Symmetrization of Functions and the Best Constant of 1-DIM L

Volume 2009, Article ID 874631,12pages doi:10.1155/2009/874631

Research Article

Symmetrization of Functions and the Best Constant of 1-DIM L

Sobolev Inequality

Kohtaro Watanabe,

Yoshinori Kametaka,

Atsushi Nagai,

Hiroyuki Yamagishi,

and Kazuo Takemura

1Department of Computer Science, National Defense Academy, 1-10-20 Hashirimizu, Yokosuka 239-8686, Japan

2Graduate School of Mathematical Sciences, Faculty of Engineering Science, Osaka University, 1–3 Matikaneyamatyo, Toyonaka 560-8531, Japan

3Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, 2-11-1 Shinei, Narashino 275-8576, Japan

4Department of Monozukuri Engineering, Tokyo Metropolitan College of Industrial Technology, 1-10-40 Higashi-ooi, Shinagawa, Tokyo 140-0011, Japan

Correspondence should be addressed to Kohtaro Watanabe,[email protected] Received 25 June 2009; Accepted 16 October 2009

Recommended by Kunquan Lan

The best constantsCm, pof Sobolev embedding ofW₀^m,p−s, sintoL^∞−s, sform1,2,3 and 1< pare obtained. A lemma concerning the symmetrization of functions plays an important role in the proof.

Copyrightq2009 Kohtaro Watanabe et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetW₀^m,p−s, sbe a Sobolev space which consists of the functions whose derivatives up to m−1 vanish atx±s, that is,

W₀^m,p−s, s:

u|uⁱ∈L^p−s, s i0, . . . , m, u^j±s 0

j0, . . . m−1

, 1.1

whereuⁱdenotesith derivative ofuin a distributional sense. The purpose of this paper is to investigate the best constantCm, pofL^pSobolev inequality

sup

−s≤x≤s|ux|

≤ C

s

−s

u^mx^pdx 1/p

, 1.2

whereu∈W₀^m,p−s, sand 1< p. The result is as follows.

Theorem 1.1. The best constant of inequality1.2is

C 1, p

2^−q−1/qs^1/q, 1.3

C 2, p

s^q1/q 2^2q−1/q

q1_1/q, 1.4

C 3, p

2^1−2q/q

α

0

x^qα−x^qdx ^s

α

x^qx−α^qdx 1/q

, 1.5

whereqsatisfies 1/p1/q1 andαin1.5is the unique solution of the equation

− ^s

α

x^qx−α^q−1dx ^α

0

x^qα−x^q−1dx0, 1.6

satisfying 0< α < s.

For special values ofp, the solution of1.6can be written explicitly, and C3, pis expressed as follows.

Corollary 1.2. One has

C3,2 s^5/2 2^7/2·5^1/2, C

3,4

3

s^9/4 2^11/4·3^1/2·7^1/2

−1−2^1/35^2/3 133√

21_−1/3

2^−1/35^1/3 133√

21_1/31/4

. 1.7 The best constants C1, p and C2, p were recently obtained by Oshime 1. This paper gives an alternative proof which simplifies the derivation process ofC1, pandC2, p and computes further constantC3, p. To compute these constants, the following lemma with respect to the symmetrization of functions plays an important role.

Lemma 1.3. Letmbe an integer satisfying 1≤m≤3 and let us define the functionalSas follows:

Su: sup_−s≤x≤s|ux|

_s

−su^mx^pdx1/p

u∈W₀^m,p−s, s, u /0

. 1.8

Then, for an arbitraryu ∈ W₀^m,p−s, s, there exists an elementu_∗ which belongs to the following sub-spaceW_∗^m,pofW₀^m,p−s, s:

W_∗^m,p :

u∈W₀^m,p−s, s| max

−a≤x≤a|ux| u0, ux u|x| −s≤x≤s

1.9

such that

Su≤Su∗. 1.10

Remark 1.4. The proof of this lemmaseeSection 3does not apply to the casem ≥ 4, since the relation

uⁱ

y−0 uⁱ

y0

1.11 may fail to hold fori ≥ 4 see3.4–3.6. Hence the problem to obtainCm, pform ≥ 4 essentiallystill remains.

The existence of the maximizer ofScan be seen in the proof ofTheorem 1.1, where we construct it concretely, but here we would like to see this briefly though the proof of the following lemma.

Lemma 1.5. Assume that the assertion ofLemma 1.3holds, then the maximizer ofSexists.

