proof of Sobolev inequality for Ω=𝐑^n

proof of Sobolev inequality for

Ω

=

𝐑

n

Step 1:

u

is smooth and

p

=

1

First suppose

u

is a compactly supported

smooth function

, and let

(

e

k

)

1

≤

k

≤

n

denote a basis of

𝐑

n

. For every

1

≤

k

≤

n

,

u

⁢

(

x

)

=

∫

-

∞

0

∂

⁡

u

∂

⁡

x

k

⁢

(

x

+

s

⁢

e

k

)

⁢

𝑑

s

.

Therefore,

|

u

⁢

(

x

)

|

≤

S

k

⁢

(

x

)

:=

∫

𝐑

|

∂

⁡

u

∂

⁡

x

k

⁢

(

x

1

,

…

,

x

k

-

1

,

s

,

x

k

+

1

,

…

,

x

n

)

|

⁢

𝑑

s

.

Note that

S

k

does not depend on

x

k

.
One also has

|

u

⁢

(

x

)

|

n

/

(

n

-

1

)

≤

∏

k

=

1

n

|

S

k

⁢

(

x

)

|

1

/

(

n

-

1

)

.

The integration of this

inequality

yields,

∫

𝐑

n

|

u

⁢

(

x

)

|

n

/

(

n

-

1

)

⁢

𝑑

x

≤

∫

𝐑

n

∏

k

=

1

n

|

S

k

⁢

(

x

)

|

1

/

(

n

-

1

)

⁢

d

⁢

x

.

Since

S

1

does not depend on

x

k

, we can apply the generalized Hölder inequality with

n

-

1

for the integration with respect to

x

1

in order to obtain:

∫

𝐑

n

|

u

⁢

(

x

)

|

n

/

(

n

-

1

)

⁢

𝑑

x

≤

∫

𝐑

n

-

1

S

1

⁢

(

x

)

⁢

∏

k

=

2

n

(

∫

𝐑

S

k

⁢

(

x

)

⁢

𝑑

x

1

)

1

/

(

n

-

1

)

⁢

d

⁢

x

1

⁢

…

⁢

d

⁢

x

n

.

The repetition of this process for the

variables

x

2

,

…

,

x

n

gives

∫

𝐑

n

|

u

⁢

(

x

)

|

n

/

(

n

-

1

)

⁢

𝑑

x

≤

∏

k

=

1

n

(

∫

𝐑

n

|

∂

⁡

u

∂

⁡

x

k

|

⁢

𝑑

x

)

1

/

(

n

-

1

)

.

By the

arithmetic-geometric means inequality

, one obtains

∫

𝐑

n

|

u

⁢

(

x

)

|

n

/

(

n

-

1

)

⁢

𝑑

x

≤

n

-

n

/

(

n

-

1

)

⁢

(

∑

k

=

1

n

(

∫

𝐑

n

|

∂

⁡

u

∂

⁡

x

k

|

⁢

𝑑

x

)

)

n

/

(

n

-

1

)

.

One finally concludes

∥

u

∥

L

n

/

(

n

-

1

)

≤

n

1

/

2

-

n

/

(

n

-

1

)

⁢

∥

∇

⁡

u

∥

L

n

/

(

n

-

1

)

.

Step 2: general

u

and

p

=

1

In general if

u

∈

W

1

,

1

⁢

(

𝐑

n

)

. It can be approximated by a

sequence

of compactly supported smooth functions

(

u

m

)

. By step 1, one has

∥

u

m

-

u

ℓ

∥

L

n

/

(

n

-

1

)

≤

n

1

/

2

-

n

/

(

n

-

1

)

⁢

∥

∇

⁡

u

m

-

∇

⁡

u

ℓ

∥

L

1

.

therefore

(

u

m

)

is a

Cauchy sequence

in

L

n

/

(

n

-

1

)

⁢

(

𝐑

n

)

. Since it converges to

u

in

L

1

⁢

(

𝐑

n

)

, the limit of

(

u

m

)

is

u

in

L

n

/

(

n

-

1

)

⁢

(

𝐑

n

)

and one has

∥

u

∥

L

n

/

(

n

-

1

)

≤

n

1

/

2

-

n

/

(

n

-

1

)

⁢

∥

∇

⁡

u

∥

L

n

/

(

n

-

1

)

.

Step 3:

1

<

p

<

n

and

u

is smooth

Suppose

1

<

p

<

n

and

u

is a smooth compactly supported function.
Let

r

=

p

⁢

(

n

-

1

)

n

-

p

and

v

=

|

u

|

r

.

Since

u

is smooth,

v

∈

W

1

,

1

(It is however not necessarily smooth), and its weak derivative is

∇

⁡

v

=

r

⁢

u

⁢

|

u

|

r

-

2

⁢

∇

⁡

u

.

One has, by the Hölder inequality,

∥

∇

v

∥

L

1

⁢

(

𝐑

N

)

≤

r

∥

|

u

|

r

∥

L

p

/

p

-

1

⁢

(

𝐑

N

)

∥

∇

u

∥

L

p

⁢

(

𝐑

N

)

=

r

∥

u

∥

L

n

⁢

p

/

(

n

-

p

)

⁢

(

𝐑

N

)

r

-

1

∥

∇

u

∥

L

p

⁢

(

𝐑

N

)

Therefore, the Sobolev inequality yields

∥

u

∥

L

n

⁢

p

/

(

n

-

p

)

⁢

(

𝐑

N

)

r

=

∥

v

∥

L

n

/

(

n

-

1

)

⁢

(

𝐑

N

)

≤

r

n

1

/

2

-

n

/

(

n

-

1

)

∥

u

∥

L

n

⁢

p

/

(

n

-

p

)

⁢

(

𝐑

N

)

r

-

1

∥

∇

u

∥

L

p

⁢

(

𝐑

N

)

.

This yields

∥

u

∥

L

n

⁢

p

/

(

n

-

p

)

⁢

(

𝐑

N

)

≤

n

1

/

2

-

n

/

(

n

-

1

)

p

⁢

(

n

-

1

)

n

-

p

∥

∇

u

∥

L

p

⁢

(

𝐑

N

)

.

Step 4:

1

<

p

<

n

and

u

∈

W

1

,

p

This is done as step 2.

This proof is due to Gagliardo and Nirenberg, who were the first to prove the inequality for

p

=

1

. This proof can be also found in

[

1

,

2

,

3

]

.

References

1

HaÃÂ¯m

Brezis

,

Analyse fonctionnelle, ThÃÂ©orie et applications

, MathÃÂ©matiques appliquÃÂ©es pour la maÃÂ®trise, Masson, Paris, 1983.

http://www.ams.org/mathscinet-getitem?mr=0697382

[MR85a:46001]

2

JÃÂ¼rgen

Jost

,

Partial Differential Equations

, Graduate Texts in Mathematics, Springer, 2002,

http://www.ams.org/mathscinet-getitem?mr=1919991

[MR:2003f:35002].

3

Michel

Willem

,

Analyse fonctinnelle ÃÂ©lÃÂ©mentaire

, Cassini, Paris, 2003.

Title

proof of Sobolev inequality for

Ω

=

𝐑

n

Canonical

name

ProofOfSobolevInequalityForOmegamathbfRn

Date of creation

2013-03-22 15:05:22

Last modified on

2013-03-22 15:05:22

Owner

vanschaf (8061)

Last modified by

vanschaf (8061)

Numerical id

14

Author

vanschaf (8061)

Entry type

Proof

Classification

msc 46E35

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