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Hindawi Publishing CorporationJournal of Inequalities and Applications Volume 2009, Article ID 734528, 11 pages doi:10.1155/2009/734528 Research Article Some Caccioppoli Estimates for Di

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Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2009, Article ID 734528, 11 pages

doi:10.1155/2009/734528

Research Article

Some Caccioppoli Estimates for Differential Forms

Zhenhua Cao, Gejun Bao, Yuming Xing, and Ronglu Li

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

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

Received 31 March 2009; Accepted 26 June 2009

Recommended by Shusen Ding

We prove the global Caccioppoli estimate for the solution to the nonhomogeneous A-harmonic equation d∗A x, u, du Bx, u, du, which is the generalization of the quasilinear equation divAx, u, ∇u Bx, u, ∇u We will also give some examples to see that not all properties of

functions may be deduced to diﬀerential forms

Copyrightq 2009 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

The main work of this paper is study the properties of the solutions to the nonhomogeneous

A-harmonic equation for diﬀerential forms

When u is a 0-form, that is, u is a function,1.1 is equivalent to

In 1, Serrin gave some properties of 1.2 when the operator satisfies some conditions

In 2, chapter 3, Heinonen et al discussed the properties of the quasielliptic equations

− div Ax, ∇u 0 in the weighted Sobolev spaces, which is a particular form of 1.2

Recently, a large amount of work on the A-harmonic equation for diﬀerential forms has been done In 1992, Iwaniec introduced the p-harmonic tensors and the relations between

quasiregular mappings and the exterior algebraor differential forms in 3 In 1993, Iwaniec and Lutoborski discussed the Poincaré inequality for differential forms when 1 < p < n in

4, and the Poincar´e inequality for diﬀerential forms was generalized to p > 1 in 5 In 1999,

Nolder gave the reverse H ¨older inequality for the solution to the A-harmonic equation in6, and diﬀerent versions of the Caccioppoli estimates have been established in 7 9 In 2004,

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Ding proved the Caccioppli estimates for the solution to the nonhomogeneous A-harmonic equation d∗A x, du Bx, du in 10, where the operator B satisfies |Bx, ξ| ≤ |ξ| p−1 In

2004, D’Onofrio and Iwaniec introduced the p-harmonic type system in 11, which is an

important extension of the conjugate A-harmonic equation Lots of work on the solution to the p-harmonic type system have been done in5,12

As prior estimates, the Caccioppoli estimate, the weak reverse H ¨older inequality, and the Harnack inequality play important roles in PDEs In this paper, we will prove some Caccioppoli estimates for the solution to1.1, where the operators A : Ω × Λ l× Λl → Λl

and B :Ω × Λl× Λl → Λlsatisfy the following conditions on a bounded convex domainΩ:

|Ax, u, ξ| ≤ a|ξ| p−1

Ω|p−1

|Bx, u, ξ| ≤ cx|ξ| p−1 p−1

ξ, Ax, u, du ≥ |ξ| p − dx|u − uΩ| − gx

1.3

for almost every x

positive constant and bx through gx are measurable functions on Ω satisfying:

b, e ∈ L m Ω, c ∈ L n/ 1−ε , d, f, g ∈ L tΩ 1.4

with some 0 < ε ≤ 1, 1/m 1 − 1/p − p − 1/χp, 1/t 1 − ε/p − p − ε/χp, and χ is the

Poincar´e constant

Now we introduce some notations and operations about exterior forms Let

e1, e2, , e n denote the standard orthogonal basis ofRn For l 0, 1, , n, we denote the linear space of all l-vectors byΛl ΛlRn , spanned by the exterior product e I e i1 ∧ e i2 ∧

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

Grassmann algebraΛ ⊕Λl is 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

α, β

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 ,

α ∧ ∗β β ∧ ∗α α, β

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

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 which center is as same as Q, and

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

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Journal of Inequalities and Applications 3

α I of α is Schwartz distribution on Ω We denote the space spanned by diﬀerential l-form on

Ω 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 pΩ 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

1/p

⎛

⎝

Ω

I

|α I|2

p/2⎞

⎠

1/p

Similarly, W k,p Ω, Λ l denotes those l-forms on Ω which all coeﬃcients belong to W k,pΩ The following definition can be found in3, page 596

Definition 1.13 We denote the exterior derivative by

d : D

Ω, Λ l

−→ DΩ, Λ l

and its formal adjointthe Hodge co-diﬀerential is the operator

d∗: D

Ω, Λ l

−→ DΩ, Λ l−1

The operators d and d∗are given by the formulas

dα

I

By 3, Lemma 2.3, we know that a solution to 1.1 is an element of the Sobolev space

