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Volume 2008, Article ID 397340, 15 pagesdoi:10.1155/2008/397340 Research Article The Reverse H ¨older Inequality for the Solution to p-Harmonic Type System Zhenhua Cao, 1 Gejun Bao, 1 Ro

Volume 2008, Article ID 397340, 15 pages

doi:10.1155/2008/397340

Research Article

The Reverse H ¨older Inequality for

the Solution to p-Harmonic Type System

Zhenhua Cao, 1 Gejun Bao, 1 Ronglu Li, 1 and Haijing Zhu 2

1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2 College of Mathematics and Physics, Shan Dong Institute of Light Industry, Jinan 250353, China

Correspondence should be addressed to Gejun Bao,baogj@hit.edu.cn

Received 6 July 2008; Revised 9 September 2008; Accepted 5 November 2008

Recommended by Shusen Ding

Some inequalities to A-harmonic equation A x, du  d∗v have been proved The A-harmonic

equation is a particular form of p-harmonic type system Ax, a  du  b  d∗v only when a 0

and b 0 In this paper, we will prove the Poincar´e inequality and the reverse H¨older inequality

for the solution to the p-harmonic type system.

Copyrightq 2008 Zhenhua Cao 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

Recently, amount of work about the A-harmonic equation for the differential forms has been done In fact, the A-harmonic equation is an important generalization of the p-harmonic

equation inRn , p > 1, and the p-harmonic equation is a natural extension of the usual Laplace

equationsee 1 for the details The reverse H¨older inequalities have been widely studied and frequently used in analysis and related fields, including partial differential equations and the theory of elasticitysee 2 In 1999, Nolder gave the reverse H¨older inequality for the

solution to the A-harmonic equation in3, and different versions of Caccioppoli estimates have been established in4 6 In 2004, D’Onofrio and Iwaniec introduced the p-harmonic

type system in7, which is an important extension of the conjugate A-harmonic equation In

2007, Ding proved the following inequality in8

Theorem A Let u, v be a pair of solutions to Ax, g  du  h  d∗v in a domain Ω ⊂ Rn If

g ∈ L p B, Λ L  and h ∈ L q B, Λ L , then du ∈ L p B, Λ L  if and only if d∗v ∈ L q B, Λ L  Moreover,

there exist constants C1, C2independent of u and v, such that

d∗vq q,B ≤ C1



h q q,B  g p

p,B  du p

p,B



,

du p p,B ≤ C2



h q q,B  g p

p,B  d∗vq

q,B



∀B ⊂ σB ⊂ Ω. 1.1

In this paper, we will prove the Poincar´e inequalityseeTheorem 2.5 and the reverse

H ¨older inequality for the solution to the p-harmonic type systemseeTheorem 3.5 Now let

us see some notions and definitions about the p-harmonic type system.

Let e1, e2, , e n denote the standard orthogonal basis ofRn For l  0, 1, , n, we

denote byΛl  ΛlRn  the linear space of all l-vectors, spanned by the exterior product e I 

e i1∧e i2∧· · ·∧e i l corresponding to all ordered l-tuples I  i1, i2, , i l , 1 ≤ i1< i2 < · · · < i l ≤ n.

The Grassmann algebraΛ  ⊕Λlis a graded algebra with respect to the exterior products For

αα I e I ∈ Λ and β β I e I ∈ Λ, then its inner product is obtained by



with the summation over all I  i1, i2, , i l  and all integers l  0, 1, , n The Hodge star

operator∗: Λ → Λ is defined by the rule

∗1  e i1∧ e i2∧ · · · ∧ e i n ,

Hence, the norm of α∈ Λ can be given by

|α|2

Throughout this paper,Ω ⊂ Rn is an open subset, for any constant σ > 1, Q denotes

a cube such that Q ⊂ σQ ⊂ Ω, where σQ denotes the cube whose center is as same as Q

and diamσQ  σ diam Q We say α  α I e I ∈ Λ is a differential l-form on Ω, if every

coefficient αI of α is Schwartz distribution on Ω The space spanned by differential l-form

on Ω is denoted by D Ω, Λ l  We write L p Ω, Λ l  for the l-form α  α I dx I onΩ with

α I ∈ L p Ω for all ordered l-tuple I Thus L p Ω, Λ l is a Banach space with the norm

