|



for |x| > 1.

This extension satisﬁes

u

1,p

(B(0,2))

≤ c

u

1,p

(B(0,1))

. (9.1.10)

9.1 General Sobolev Spaces. Embedding Theorems 225

Now let η ∈ C

∞

(B(0, 2)) with

η ≥ 0,η≡ 1onB(0, 1), |Dη|≤2.

Then v = ηu ∈ H

1,p

(B(0, 2)), and by (9.1.2),





B(0,2)

|v|

d−p



d−p

≤ c





B(0,2)

|Dv|



. (9.1.11)

Since

Dv = ηDu + uDη,

from the properties of η, we deduce

|Dv|

≤ c

(|Du|

+ |u|

) , (9.1.12)

and hence with (9.1.10),



B(0,2)

|Dv|

≤ c





B(0,1)

|Du|



B(0,1)

|u|



. (9.1.13)

Since on the other hand



B(0,1)

|u|

d−p

≤



B(0,2)

|v|

d−p

(9.1.9) follows from (9.1.11) and (9.1.13). 

Later on (in Section 12.1), we shall need the following result of John and

Nirenberg:

Theorem 9.1.2: Let B(y

) be a ball in R

, u ∈ W

1,1

(B(y

)),and

suppose that for all balls B(y, R) ⊂ R



B(y,R)∩B(y

)

|Du|≤R

d−1

. (9.1.14)

Then there exist α>0 and β

< ∞ satisfying



B(y

)

α|u−u

≤ β

(9.1.15)

with



B(y

)

u (mean of u on B(y

)).

In particular,



B(y

)

αu



B(y

)

−αu



B(y

)

α(u−u

)



B(y

)

−α(u−u

)

≤ β

(9.1.16)

226 9. Sobolev Spaces and L

Regularity Theory

More generally, for a measurable set B ⊂ R

,andu ∈ L

(B), we denote

the mean by

|B|



u(y)dy, (9.1.17)

|B| being the Lebesgue measure of B. In order to prepare the proof of The-

orem 9.1.2, we start with a lemma:

Lemma 9.1.3: Let Ω ⊂ R

be convex, B ⊂ Ω measurable with |B| > 0,

u ∈ W

1,1

(Ω). Then we have for almost all x ∈ Ω,

|u(x) − u

|≤

(diam Ω)

d |B|



|x − z|

1−d

|Du(z)|dz. (9.1.18)

Proof: As before, it suﬃces to prove the inequality for u ∈ C

(Ω). Since Ω

is convex, if x and y are contained in Ω, so is the straight line joining them,

and we have

u(x) − u(y)=−



|x−y|

∂

∂r



x + r

y − x

|y − x|



dr,

and thus

u(x) − u

|B|



(u(x) − u(y))dy

= −

|B|



|x−y|

∂

∂r



x + r

y − x

|y − x|



dr dy.

This implies

|u(x) − u

|≤

|B|

(diam Ω)





|ω|=1

x+rω∈Ω



|x−y|

∂

∂r

u(x + rω)dr dω



, (9.1.19)

if instead of over B, we integrate over the ball B(x, diam Ω)) ∩ Ω,write

dy = 

d−1

dω d in polar coordinates, and integrate with respect to .Thus,

as in the proofs of Theorems 1.2.1 and 8.2.2,

|u(x) − u

|≤

|B|

(diam Ω)



|x−y|



∂B(x,r)∩Ω

d−1

∂u

∂ν

(z)dσ(z)dr



|B|

(diam Ω)





|x − z|

d−1



i=1

∂

∂z

u(z)

− z

|x − z|



≤

(diam Ω)

d |B|



|x − z|

d−1

|Du(z)|dz.