Proof. LetRbe sufficiently large, and letWandWbe as W:

u∈W₀^m,p−s, s|u0 1 , W:

u∈W₀^m,p−s, s|u^m

L^p−s,s≤R

.

1.12

FromLemma 1.3, we see that if the maximizer exists, it is the element ofW :W∩W. So, we can restrict the definition domain ofStoW. SinceWis convex and strongly closedby Sobolev inequalityinW₀^m,p−s, s, it is weakly closed. In addition,Wis weakly compact, soWis also weakly compact. Moreover, · is weakly lower-semicontinuous inW₀^m,p−s, s, and hence 1/Sattains its minimum inW. This proves the lemma.

Finally, we introduce some studies related to the present paper. Whenp2Hilbertian Sobolev space case, the best constants for the embeddings ofW^m,2a, binto L^∞a, b for various conditions were treated in Richardson2, Kalyabin3, and4–8; see also references of these literatures. On the other hand, for the casep /2, few literature seems to be available.

In 9, Kametaka, Oshime, Watanabe, Yamagishi, Nagai, and Takemura obtained the best constant of1.2whenubelongs to a subspaceW_P^m,pofW^m,p0,1which consists of periodic functions

W_P^m,p:

u∈W^m,p0,1|uⁱ0 uⁱ1 0≤i≤m−1, ¹

0

uxdx0

, 1.13

as

C m, p

⎧⎨

⎩

bm·_L^p/p−1_0,1 m2n−1, n1,2,3, . . .,

bmα;·_L^p/p−1_0,1 m2n, n1,2,3, . . ., 1.14

wherebm·is a Bernoulli polynomial,bmα;· bm·−bmα,andαis an unique solution of the equation

α

0

−1^m−1bmα;xdx_1/p−1

− ^1/2

α

−1^mbmα;xdx_1/p−1

0 1.15

in the interval 0< α < 1/2. Moreover, in1, Oshime obtained the best constantC1, pand C2, p. Other topics on this subject, especially the best constant of Sobolev inequalities on Riemannian manifolds, are seen in Hebey10.

2. Proof of Theorem 1.1

First, we prepare the following lemma.

Lemma 2.1. Letu∈W₀^m,p−s, sandHbe a function satisfying

Hx:

⎧⎪

⎪⎨

⎪⎪

⎩ 1

2, −s≤x <0,

−1

2, 0≤x≤s, m1

⎧⎪

⎪⎨

⎪⎪

⎩

−1 2

x s 2

, −s≤x <0, 1

2

x− s 2

, 0≤x≤s, m2

⎧⎪

⎪⎨

⎪⎪

⎩ 1

4xxα, −s≤x <0,

−1

4xx−α, 0≤x≤s, m3

2.1

then, it holds that

u0 ^s

−su^mxHxdx, 2.2

whereαis an arbitrary constant (later, inLemma 2.3, one fixes the value ofαto satisfy1.6).

Proof. By integration by parts, we obtain the result.

From Lemma 1.3, to obtain the best constant of 1.2, we can restrict the definition domain of the functionalSto the nonzero element of W_∗^m,p. Now, letu ∈ W_∗^m,p, then from Lemma 2.1and H ¨older’s inequality, we have form1,2,3,

−s≤x≤ssup|ux|u0≤ HL^q−s,su^m

L^p−s,s. 2.3

So, we have

C m, p

≤ HL^q−s,s, 2.4

and the equality holds in2.4if and only if there existsu∈W_∗^m,psatisfying u^mx

sgnHx

|Hx|^q/p

sgnHx

|Hx|^q−1. 2.5

To confirm the existence of suchu, we use the following lemmas.

Lemma 2.2. Letf ∈C−s, ssatisfy

s

−sxⁱfxdx0 i0,1, . . . , m−1, 2.6 then the solutionuof

u^mx fx 2.7

exists inW₀^m,p−s, s.

Proof . Let us defineuas

ux: ^x

−s

x−t^m−1

m−1! ftdt. 2.8

ClearlyuisC^m−s, s,and

uⁱx

⎧⎪

⎪⎨

⎪⎪

⎩

x

−s

x−t^m−1−i

m−1−i!ftdt 0≤i≤m−1,

fx im.

2.9

Moreover, from the assumption, it holds thatuⁱ±s 00≤i≤m−1.

Lemma 2.3. The solutionαof1.6uniquely exists.

Proof. Let

fα:− ^s

α

x^qx−α^q−1dx ^α

0

x^qα−x^q−1dx. 2.10

Since

fα q−1

s

α

x^qx−α^q−2dx q−1

α

0

x^qα−x^q−2dx >0, f0 − ^s

0

x^2q−1dx <0,

fs ^s

0

x^qs−x^q−1dx >0

2.11

the assertion is proved.