α p,Ω



Ω|α| p dx

1/p



Ω



I

|α I|2

p/2

dx

1/p

Similarly W k,p Ω, Λ l  denotes those l-forms on Ω with all coefficients in W k,pΩ We denote the exterior derivative by

d : D

Ω, Λ l

−→ DΩ, Λ l1

, for l  0, 1, 2, , n, 1.6 and its formal adjoint operatorthe Hodge codifferential operator

d∗: D

Ω, Λ l

−→ DΩ, Λ l−1

The operators d and d∗are given by the formulas

dα

I

dα I ∧ dx I , d∗ −1nl1∗d∗. 1.8

The following two definitions appear in7.

Definition 1.1 The Hodge system holds:

A x, a  du  b  d∗v, 1.9

where a ∈ L p Ω, Λ l  and b ∈ L q Ω, Λ l , is a p-harmonic type system if A is a mapping from

Ω × ΛltoΛlsatisfying

1 x → Ax, ξ is measurable in x ∈ Ω for every ξ ∈ Λ l;

2 ξ → Ax, ξ is continuous in ξ ∈ Λ l for almost every x∈ Ω;

3 Ax, tξ  t p−1A x, ξ for every t ≥ 0;

2|ξ|  |ζ| p−2;

5 |Ax, ξ − Ax, ζ| ≤ K|ξ − ζ||ξ|  |ζ| p−2

for almost every x ∈ Ω and all ξ, ζ ∈ Λ l , where K ≥ 1 is a constant It should be noted that

A x, ∗ : Ω × Λ l → Λl is invertible and its inverse denoted by A−1satisfies similar conditions

as A but with H ¨older conjugate exponent q in place of p.

Definition 1.2 If1.9 is a p-harmonic type system, then we say the equation

d∗A x, a  du  d∗b 1.10

is a p-harmonic type equation.

The following definition appears in9

Definition 1.3 A di fferential form u is a weak solution for 1.10 in Ω if u satisfies



Ω

A x, a  du, dϕ d∗b, ϕ ≡ 0 1.11

for every ϕ ∈ W k,p Ω, Λ l−1 with compact support

We can find that if we let a  0 and b  0, then the p-harmonic type system

A x, a  du  b  d∗v 1.12 becomes

A x, du  d∗v. 1.13

It is the conjugate A-harmonic equation, where the mapping A : Ω × Λl → Λlsatisfies the following conditions:

A x, ξ ≤ a|ξ| p−1,

A x, ξ, ξ ≥ |ξ| p 1.14

If we let Ax, ξ  |ξ| p−2ξ, then the conjugate A-harmonic equation becomes the form

|du| p−2du  d∗v. 1.15

It is the conjugate p-harmonic equation.

So we can see that the conjugate p-harmonic equation and the conjugate A-harmonic equation are the specific p-harmonic type system.

Remark 1.4 It should be noted that the mapping A x, ∗ in p-harmonic system Ax, a  du 

b  d∗v, is invertible If we denote its inverse by A−1x, ∗, then the mapping A−1x, ∗ : Λ l →

Λl satisfies similar conditions as A but with H ¨older conjugate exponent q in place of p.

2 The Poincar ´e inequality

In this section, we will introduce the Poincar´e inequality for the differential forms

Now first let us see a lemma, which can be found in9, Section 4 for the details

Lemma 2.1 Let D be a bounded, convex domain in R n To each y ∈ D there corresponds a linear

operator K y : C∞D, Λ l  → C∞D, Λ l−1 defined by



K y ω

x; ξ1, , ξ l−1



1

0

t l−1ω

tx  y − ty; x − y, ξ1, , ξ l−1

dt, 2.1

and the decomposition

ω  dK y ω

holds at any point y ∈ D.

We construct a homotopy operator T : C∞D, Λ l  → C∞D, Λ l−1 by averaging K y over all points y ∈ D:

Tω



where ϕ form C∞D is normalized so thatϕ ydy  1 It is obvious that ω  dK y ω   K y dω

remains valid for the operator T :

We define the l-forms ωD∈ D D, Λ l  by ωD |D|−1

Dω ydy for l  0 and ωD dTω for

l  1, 2, , n, and all ω ∈ W 1,p D, Λ l , 1 < p < ∞.

