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Volume 2008, Article ID 571417, 10 pagesdoi:10.1155/2008/571417 Research Article Sharp Integral Inequalities Involving High-Order Partial Derivatives C.-J Zhao 1 and W.-S Cheung 2 1 Depa

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Volume 2008, Article ID 571417, 10 pages

doi:10.1155/2008/571417

Research Article

Sharp Integral Inequalities Involving High-Order Partial Derivatives

C.-J Zhao 1 and W.-S Cheung 2

1 Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou 310018, China

2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

Received 28 November 2007; Accepted 10 April 2008

Recommended by Peter Pang

The main purpose of the present paper is to establish some new sharp integral inequalities involving higher-order partial derivatives Our results in special cases yield some of the recent results on Agarwal, Wirtinger, Poincar´e, Pachpatte, Smith, and Stredulinsky’s inequalities and provide some new estimates on such types of inequalities.

the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

Inequalities involving functions of n independent variables, their partial derivatives, integrals

play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial diﬀerential equations as well as diﬀerence equations

1 10 Especially, in view of wider applications, inequalities due to Agarwal, Opial, Pachpatte, Wirtinger, Poincar´e and et al have been generalized and sharpened from the very day of their discover As a matter of fact, these now have become research topic in their own right11–14

In the present paper, we will use the same method of Agarwal and Sheng15, establish some new estimates on these types of inequalities involving higher-order partial derivatives We further generalize these inequalities which lead to result sharper than those currently available

An important characteristic of our results is that the constant in the inequalities are explicit

2 Main results

Let R be the set of real numbers and R n the n-dimensional Euclidean space Let E, E

be a bounded domain in R n defined by E × E n

i1 a i , b i × c i , d i , i 1, , n For

x i , y i ∈ R, i 1, , n, x, y x1, , x n , y1, , y n is a variable point in E × E and

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2 Journal of Inequalities and Applications

E

b1

a1

· · ·

b n

a n

d1

c1

· · ·

d n

c n

dx1· · · dx n dy1· · · dy n , 2.1

and for anyx, y ∈ E × E,

Ex

x1

a1

· · ·

x n

a n

y1

c1

· · ·

y n

c n

dx1· · · ds n dt1· · · dt n 2.2

We represent by FE × E the class of continuous functions ux, y : E × E→ R for which

∂y n 2.3

exists and that for each i, 1 ≤ i ≤ n,

x i a i 0, ux, y

y i c i 0, ux, y

x i b i 0, ux, y

y i d i 0, i 1, , n

2.4

the class FE × E is denoted as GE × E

Theorem 2.1 Let μ ≥ 0, λ ≥ 1 be given real numbers, and let px, y ≥ 0, x, y ∈ E × E be a

E

dx dy

≤

E

dx dy

μ/λ

,

2.5

where

2n1

n

i1

x i − a ib i − x iy i − c id i − y i λ−1/2

μ/λ

. 2.6

card B card B  n − k, 0 ≤ k ≤ n It is clear that there are 2 n1 such partitions The set

of all such partitions we will denote as Z and Z, respectively For fixed partition π, π and

E π x

Ax

Bx

Ay

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Ax ,

Ay denote the k-fold integral,

Bx ,

Byrepresent then − k-fold integral Thus from the assumptions it is clear that for each π ∈ Z, π∈ Z

E π x

Eπ y

In view of H ¨older integral inequality, we have

i∈A

i∈B

i∈A

i∈B

λ−1/λ

×

E π x

Eπ y

ds dt

1/λ

.

2.9

A multiplication of these 2n1 inequalities and an application of the Arithmetic-Geometric mean inequality give

i1

x i − a ib i − x iy i − c id i − y i λ−1/2

μ/λ

π∈Z, π∈Z

E π x

Eπ y

ds dt

1/2 n1μ/λ

≤ 1

2n1

n

i1

x i − aib i − xiy i − cid i − yi λ−1/2

μ/λ

π∈Z, π∈Z

E π x

Eπ y

ds dt

μ/λ

 qx, y, λ, μ

E

ds dt

μ/λ

.