UsingLemma 2.2and 5, we obtain the following lemma.

Lemma 2.4. Letαbe a solution of 1.6(whenm 3), then the solution of 2.5belongs toW_∗^m,p form1,2,3.

Proof. First, we prove the case m 2 and 3. For simplicity, let us put Hx sgnHx|Hx|^q−1. Note that in these casesHis a continuous function on−s, s.

1In the casem2,His an even function, so integration ofxHx over the interval

−s, svanishes. In addition,

s

s/2

1 2^q−1

xs

2 _q−1

dx− ^s/2

0

1 2^q−1

−x− s 2

_q−1

dx0 2.12

holds, so the integration ofHx over the interval−s, salso vanishes. Hence, byLemma 2.2, the solutionuof2.5belongs toW₀^m,p−s, s. Properties max−s≤x≤s|ux| u0andux u−xfollow from the fact thatHis an even function and2.12. So, we have provenu∈W_∗^m,p. 2 In the case m 3,H is an odd function, so integrations ofHx andx²Hx over the interval−s, s vanish. Moreover, from 1.6, the integration ofxHx over the interval−s, salso vanishes. Hence, again byLemma 2.2, we have the solution of2.5which belongs toW₀^m,p−s, s. The remaining part is the same as casei.

3In the casem1, let us defineuas follows:

ux

⎧⎪

⎪⎨

⎪⎪

⎩ 1

2^q−1xs −s≤x <0,

− 1

2^q−1x−s 0≤x≤s.

2.13

Clearly it holds thatusatisfies2.5 a.e.,u±s 0, max_−s≤x≤s|ux|u0andux u−x.

To seeu∈W^1,p−s, s, letϕbe an arbitrary element ofC^∞₀ −s, s. Since

s

−suxϕxdx ⁰

−s− 1

2^q−1xsϕxdx ^s

0

1

2^q−1x−sϕxdx ⁰

−s

1

2^q−1ϕxdx

− ^s

0

1

2^q−1ϕxdx,

2.14

we have, in a distributional sense,

ux

⎧⎪

⎪⎨

⎪⎪

⎩ 1

2^q−1 −s≤x <0,

− 1

2^q−1 0≤x≤s.

2.15

Therefore,u∈W^1,p−s, s. This proves the casem1.

Proof ofTheorem 1.1. From Lemma 1.3, and the argument of this section, especially Lemma 2.4,Cm, p HL^q−s,s. This provesTheorem 1.1.

Proof of Corollary 1.2 . In this case,1.6becomes

35s³−120αs²140α²s−56α³0. 2.16 So, we can explicitly solve this equation with respect toα. Substituting to1.5, we obtain the result.

3. Proof of Lemma 1.3

Now, all we have to do is to proveLemma 1.3.

Proof ofLemma 1.3. To avoid the complexity of notation, in the follwings, we fixs 1. Letu be an arbitrary element ofW₀^m,p−1,1 1≤m≤3, and let

−1≤x≤1max|ux|u y

. 3.1

Here, we can assumey≥0, since if it does not,uxcan be replaced byu−x.

iThere is the case

y

−1

u^mx^pdx≥ ¹

y

u^mx^pdx. 3.2

Let us defineuas

ux:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0

−1≤x <−12y , u

2y−x −12y≤x < y ,

ux

y≤x≤1 .

3.3

We have

u

y−0 u

y0 u

y

, 3.4

u

y−0 u

y0

0, 3.5

u y−0

u y0

u y

, 3.6

whenm3,3.4and3.5whenm2, and3.4whenm1. Further, let us define

u∗x:

⎧⎨

⎩ u

xy −1y≤x≤1−y ,

0

1−y <|x| ≤1

. 3.7

Thenu∗ ∈W_∗^m,p, since max_−1≤x≤1|u∗x|u∗0,uⁱ_∗ ±1 00 ≤i≤m−1. Moreover, from 3.2, we haveu^m_∗ L^p−1,1 ≤ u^mL^p−1,1. In addition, clearlyu_∗0 max_−1≤x≤1|u_∗x|

uy max_−1≤x≤1|ux|. So, in the casei, we have proven the lemma.