The following definition can be found in [ 9 , page 34].

Definition 2.2 For ω ∈ D D, Λ l, the vector valued differential form

∇ω 



∂ω

∂x1, ,

∂ω

∂x n

2.5

consists of differential forms ∂ω/∂xi ∈ D D, Λ l, where the partial differentiation is applied

to coefficients of ω

The proof of9, Proposition 4.1 implies the following inequality

Lemma 2.3 For any ω ∈ L p D, Λ l , it holds that

∇Tω p,D≤ Cn, pω p,D 2.6

for any ball or cubeD ∈ Rn

The following Poincar´e inequality can be found in [ 2 ].

Lemma 2.4 If u ∈ W 1,p

0 Ω, then there is a constant C  Cn, p > 0 such that

 1

|B|



B

|u| pχ dx

1/pχ

≤ Cr

 1

|B|



B

|∇u| p dx

1/p

whenever B  Bx0, r  is a ball in R n , where n ≥ 2 and χ  2 for p ≥ n, χ  np/n − p for p < n.

Theorem 2.5 Let u ∈ D D, Λ l , and du ∈ L p D, Λ l1 Then, u − uDis in L χp D, Λ l  and



1

|D|



D|u − uD|pχ dx

1/pχ

≤ Cn, p, ldiamD

 1

|D|



D|du| p dx

1/p

2.8

for any ball or cubeD ∈ Rn , where χ  2 for p ≥ n and χ  np/n − p for 1 < p < n.

Proof We know T du  u − uD Now we suppose u − u Q  Tdu  I u I dx I , where I 

i1, , i l1 take over all l  1-tuples So we have

∇Tdu 



∂u

∂x1, ,

∂u

∂x n

 

I

∂u I

∂x1dx I , ,



I

∂u I

∂x n dx I

So we have

 1

|D|



D

u − uDpχ

dx

1/pχ

 |D|1



D









I

u I dx I





pχ

dx

1/pχ

 |D|1



D



I

u I2

pχ/2

dx

1/pχ

2.10

By the inequality

n



i1



a i2

1/2

≤n

i1

a i ≤ n 1/2 n

i1



a i2

1/2

2.11

for any a i≥ 0, and the Minkowski inequality, we have

1

|D|



D



I

u I2

pχ/2

dx

1/pχ

≤

I

 1

|D|



D

u Ipχ

dx

1/pχ

. 2.12

According to the Poincar´e inequality, we have



I

 1

|D|



D|u I|pχ dx

1/pχ

≤ C1n, pdiamD

I

 1

|D|



D|∇u I|p dx

1/p

. 2.13

Combining2.10, 2.12, and 2.13, we can obtain



1

|D|



D

u − uDpχ

dx

1/pχ

≤ C1n, pdiamD

I

 1

|D|



D|∇u I|p dx

1/p 2.14

By2.9 we have

∇Tdu p,D

∂x ∂u1, , ∂u

∂x n





p,D





D



∂x ∂u1, , ∂u

∂x n



p dx

1/p



D

n



i1



∂x ∂u i2

p/2

dx

1/p



D

n



i1



I



∂u I

∂x i



2

p/2

dx

1/p



D



I

n



i1



∂u I

∂x i



2

p/2

dx

1/p

2.15

Combining2.11 and 2.15, then we have

∇Tdu p,D

D



I

n



i1



∂u I

∂x i



2

p/2

dx

1/p

≥C l1 n −1/2

D



I

n



i1



∂u I

∂x i



2

1/2 p

dx

v 1/p

≥C l1 n −1/2

D



I

n



i1



∂u I

∂x i



2

p/2

dx

1/p

≥C l1 n −1/2

C l1 n −p−1/p

n



i1



∂u I

∂x i



2

p/2

dx

1/p

≥C2n, p, l−1

I



D∇u Ip

dx

1/p ,

2.16

where C2n, p, l  C l1 n 1/2p−1/p Now combining2.14, 2.16, and 2.6, we can get



1

|D



D

u − uD|pχ dx

1/pχ

≤ C1n, pdiamD

I

 1

|D|



D∇u Ip

dx

1/p

≤ C1n, p, lC2n, p, l

 1

|D|

1/p

∇Tdu p,D

≤ C3n, p, ldiamD

 1

|D|



D|du| p dx

1/p

2.17

3 The reverse H ¨older inequality

In this section, we will prove the reverse H ¨older inequality for the solution of the p-harmonic

type system Before we prove the reverse H ¨older inequality, let us first see some lemmas