Wloc1,p Ω, Λ l−1 such that

Ω

for all ϕ ∈ W 1,p

0 Ω, Λ l−1 with compact support

Remark 1.2 In fact, the usual A-harmonic equation is the particular form of the equation1.1

when B 0 and A satisfies

We notice that the nonhomogeneous A-harmonic equation d∗A x, du Bx, du and the

p-harmonic type equation are special forms of1.1

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2 The Caccioppoli Estimate

In this section we will prove the global and the local Caccioppoli estimates for the solution

to 1.1 which satisfies 1.3 In the proof of the global Caccioppoli estimate, we need the following three lemmas

Lemma 2.1 1 Let α be a positive exponent, and let α i , β i , i 1, 2, , N, be two sets of N real

numbers such that 0 < α i < ∞ and 0 ≤ β i < α Suppose that z is a positive number satisfying the inequality

then

where C depends only on N, α, β i , and where γ i α − β i−1.

By the inequalities2.13 and 3.28 in 5, One has the following lemma

Lemma 2.2 5 Let Ω be a bounded convex domain in R n , then for any diﬀerential form u, one has

|d|u − uΩ|| ≤ Cn, p

Lemma 2.3 5 If f, g ≥ 0 and for any nonnegative η ∈ C∞

0 Ω, one has

Ωηf dx≤

then for any h ≥ 0, one has

Ωηfh dx≤

Theorem 2.4 Suppose that Ω is a bounded convex domain in R n , and u is a solution to1.1 which

satisfies1.3, and p > 1, then for any η ∈ C∞

0Ω, there exist constants C and k, such that

ηdu

p,Ω ≤ Cdiam Ωs p−1 u − u

Ωdη

p,Ω s p/ε−1η u − uΩ

p,Ω

diam Ωs p−1 dη

,

2.6

where s 1/p−1 1/p , and χ is the Poincar´e constant (i.e., χ 2 when p ≥ n, and χ np/n − p when 1 < p < n).

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Proof We assume that B x, u, du I ω I dx I For any nonnegative η ∈ C∞

0 Ω, we let

ϕ1 −I η sign ω I dx I , then we have dϕ1 −IsignωI dη ∧ dx I By using ϕ ϕ1 in the equation1.12, we can obtain

Ω

B x, u, du,

I

η sign ω I dx I

dx

Ω

A x, u, du, −

I

signωI dη ∧ dx I

dx, 2.7

that is,

Ω

I

η |ω I |dx

Ω

A x, u, du, −

I

By the elementary inequality

n

i1

a i2

1/2

≤n

i1

2.8 becomes

Ωη |Bx, u, du|dx

Ωη

I

ω2

I

1/2

dx

≤

Ωη

I

|ω I |dx

Ω

A x, u, du, −

I

dx

≤

Ω

A x, u, du, −

I

dx.

2.10

Using the inequality

then2.10 becomes

Ωη |Bx, u, du|dx ≤

Ω|Ax, u, du|

I

dx

≤

Ω|Ax, u, du| ·

I

signωI dη ∧ dx Idx

Ω|Ax, u, du|

I

dηdx.

2.12

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Since Bx, u, du ∈ Λ l−1,so we can deduce

Ωη |Bx, u, du|dx ≤ C l−1

n

Now we let ϕ2 −u − uΩη p , then dϕ2 −pη p−1dη ∧ u − uΩ − η p du We use ϕ ϕ2in1.12, then we can obtain

−

Ω

A x, u, du, pη p−1dη p du

dx−

Ω

B x, u, du, η p u

So we have

Ω

A x, u, du, η p du

dx −

Ω

A x, u, du, pη p−1dη ∧ u − uΩdx

−

Ω

B x, u, du, η p u

dx.

2.15

By1.3, 2.13, 2.15 andLemma 2.2, we have

0≤

Ωη p |du| p dx≤

Ω

A x, u, du, η p u

dx

Ω

|dx| |u − uΩ|p gdx

≤

Ω

A x, u, du, pη p−1dη ∧ u − uΩ

Ω

B x, u, du, η p u − uΩdx

Ωη p

≤

Ω|Ax, u, du|pη p−1dη |u − uΩ

Ω|Bx, u, du|η p |u − uΩ|dx

Ωη p

|dx| |u| p gdx

≤C l−1

n

Ω|Ax, u, du|η p−1dη |u − uΩ

Ωη p

≤ C1

Ωη p−1dη |u − uΩ||du| p−1dx

Ωη p−1dη |u − uΩ||bx| |u − uΩ|p−1

dx

Ωη p

|dx| |u − uΩ|p gdx,

2.16

where C1 C l−1

n

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We suppose that u − uΩ I u I dx I , k e 1/p−1 1/p and let u1 I u I