2.10

Now, multiplying both the sides of2.10 by px, y and integrating the resulting inequality on

E

dx dy ≤

E

ds dt

μ/λ

,

2.11 where

2n1

n

i1

x i − aib i − xiy i − cid i − yi λ−1/2

μ/λ

. 2.12

E

dx dy

μ/λ

, 2.13 where

1 2

μ/λ

1 μ

2 − μ

μ

2 − μ

2λ

n n

i1

b i − aid i − ci 1μ−μ/λ, 2.14

and B is the Beta function.

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Taking for λ μ 2 in 2.13 reduces to

E

dx dy ≤

8

n

E

dx dy

, 2.15 where

i1

Let ux, y reduce to ux in 2.15 and with suitable modifications, then 2.15 becomes the following two Wirting type inequalities:

E

ux2

dx ≤

π

4

n

E

D n ux2

dx

where

M 

n

i1

Similarly

E

ux4

dx ≤

3π

16

n

E

D n ux4

dx

where M is as in 2.17

For n 2, the inequalities 2.17 and 2.19 have been obtained by Smith and Stredulinsky 16, however, with the right-hand sides, respectively, multiplies 4/π2 and

16/3π4 Hence, it is clear that inequalities2.17 and 2.19 are more strengthed

becomes the following result:

E

dx ≤

E

pxqx, λ, μdx

E

D n uxλ

dx

μ/λ

, 2.20 where

2n

n

i1

b i − xi λ−1/2

μ/λ

. 2.21 This is just a new result which was given by Agarwal and Sheng15

Theorem 2.4 Let px, y ≥ 0, x, y ∈ E × E be a continuous function Further, let for k 

1, , r, μk ≥ 0, λk ≥ 1, be given real numbers such thatr

k1 μk /λ k 1, and ukx, y ∈ GE × E.

Then the following inequality holds

E

px, y

r

k1

u k x, yμ k

dx dy

≤

E

px, y

r

k1

dx dy

r

k1

E

D 2n u k x, yλ k

dx dy.

2.22

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Proof Setting μ μ k , λ λ k and ux, y uk x, y, 1 ≤ k ≤ r in 2.10, multiplying the r

inequalities, and applying the extended Arithmetic-Geometric means inequality,

r

k1

k ≤r

k1

λ k a k , a k ≥ 0, 2.23

to obtain

r

k1

u kx, yμ k ≤r

k1

E

D 2n u k s, tλ k

ds dt

μ k /λ k

≤r

k1

r k1

E

D 2n u k s, tλ k

ds dt.

2.24

Now multiplying both sides of2.24 by px, y and then integrating over E × E, we obtain

2.22

Corollary 2.5 Let the conditions of Theorem 2.4 be satisfied Then the following inequality holds

E

px, y

r

k1

u kx, yμ k

E

px, ydx dy

r

k1

E

dx dy,

2.25

where

1

2n1

r

k1 μ k n i1

. 2.26

This is just a general form of the following inequality which was established by Agarwal and Sheng15:

E

px

r

k1

u k xμ k

E

pxdx

r

k1

E

D n u k xλ k

dx, 2.27 where

1

2n

r

k1 μ k n i1

E

r

k1

u k x, yμ k

r

k1

E

where

1

2B2

1r

k1 μ k

2 ,1r

k1 μ k

2

n n

i1

. 2.30

For ux, y ux, the inequality 2.29 has been obtained by Agarwal and Sheng 15

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Theorem 2.7 Let λ and ux, y be as in Theorem 2.1 , μ ≥ 1 be a given real number Then the following inequality holds

E

grad ux, yλ

where

K3λ, μ 1

2n B

2

λ 1

2

K

λ μ

n

i1

,

grad ux, y

μ n

i1

μ

1/μ

,

2.32

Proof For each fixed i, 1 ≤ i ≤ n, in view of

x i a i 0, ux, y

y i c i 0, ux, y

x i b i 0, ux, y

y i d i 0, i 1, , n,

2.33

we have

ux, y 

x i

a i

y i

c i

ux, y 

b i

x i

d i

y i

2.34

where

. 2.35

Hence from H ¨older inequality with indices λ and λ/1 − λ, it follows that

a i

y i

c i

∂s ∂ i2∂t i u

x, y; s i , t iλ

x i

d i

y i

∂s ∂ i2∂t i u

x, y; s i , t iλ

2.36

Multiplying2.36, and then applying the Arithmetic-Geometric means inequality, to obtain