Lemma 3.1 If f, g ≥ 0 and for any nonnegative η ∈ C∞

0 Ω, it holds



Ωηf dx≤



then for any h ≥ 0:



Ωηfh dx≤



Proof Let μ be a measure in X, f be a nonnegative μ-measurable function in a measure space

X, using the standard representation theorem, we have



X

f q dμ  q

∞

0

t q−1μ x : fx > tdt 3.3

for any 0 < t < q Now, we let μE E ηf dx and ν E E g dx then, we can obtain



Ωηfh dx

∞

0



h>t

ηf dx dt≤

∞

0



h>t

g dx dt



SoLemma 3.1is proved

Lemma 3.2 If u, v is a pair of solution to the p-harmonic type system 1.9, then it holds



Ω|η da| p dx ≤ C



Ω|a  dudη| p dx 3.5

for any nonnegative η ∈ C∞

0 Ω and where C  C l1

n p Proof Since u, v is a pair of solutions to Ax, a  du  b  d∗v, it is also the solution to

A−1x, b  d∗v   a  du, where A−1x, ∗ is the inverse Ax, ∗ Now, we suppose that da 



I ω I dx I and let ϕ1 −I η sign ω I dx I By using ϕ  ϕ1and dϕ1IsignωI dη ∧ dx I in

1.11, we can obtain



Ω

A−1

x, b  d∗v

That is,



Ω



da,

I

η sign

ω I

dx I



dx



Ω



A−1

x, b  d∗v

,−

I

sign

ω I

dη ∧ dx I



dx. 3.7

In other words,



Ω



I

ηω Idx

Ω



A−1

x, b  d∗v

,−

I

sign

ω I



dη ∧ dx I



dx. 3.8

By the elementary inequality

n



i1

a i2

1/2

≤n

i1

a i, 3.9

we have



Ωη |da|dx 



I

ω I2

1/2

dx≤



Ω



I

η |ω I |dx





Ω



A−1

x, b  d∗v

,−

I

sign

ω I

dη ∧ dx I



dx.

3.10

Using the inequality

3.11

3.10 becomes



Ωη |da|dx ≤



Ω

A−1

x, b  d∗v







I

sign

ω I

dη ∧ dx I







≤



Ω

A−1

x, b  d∗v

I

sign

ω I |dη|dx

 C l1

n



Ω

A−1

x, b  d∗v |dη|dx

 C l1

n



Ω|a  du||dη|dx,

3.12

where I takes over all l  1-tuples for dη ∈ Λ l1, thus it has C l1

n numbers at most Now we

let f  |da| and g  C l1

n |a  du||dη| In the subset {x : fη  g}, we have



{x:fηg} |ηda| p dx≤



In the subset{x : fη / g}, let h  |fη| p − |g| p /fη − g, then we easily obtain h > 0 So by

Lemma 3.1, we have



{x:fη / g} hfη dx≤



That is to say



that is,



{x:fη / g} |fη| p dx≤



fη /  g |g| p dx. 3.16 Combining3.13 and 3.16, we have



Ω|fη| p dx≤



that is,



Ω|η da| p dx≤



Ω

C l1

n a  dudηp

SoLemma 3.2is proved

The following lemma appears in2

Lemma 3.3 Suppose that 0 < q < p < s ≤ ∞, ξ ∈ R, and that B  Bx0, r  is a ball If a nonnegative

function v ∈ L p B, dμ satisfies

1

μ

λB



λB

v s dμ

1/s

≤ C1 − λ ξ 1

μ

B



B

v p dμ

1/p

3.19

for each ball B  Bx0, r  with r ≤ r and for all 0 < λ < 1, then



1

μ λB



λB

v s dμ

1/s

≤ C1 − λ ξ/θ

1

μ B



B

v q dμ

1/q

∀0 < λ < 1. 3.20

Here C > 0 is a constant depending on p, q, s and θ ∈ 0, 1 is such that 1/p  θ/q  1 − θ/s.

