Sharp Integral Inequalities Involving High Order Partial Derivatives

Volume 2008, Article ID 571417,10pages doi:10.1155/2008/571417

Research Article

Sharp Integral Inequalities Involving High-Order

Partial Derivatives

C.-J Zhao1 _{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}

Correspondence should be addressed to C.-J Zhao,[email protected]

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.

Copyrightq2008 C.-J Zhao and W.-S Cheung. 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

Inequalities involving functions ofnindependent 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 Rn _{the n-dimensional Euclidean space. Let E, E} be a bounded domain in Rn _{defined by E × E} n

dxdy dx1· · ·dxndy1· · ·dyn. For any continuous real-valued function ux, y defined on E×E, we denote by_EEux, ydxdythe 2n-fold integral

_b1

a1

· · ·

_bn

an

_d1

c1

· · ·

_dn

cn

ux1, . . . , xn, y1, . . . , yn

dx1· · ·dxndy1· · ·dyn, 2.1

and for anyx, y∈E×E, _ExExus, tdsdtis the 2n-fold integral

_x1

a1

· · ·

_xn

an

_y1

c1

· · ·

_yn

cn

us1, . . . , sn, t1, . . . , tn

dx1· · ·dsndt1· · ·dtn. 2.2

We represent by FE×Ethe class of continuous functions ux, y : E×E→R for which D2n_{ux, y D}

1· · ·D2nux, y, where

D1 ∂

∂x1, . . . , Dn ∂

∂xn, Dn1 ∂

∂y1, . . . , D2n ∂

∂yn 2.3

exists and that for eachi, 1≤i≤n,

ux, y_xiai 0, ux, y_yici 0, ux, y_xibi 0, ux, y_yidi0, i1, . . . , n

2.4

the classFE×Eis denoted asGE×E.

Theorem 2.1. Let μ ≥ 0, λ ≥ 1 be given real numbers, and let px, y ≥ 0, x, y ∈ E×E be a continuous function. Further, letux, y∈GE×E. Then, the following inequality holds

E

E

px, yux, yμdx dy

≤

E

Ep

x, yqx, y, λ, μdx dy

E

E

_D2n_{ux, y}λ_{dx dy} μ/λ_,

2.5

where

qx, y, λ, μ

1 2n1

n

i1

xi−aibi−xiyi−cidi−yiλ−1/2

_μ/λ

. 2.6

Proof. For the set{1, . . . , n}, letπ A∪B, πA∪Bbe partitions, whereA j1, . . . , jk, B

jk1, . . . , jn, A i1, . . . , ik, and B ik1, . . . , inare such that cardA cardA k and cardB cardB n−k,0 ≤ k ≤ n. It is clear that there are 2n1 such partitions. The set of all such partitions we will denote as Z andZ, respectively. For fixed partition π, π and x∈E, y∈E, we define

Eπx

E_πy

us, tds dt

Ax

Bx

Ay

By

where_Ax, _Aydenote thek-fold integral,

Bx,

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

_{ux, y≤}

Eπx

E_πy

_D2n_{us, tds dt. 2.8}

In view of H ¨older integral inequality, we have

_{ux, y≤}

i∈A

xi−ai i∈B

bi−xi i∈A

yi−ci i∈B

di−yi

λ−1/λ

×

Eπx

E_πy

_D2n_{us, t}λ ds dt

1/λ .

2.9

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

_{ux, y}μ_≤

_n

i1

xi−aibi−xiyi−cidi−yiλ−1/2

_μ/λ

×

π∈Z, π_∈Z

Eπx

E_πy

_D2n_{us, t}λ ds dt

1/2n1μ/λ

≤

1 2n1

n

i1

xi−aibi−xiyi−cidi−yiλ−1/2

_μ/λ

×

π∈Z, π_∈Z

Eπx

E_πy

_D2n_{us, t}λ_{ds dt}

μ/λ

qx, y, λ, μ

E

E

_D2n_{us, t}λ ds dt

μ/λ .

2.10

Now, multiplying both the sides of2.10bypx, yand integrating the resulting inequality on E×E, we have

E

Ep

x, yux, yμdx dy≤

E

Ep

x, yqx, y, λ, μdx dy

E

E

_D2n_{us, t}λ ds dt

μ/λ ,

2.11 where

qx, y, λ, μ

1 2n1

n

i1

xi−aibi−xiyi−cidi−yiλ−1/2

_μ/λ

. 2.12

Remark 2.2. Taking forpx, y 1 in2.5,2.5reduces to

E

E

_{ux, y}μ

dx dy≤K0

E

E

_D2n_{ux, y}λ dx dy

μ/λ

, 2.13

where

K₀

1 2

μ/λ B2

1 μ 2 −

μ 2λ,1

μ 2 −

μ 2λ

_{n n}

i1

bi−aidi−ci1μ−μ/λ, 2.14

Taking forλμ2 in2.13reduces to

E

E

_{ux, y}2

dx dy≤

π2 8

n M2

E

E

_D2n_{ux, y}2

dx dy , 2.15

where

M n

i1

bi−aidi−ci

4 . 2.16

Let ux, yreduce touxin 2.15and with suitable modifications, then 2.15becomes the following two Wirting type inequalities:

E

_ux2 dx≤

π 4

n M2

E

_Dn_ux2

dx , 2.17

where

M

n

i1

bi−ai

2 . 2.18

Similarly

E

_ux4 dx≤

3π 16

n M4

E

_Dn_ux4

dx , 2.19

whereMis as in2.17.