2

x i − aiy i − cib i − xid i − yi λ−1/2 ×

b i

a i

d i

c i

∂s ∂ i2∂t i u

x, y; s i , t iλ

2.37 and now integrating2.37 on E × E, we arrive at

E

dx dy ≤

b i

a i

d i

c i

1 2

x i − a iy i − c ib i − x id i − y i λ−1/2 dx i dy i

×

E

2.38

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Next, multiplying the inequality 2.38 for 1 ≤ i ≤ n, and using the Arithmetic-Geometric

means inequality, and in view of the following inequality:

n

i1

a α i ≤ Kα n

i1

α

where Kα 1 if α ≥ 1, and Kα n1−αif 0≤ α ≤ 1, we get

E

dx dy ≤

n

i1

b i

a i

d i

c i

1 2

1/n

×n

i1

E

1/n

≤ 1

2n

n

i1

b i

a i

d i

c i

1 2

1/n

×n

i1

E

≤ 1

2n B

2

λ 1

2

n

i1

×

E

grad ux, yλ

≤ K

3λ, μ

E

grad ux, yλ

2.40 where

K3λ, μ 1

2n B

2

λ 1

2

K

λ μ

n

i1

,

grad ux, y

μ n

i1

μ

1/μ

,

2.41

and where Kλ/μ 1 if λ ≥ μ, and Kλ/μ n1−λ/μ if 0≤ λ/μ ≤ 1.

E

uxλ

E

grad uxλ

μ dx. 2.42 This is just a better inequality than the following inequality which was given by Pachpatte17

E

uxλ

n

β

2

λ

E

grad uxλ

μ dx. 2.43

Because for λ ≥ 2, it is clear that K3λ, 2 < 1/nβ/2 λ , where β max1≤i≤nb i − a i

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On the other hand, taking for μ 2, λ 2 or μ 2, λ 4 in 2.31 and let ux, y reduce

to ux with suitable modifications, it follows the following Poincar´e-type inequalities:

E

ux2

16n β

2

E

grad ux2

2dx,

E

ux4

256n β

4

E

grad ux4

2dx.

2.44

The inequalities 2.44 have been discussed in 18 with the right-hand sides, respectively,

multiplied by 4/π and 16/3π Hence inequalities 2.44 are more strong results on these types

of inequalities

If μ ≥ λ, in the right sides of 2.31 we can apply H¨older inequality with indices μ/λ and

μ/μ − λ, to obtain the following corollary.

Corollary 2.9 Let the conditions of Theorem 2.7 be satisfied and μ ≥ λ Then

E

grad ux, yμ

λ/μ

, 2.45

where

K4λ, μ K

3λ, μn i1

b i − a id i − c i μ−λ/μ 2.46

reduces to the following result which was given by Agarwal and Sheng15:

E

uxλ

E

grad uxμ

μ dx

λ/μ

, 2.47 where

K6λ, μ K5λ, μn

i1

b i − a iμ−λ/μ ,

K5λ, μ 1

2n B

1 λ

2 ,1 λ

2

K

λ μ

n

i1

,

2.48

and Kλ/μ is as inTheorem 2.7

Taking λ 1, μ 2 the inequality 2.45, 2.45 reduces to

E

≤ K

41, 2

E

grad ux, y2

This is just a general form of the following inequality which was given by Agarwal and Sheng

15

E

uxdx 2

≤K61, 2 2

E

grad ux2

Similar to the proof ofTheorem 2.7, we have the following theorem

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Theorem 2.11 For u kx, y ∈ GE × E, μk ≥ 1, 1 ≤ k ≤ r Then the following inequality holds

E

n

i1

u k x, yμ k

1/r

E

r

k1

grad uk x, yμ k

where

2nr B

2

1 1/rr

k1 μ k

2 ,1 1/rr

k1 μ k

2

n

i1

. 2.52

reduces to the following result:

E

n

i1

u k xμ k

1/r

E

r

k1

grad uxμ k

μ k dx, 2.53 where

2nr B

1 1/rr

k1 μ k

2 ,1 1/rr

k1 μ k

2

n

i1

. 2.54

In19, Pachpatte proved the inequality 2.53 for μ k ≥ 2, 1 ≤ k ≤ r with K9replaced by

1/nrβ/2r k1 μ k /r , where β is as inRemark 2.8 It is clear that K9 < 1/nrβ/2r k1 μ k /r , and

hence2.53 is a better inequality than a result of Pachpatte

Similarly, all other results in15 also can be generalized by the same way Here, we omit the details

Acknowledgments

Research is supported by Zhejiang Provincial Natural Science Foundation of ChinaY605065, Foundation of the Education Department of Zhejiang Province of China20050392 Research

is partially supported by the Research Grants Council of the Hong Kong SAR, ChinaProject No.:HKU7016/07P

References

1 R P Agarwal and P Y H Pang, Opial Inequalities with Applications in Diﬀerential and Diﬀerence

Equations, vol 320 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The

Netherlands, 1995.

2 R P Agarwal and V Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Diﬀerential

Equations, vol 6 of Series in Real Analysis, World Scientific, Singapore, 1993.

3 R P Agarwal and E Thandapani, “On some new integro-diﬀerential inequalities,” Analele S¸tiint¸ifice

ale Universit˘at¸ii “Al I Cuza” din Ias¸i, vol 28, no 1, pp 123–126, 1982.

4 D Ba˘ınov and P Simeonov, Integral Inequalities and Applications, vol 57 of Mathematics and Its

Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.

5 J D Li, “Opial-type integral inequalities involving several higher order derivatives,” Journal of

Mathematical Analysis and Applications, vol 167, no 1, pp 98–110, 1992.

6 D S Mitrinoviˇc, J E Peˇcari´c, and A M Fink, Inequalities Involving Functions and Their Integrals ang

Derivatives, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.

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7 W.-S Cheung, “On Opial-type inequalities in two variables,” Aequationes Mathematicae, vol 38, no 2-3,

pp 236–244, 1989.

8 W.-S Cheung, “Some new Opial-type inequalities,” Mathematika, vol 37, no 1, pp 136–142, 1990.

9 W.-S Cheung, “Some generalized Opial-type inequalities,” Journal of Mathematical Analysis and

Applications, vol 162, no 2, pp 317–321, 1991.

10 W.-S Cheung, “Opial-type inequalities with m functions in n variables,” Mathematika, vol 39, no 2,

pp 319–326, 1992.

11 P S Crooke, “On two inequalities of Sobolev type,” Applicable Analysis, vol 3, no 4, pp 345–358,

1974.

12 B G Pachpatte, “Opial type inequality in several variables,” Tamkang Journal of Mathematics, vol 22,

no 1, pp 7–11, 1991.

13 B G Pachpatte, “On some new integral inequalities in two independent variables,” Journal of

Mathematical Analysis and Applications, vol 129, no 2, pp 375–382, 1988.

14 X.-J Wang, “Sharp constant in a Sobolev inequality,” Nonlinear Analysis: Theory, Methods & Application,

vol 20, no 3, pp 261–268, 1993.

15 R P Agarwal and Q Sheng, “Sharp integral inequalities in n independent variables,” Nonlinear

Analysis: Theory, Methods & Applications, vol 26, no 2, pp 179–210, 1996.

16 P D Smith and E W Stredulinsky, “Nonlinear elliptic systems with certain unbounded coeﬃcients,”

Communications on Pure and Applied Mathematics, vol 37, no 4, pp 495–510, 1984.

17 B G Pachpatte, “A note on Poincar´e and Sobolev type integral inequalities,” Tamkang Journal of

Mathematics, vol 18, no 1, pp 1–7, 1987.

18 B G Pachpatte, “On multidimensional integral inequalities involving three functions,” Soochow

Journal of Mathematics, vol 12, pp 67–78, 1986.

19 B G Pachpatte, “On Sobolev type integral inequalities,” Proceedings of the Royal Society of Edinburgh A,

vol 103, no 1-2, pp 1–14, 1986.

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7 W.-S Cheung, “On Opial-type inequalities. .. obtained by Agarwal and Sheng 15

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Theorem... i − a i

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