k sign u I dx I , then we have du1 du and

I

1/2

Combining2.16 and 2.17, we have

Ωη p |du1|p dx

≤ C1

Ωη p−1dη |u − uΩ| |du| p−1dx

Ωη p−1dη |u − uΩ||bx| |u − uΩ|p−1

dx

Ωη p

≤ C1

Ωη p−1dη |u1| |du1|p−1dx

Ωη p−1dη |u1| 1−p|e||u − uΩ|p−1 p−1

dx

Ωη p −pg |u − uΩ|p p

dx

≤ C2

Ωη p−1dη |u1| |du1|p−1dx

Ωη p−1dη |u1| 1−p|e||u − uΩ p−1dx

Ωη p −pg |u − uΩ p dx

≤ C2

Ωη p−1dη |u1||du1|p−1dx

Ωη p−1b1xdη |u1|p dx

Ωη p d1x|u1|p dx

,

2.18

where C2 C12p−1, b1 1−p|e| and d1 −p |g| By simple computations,

The terms on the right-hand side of the preceding inequality can be estimated by using the H ¨older inequality, Minkowski inequality, Poincar´e inequality andLemma 2.2 Thus

Ωη p−1dη |u1| |du1|p−1dx ≤ u1dηp,Ωηdu1p p,−1Ω, 2.19

Ωη p−1b1 xdη |u1|p dx

Ωη p−1b1 xdη |u1||u1|p−1dx

≤ b1x m,Ω

Ω

η p−1dη |u1||u1|p−1m/ m−1

dx

1−1/m

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≤ b1x m,Ωu1dηp,Ωu1ηp χp,−1Ω

≤ C3b1x m,Ωdiam Ωs p−1 u1dηp,Ωd|u1η|p−1

p,Ω

≤ C4diam Ωs p−1 u1dηp,Ωu1dηp,Ω 1|η p,Ωp−1

≤ C5diam Ωsp−1u1dηp,Ωu1dηp−1

p,Ω

p−1

p,Ω

C5diam Ωsp−1u1dηp,Ωu1dηp−1

p,Ω

C5diam Ωsp−1u1dηp

p,Ω 1dηp,Ωηdu1p−1

p,Ω

.

2.20

By the similar computation, we can obtain

Ωη p d1 x|u1|p dx

Ωη p d1 x|u1|p −ε |u1|ε dx

≤ C6diam Ωsp −ε u1ηε

p,Ω

u

1dηp −ε

p,Ω ηdu1p −ε

p,Ω

.

2.21

We insert the three previous estimates 2.19, 2.20 and 2.21 into the right-hand side of

2.15, and set

z ηdu1p,Ω

u1dηp,Ω, ζ

ηu1p,Ω

the result can be written

z p ≤ C2zp−1 5diam Ωsp−11 p−1

6diam Ωsp −ε ζ ε

1 p −ε

≤ C7

diam Ωsp−1

1 p−1

sp −ε ζ ε

ApplyingLemma 2.1and simplifying the result, we obtain

z ≤ C7

diam Ωsp−1

or in terms of the original quantities

ηdu1p,Ω ≤ C7

diam Ωsp−1

u1dηp,Ω sp/ε−1ηu1p,Ω. 2.25

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Journal of Inequalities and Applications 9 Combining2.17 and 2.25, we can obtain

ηdu p,Ω ≤ C7

diam Ωsp−1

s p/ε−1 ηu − uΩp,Ω diam Ωsp−1

dη p,Ω.

2.26

If 1 < p < n inTheorem 2.4, we can obtain the following

Corollary 2.5 Suppose that Ω is a bounded convex domain in R n , and u is a solution to1.1 which

satisfies1.3, and 1 < p < n, then for any η ∈ C∞

0 Ω, there exist constants C and k, such that

ηdu p,Ω≤ Cu − u

Ωdη

p,Ω η u − uΩ

p,Ω dη

p,Ω

where C Cn, p, l, a, b, d, ε and k e 1/p−1 1/p

When u is a 0-diﬀerential form, that is, u is a function, we have |d|u|| ≤ |du| Now we use u in place of u − uΩ in1.3, then 1.1 satisfying 1.3 is equivalent to 5 which satisfies

6 in 1, we can obtain the following result which is the improving result of 1, Theorem 2

Corollary 2.6 Let u be a solution to the equation div Ax, u, ∇u Bx, u, ∇u in a domain Ω.