iiThere is the case

y

−1

u^mx^pdx <

1 y

u^mx^pdx. 3.8

Lettbe an element satisfying 0≤t≤y, and let

x−1 t1

y1x1 −1ax1. 3.9

Further, letUxbe

U x

:u x1

a −1 −1≤x≤t

. 3.10

So,

∂^m_xU x

a^−m∂^m_xux|_xx1/a−1a^−mu^m x1

a −1

, 3.11

and hence we obtain

t

−1

∂^m_xU

x^pdxa^−mp

t

−1

u^m x1

a −1

^pdx. 3.12

By puttingx−1ax1the right-hand side of3.12becomes

a^−mp1

y

−1

u^mx^pdx. 3.13

Similarly, let us put

x1 1−t

1−yx−1 1bx−1 3.14

and defineUxas

U x

:u x−1

b 1

t≤x≤1

. 3.15

So,

∂^m_xU x

b^−m∂^m_xux|_xx−1/b1b^−mu^m x−1

b 1

, 3.16

and hence we obtain

1 t

∂^m_xU

x^pdxb^−mp

a

t

u^m x−1

b 1

^pdx. 3.17

By puttingx1bx−1the right-hand side of3.17becomes

b^−mp1

1 y

u^mx^pdx. 3.18

Let us put

A: ^y

−1

u^mx^pdx, B: ¹

y

u^mx^pdx 3.19

and define

ft: t1

y1 _−mp1

A 1−t

1−y _−mp1

B. 3.20

Note that

f y

ABu^mp

L^p−1,1. 3.21

The derivative offis

ft

mp−1

− 1 y1

t1 y1

_−mp A 1

1−y 1−t

1−y _−mp

B

. 3.22

aThe case

1≤ 1−y

1y _mp−1

B

A. 3.23

In this case, we have

ft≥0

←→

y1_mp−1

t1^−mpA≤

1−y_mp−1

1−t^−mpB

←→

1−t 1t

_mp

≤ 1−y

1y _mp−1

B A.

3.24

Since

max0≤t≤y

1−t 1t

_mp

1, 3.25

from the assumption3.23,fis monotone increasing. So, we have

0≤t≤yminft f0 ⁰

−1

U^mx^pdx ¹

0

U^mx^pdx. 3.26

But, from3.23, it holds that

0

−1

U^mx^pdx 1

y1 _−mp1

A≤ 1

1−y _−mp1

B ¹

0

U^mx^pdx. 3.27

So, if we put

u∗x:

⎧⎨

⎩

Ux −1≤x≤0,

U−x 0≤x≤1, 3.28

as casei, we haveu∗ ∈ W_∗^m,p andu^m_∗ ^p_Lp−1,1 ≤ f0 ≤ fy u^mp_Lp−1,1. In addition, u_∗0 max_−1≤x≤1|u_∗x|uy max_−1≤x≤1|ux|. So, we have proven the caseii-a.

bThe case

1−y 1y

_mp−1 B

A <1. 3.29

In this case, we have

ft mp

mp−1 1 y1

t1 y1

_−mp−1 A 1

1−y 1−t

1−y _−mp−1

B

>0. 3.30

Moreover

f0

mp−1

−

1y_mp−1 A

1−y_mp−1 B

<0, f

y

mp−1

− 1

y1A 1 1−yB

>0,

3.31

since we have3.29and the assumption3.8 A < B, respectively. Therefore, there exists t0 0< t0< ysuch thatft0 0. Let us define the constantMas

M: 1−y

1y

_mp−1/mp

B A

_1/mp

, 3.32

thent0 1−M/1M. Now we have

1 t0

U^mx^pdx <

t0

−1

U^mx^pdx, 3.33

since

1 t0

U^mx^pdx <

t0

−1

U^mx^pdx

←→

1−t0

1−y _−mp1

B <

t01 y1

_−mp1 A←→

1−y 1y

_mp−1 B A <

1−t0

1t0

_mp−1

←→M^mp< M^mp−1←→M <1←→3.29.

3.34

Let us defineuas

ux:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0 −1≤x≤2t0−1, U2t0−x 2t0−1≤x≤t0, Ux t0≤x≤1,

u∗:

⎧⎨

⎩

uxt0 −1t0≤x≤1−t0, 0 1−t0<|x| ≤1.

3.35

Then, again as case i, we have u∗ ∈ W_∗^m,p and by3.33,u^m_∗ ^p_Lp−1,1 u^mp_Lp−1,1 ≤ U^mp_Lp−1,1 ft0 < fy u^mp_Lp−1,1. In addition,u_∗0 max_−1≤x≤1|u_∗x| uy max_−1≤x≤1|ux|. So, we have proven the caseii-b. This completes the proof.

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