The following lemma appears in10

Lemma 3.4 Let u, v be a pair of solutions of the p-harmonic type system on domain Ω, then we

have a constant C only depending on K, n, p, and l, such that

η du p,Ω≤ Cu − cdη p,Ω ηa p,Ω

where c is any closed form (i.e., dc  0) and for any η ∈ C∞

0Ω Also we have a constant C only depending on K, n, q, such that

η d∗v

q,Ω≤ Cv − c

dη

q,Ω ηb q,Ω

where c is any coclosed form (i.e., d∗c  0) and q is the conjugate exponent of p.

Theorem 3.5 If u, v is a pair of solutions to the p-harmonic type system, then there exists a constant

C > 0 dependent on K, p, n, and l, such that



1

|Q|



Q

u − u Q   a ∞,Qs

dx

1/s

≤ C1− σ−1−tχ/pχ−1

diam Q  1 χ/ χ−1

×

 1

|σQ|



σQ

u − u σQ   a ∞,σQt

dx

1/t 3.23

for any 0 < s, t < ∞, σ > 1 and all cubes with Q ⊂ σQ ⊂ Ω, where χ > 1 is the Poincar´e constant.

Proof Suppose that the center of Q is x0and diam Q  r, 0 < λ  σ−1< 1 Let

r m  λ  1 − λ2 −m , m  0, 1, 2, 3.24

Then r m is decreasing and λ < r m < 1 So we have u Q|r m Q  u r m Q , for any m ∈ 0, 1, 2, Let η m ∈ C∞

0 r m Q  be a nonnegative function such that η m  1 in r m1Q, 0 ≤ η m ≤ 1 in

r m Q − r m1Q |dη m | ≤ 1 − λ−12m r−1 Given any t ≥ 0 and let ω m  |u − u Q |  a ∞,Q1t/pη m, then we have

du m



1 t

p

u − u

Q   a ∞,Qt/p

η m du − u Qu − u Q   a ∞,Q1t/p

dη m 3.25

By the Minkowski inequality, we can obtain



r m Q

du mp dx

1/p

≤



r m Q

u − u Q   a ∞,Qp tdη mp dx

1/p

p  t

p



r m Q

du − u Qpu − u Q   a ∞,Qtη mp

dx

1/p

.

3.26

We assume that u − u Q I a I dx I, then we have|u − u Q|  I a2I1/2 If u − u Qis zero, then

we have|d|u − u Q ||  0  |∇Tdu| If u − u Qis not equal zero, and the proof of2.15 implies

that|∇Tdu|  In

i1|∂a I /∂x i|21/2

du − u Q   ∇u − u Q ∂u − u Q

∂x1

, , ∂u − u Q

∂x n





 n

i1



∂u − u Q

∂x i



2

1/2

 n

i1



∂u − u Q

∂x i



2 1/2

 n

i1







∂I a2I1/2

∂x i







21/2

 n

i1

1



I a2

I









I

a I ∂a I

∂x i







21/2

≤ n

i1

1



I a2I



I

a2I

I



∂a I

∂x i

21/2

 n

i1



I



∂a I

∂x i

21/2

 n

i1



I



∂a I

∂x i



2

1/2

∇Tdu  ∇u − u Q.

3.27

So we have

du − u Q  ≤ ∇Tdu. 3.28

For any η ∈ C∞

0Ω, according to 2.6, we have

η∇T dω p,D≤ Cn, p max

x∈Dηdω p,D. 3.29

By the similar method asLemma 3.1, we can prove the following inequality:



r m Q

du − u Qpu − u Q   a ∞,Qtη mp

dx

1/p

≤



r m Q

η mp ∇Tdu pu − u Q   a ∞,Qt

dx

1/p

≤ Cn, pmax

x∈D



η p m

r Q

η mp |du| pu − u Q   a ∞,Qt

dx

1/p

3.30

3 The reverse H ăolder inequality< /b>

In this section, we will prove the reverse H ăolder inequality for the solution of the p-harmonic< /i>

type system Before we prove the reverse. .. coclosed form (i.e., d∗c  0) and q is the conjugate exponent of p.

Theorem 3.5 If u, v is a pair of solutions to the p-harmonic type system, then there exists...