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 inequalities2.17and2.19are more strengthed.

Remark 2.3. Let ux, y reduce to ux in 2.5 and with suitable modifications, then 2.5 becomes the following result:

E

pxuxμdx≤

E

pxqx, λ, μdx

E

_Dn_uxλ dx

μ/λ

, 2.20

where

qx, λ, μ

1 2n

n

i1

x−aibi−xiλ−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_k1μk/λk 1, andukx, y∈GE×E. Then the following inequality holds

E

E

px, y r

k1

_{ukx, y}μk dx dy

≤

E

E

px, y r

k1

qx, y, λk, μkdx dy r

k1 μk λk

E

E

_D2n_{ukx, y}λk dx dy.

Proof. Settingμ μk, λ λk andux, y ukx, y, 1 ≤ k ≤ r in 2.10, multiplying the r inequalities, and applying the extended Arithmetic-Geometric means inequality,

r

k1 aμk/λk

k ≤ r

k1 μk

λkak, ak ≥0, 2.23

to obtain

r

k1

_{ukx, y}μk _≤r

k1

qx, y, λk, μk E

E

_D2n_{uks, t}λk ds dt

μk/λk

≤r

k1

qx, y, λk, μk r

k1 μk λk

E

E

_D2n_{uks, t}λk ds dt.

2.24

Now multiplying both sides of2.24bypx, yand then integrating overE×E, we obtain

2.22.

Corollary 2.5. Let the conditions ofTheorem 2.4be satisfied. Then the following inequality holds

E

E

px, y r

k1

_{ukx, y}μk

dx dy < K₁

E

E

px, ydx dy r

k1 μk λk

E

E

_D2n_{ukx, y}λk dx dy,

2.25

where

K1

1 2n1

r k1μkn

i1

bi−aibi−ai−1

r k1μk

. 2.26

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

E px

r

k1

_ukxμk

dx < K1

E

pxdx r

k1 μk λk

E

_Dn_u

kxλkdx, 2.27

where

K1

1 2n

r k1μkn

i1

bi−ai−1

r k1μk

. 2.28

Remark 2.6. Forpx, y 1,the inequality2.22becomes

E

E r

k1

_{ukx, y}μk

dx dy≤K2 r

k1 μk λk

E

E

_D2n_{ukx, y}λk

dx dy, 2.29

where

K₂

1 2B

2

_1r

k1μk

2 ,

1r_k1μk 2

n n

i1

bi−aidi−ci

r k1μk

. 2.30

Theorem 2.7. Letλandux, ybe as inTheorem 2.1,μ≥1be a given real number. Then the following inequality holds

E

E

_{ux, y}λ

dx dy≤K₃λ, μ

E

E

_{gradux, y}λ

μdx dy, 2.31

where

K₃λ, μ 1 2nB

2

λ1 2 ,

λ1

2 K

λ μ

n

i1

bi−aidi−ciλ/n,

_{gradux, y}

μ

_n

i1

_∂xi∂yi∂2 ux, y μ1/μ

,

2.32

and whereKλ/μ 1ifλ≥μ, andKλ/μ n1−λ/μ_{if0≤λ/μ≤1.}

Proof. For each fixedi, 1≤i≤n,in view of

ux, y_xiai 0, ux, y_yici 0, ux, y_xibi0, ux, y_yidi 0, i1, . . . , n,

2.33

we have

ux, y

_xi

ai

_yi

ci ∂2 ∂si∂tiu

x, y;si, tidsidti,

ux, y

_bi

xi

_di

yi ∂2 ∂si∂tiu

x, y;si, tidsidti,

2.34

where

ux, y;si, tiux1, . . . , xi−1, si, xi1, . . . , xn, y1, . . . , yi−1, ti, yi1, . . . , yn

. 2.35

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

_{ux, y}λ_≤

xi−ai

yi−di

_λ−1xi

ai

_yi

ci

_∂si∂ti∂2 ux, y;si, ti λ

dsidti,

_{ux, y}λ_≤

bi−xidi−yiλ−1

_bi

xi

_di

yi

_∂si∂ti∂2 ux, y;si, ti λ

dsidti.

2.36

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

_{ux, y}λ _≤ 1 2

xi−aiyi−cibi−xidi−yiλ−1/2 ×

_bi

ai

_di

ci

_∂si∂ti∂2 ux, y;si, ti λ

dsidti,

2.37

and now integrating2.37onE×E, we arrive at

E

E

_{ux, y}λ

dx dy≤

_bi

ai

_di

ci 1 2

xi−aiyi−cibi−xidi−yiλ−1/2dxidyi

×

E

E

_∂xi∂yi∂2 ux, y λ

dx dy.

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

aα_i ≤Kα

_n

i1 ai

_α

, ai>0, 2.39

whereKα 1 ifα≥1, andKα n1−α_{if 0≤α≤1,we get}

E

E

_{ux, y}λ

dx dy ≤ n

i1

_bi

ai

_di

ci 1 2

xi−ai

yi−ci

bi−xi

di−yi

_λ−1/2

dxidyi 1/n

×n

i1

E

E

_∂x∂_i∂y2 _iux, y λ

dx dy 1/n

≤ 1

2n n

i1

_bi

ai

_di

ci 1 2

xi−aiyi−cibi−xidi−yiλ−1/2dxidyi 1/n

×n

i1

E

E

_∂xi∂yi∂2 ux, y λ

dx dy

≤ 1

2nB 2

λ1 2 ,

λ1 2

n

i1

bi−aidi−ciλ/n

×

E

E

_{gradux, y}λ

λdx dy,

≤K₃λ, μ

E

E

_{gradux, y}λ

μdx dy,

2.40

where

K₃λ, μ 1 2nB

2

λ1 2 ,

λ1

2 K

λ μ

n

i1

bi−aidi−ciλ/n,

_{gradux, y}

μ

_n

i1

_∂x∂_i∂y2 _iux, y μ1/μ

,

2.41

and whereKλ/μ 1 ifλ≥μ, andKλ/μ n1−λ/μ if 0≤λ/μ≤1.

Remark 2.8. Letux, yreduce touxin2.31and with suitable modifications, and letλ ≥2, μ2, then2.31becomes

E

_uxλ

dx≤K3λ,2

E

_graduxλ

μdx. 2.42

This is just a better inequality than the following inequality which was given by Pachpatte17

E

_uxλ dx≤ 1

n

β 2

λ

E

_graduxλ

μdx. 2.43

On the other hand, taking forμ2, λ2 orμ2, λ4 in2.31and letux, yreduce touxwith suitable modifications, it follows the following Poincar´e-type inequalities:

E

_ux2

dx≤ π 16nβ

2

E

_gradux2

2dx,

E

_ux4

dx≤ 3π 256nβ

4

E

_gradux4

2dx.

2.44

The inequalities 2.44 have been discussed in 18 with the right-hand sides, respectively, multiplied by 4/πand 16/3π. Hence inequalities2.44are more strong results on these types of inequalities.

Ifμ≥λ, in the right sides of2.31we can apply H ¨older inequality with indicesμ/λand μ/μ−λ, to obtain the following corollary.

Corollary 2.9. Let the conditions ofTheorem 2.7be satisfied andμ≥λ. Then

E

E

_{ux, y}λ

dx dy≤K4λ, μ

E

E

_{gradux, y}μ

μdx dy λ/μ

, 2.45

where

K4λ, μ K3λ, μ n

i1

bi−aidi−ciμ−λ/μ. 2.46

Remark 2.10. Taking ux, y ux and with suitable modifications, the inequality 2.45

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

E

_uxλ

dx≤K6λ, μ

E

_graduxμ

μdx λ/μ

, 2.47

where

K6λ, μ K5λ, μ n

i1

bi−aiμ−λ/μ,

K5λ, μ 1 2nB

1λ 2 ,

1λ

2 K

λ μ

n

i1

bi−aiλ/n,

2.48

andKλ/μis as inTheorem 2.7.

Takingλ1,μ2 the inequality2.45,2.45reduces to

E

E

_{ux, ydx dy} 2

≤K₄1,2

E

E

_{gradux, y}2

2dx dy. 2.49

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

gradux2₂dx dy. 2.50

Theorem 2.11. Forukx, y∈GE×E,μk ≥1,1≤k≤r. Then the following inequality holds

E

E

n

i1

_{ukx, y}μk

1/r

dx dy≤K5

E

E r

k1

_{gradukx, y}μk

μkdx dy, 2.51

where

K5 1 2nrB

2

₁

1/rr_k1μk

2 ,

1 1/rr_k1μk 2

n

i1

bi−aidi−ci

r k1μk/nr

. 2.52

Remark 2.12. Taking ux, y ux and with suitable modifications, the inequality 2.51

reduces to the following result:

E

_n

i1

_ukxμk

1/r

dx≤K9

E r

k1

_graduxμk

μkdx, 2.53

where

K9 1 2nrB

₁

1/rr_k1μk

2 ,

1 1/rr_k1μk 2

n

i1

bi−ai

r k1μk/nr

. 2.54

In19, Pachpatte proved the inequality2.53forμk ≥2, 1≤k≤rwithK9replaced by

1/nrβ/2rk1μk/r_{,whereβis as inRemark 2.8. It is clear thatK}

9 <1/nrβ/2

r

k1μk/r_,and hence2.53is a better inequality than a result of Pachpatte.

Similarly, all other results in15also 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.

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