For any 1 < p < n, one denotes χ n/n − p Suppose that the following conditions hold

1

ii |Bx, u, ξ| ≤ c|ξ| p−1 p−1

iii ξ · Ax, u, ξ ≥ |ξ| p − d|u| p − g,

where b ∈ L n/ p−1 ; c ∈ L n/ 1−ε ; d, f, g ∈ L n/ p−ε with for some ε ∈ 0, 1 Then for any σ > 1 and

any cubes or balls Q such that Q ⊂ σQ ⊂ Ω, one has

∇u p,Q ≤ Cr−1

u p,σQ n/p

where C and k are constants depending only on the above conditions and r is the diameter of Q One can write them

C Cp, n, σ, ε; a, b, d,

k e 1/p−1 1/p

2.29

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If we let η ∈ C∞

0 σQ and η is a bump function, then we have the following.

Corollary 2.7 Suppose that Ω is a bounded convex domain in R n , and u is a solution to1.1 which

satisfies1.3, and p > 1, then for any σ > 1 and any cubes or balls Q such that Q ⊂ σQ ⊂ Ω, there

exist constants C and k, such that

where C Cn, p, l, a, b, d, ε, diam Q, k e 1/p−1 1/p , and χ is the Poincar´e constant.

3 Some Examples

Example 3.1 The Sobolev inequality cannot be deduced to di ﬀerential forms For any η ∈

C∞0B, we only let

u

B

∂η

∂y dx

dy

B

∂η

∂z dx

then u ∈ C∞

0 B, Λ1, and

du

∂η

∂x dx

∂η

∂y dy

∂η

∂z dz

∧ dx

∂η

∂y dx

B

∂2η

∂y2dx

dy

B

∂2η

∂y∂z dx

dz

∧ dy

∂η

∂z dx

B

∂2η

∂y∂z dx

dy

B

∂2η

∂z2dx

dz

∧ dz

0.

3.2

So we cannot obtain

1

|B|

B

|u| pχ dx

1/pχ

≤ Cdiam B

1

|B|

B

|du| p dx

1/p

Example 3.2 The Poincar´e inequality can be deduced to diﬀerential forms We can see the following lemma

Lemma 3.3 5 Let u ∈ D D, Λ l , and du ∈ L p D, Λ l , then u − uDis in L χp D, Λ l and

1

|D|

D|u − uD|pχ dx

1/pχ

≤ Cn, p, l

diamD

1

|D|

D|du| p dx

1/p

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

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Acknowledgment

This work is supported by the NSF of Chinano.10771044 and no.10671046

References

1 J Serrin, “Local behavior of solutions of quasi-linear equations,” Acta Mathematica, vol 111, no 1, pp.

247–302, 1964

2 J Heinonen, T Kilpel¨a`ınen, and O Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,

Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1993

3 T Iwaniec, “p-harmonic tensors and quasiregular mappings,” Annals of Mathematics, vol 136, no 3,

pp 589–624, 1992

4 T Iwaniec and A Lutoborski, “Integral estimates for null Lagrangians,” Archive for Rational Mechanics and Analysis, vol 125, no 1, pp 25–79, 1993.

5 Z Cao, G Bao, R Li, and H Zhu, “The reverse H¨older inequality for the solution to p-harmonic type system,” Journal of Inequalities and Applications, vol 2008, Article ID 397340, 15 pages, 2008.

6 C A Nolder, “Hardy-Littlewood theorems for A-harmonic tensors,” Illinois Journal of Mathematics,

vol 43, no 4, pp 613–632, 1999

7 G Bao, “A r λ-weighted integral inequalities for A-harmonic tensors,” Journal of Mathematical Analysis and Applications, vol 247, no 2, pp 466–477, 2000.

8 S Ding, “Weighted Caccioppoli-type estimates and weak reverse H¨older inequalities for A-harmonic tensors,” Proceedings of the American Mathematical Society, vol 127, no 9, pp 2657–2664, 1999.

9 X Yuming, “Weighted integral inequalities for solutions of the A-harmonic equation,” Journal of Mathematical Analysis and Applications, vol 279, no 1, pp 350–363, 2003.

10 S Ding, “Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds,” Proceedings of the American Mathematical Society, vol 132, no.

8, pp 2367–2375, 2004

11 L D’Onofrio and T Iwaniec, “The p-harmonic transform beyond its natural domain of definition,” Indiana University Mathematics Journal, vol 53, no 3, pp 683–718, 2004.

12 G Bao, Z Cao, and R Li, “The Caccioppoli estimate for the solution to the p-harmonic type system,”

in Proceedings of the 6th International Conference on Di ﬀerential Equations and Dynaminal Systems (DCDIS

’09), pp 63–67, 2009.

p-harmonic type equation are special forms of1.1

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2 The Caccioppoli Estimate

In... np/n − p when < p < n).

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Proof...

dηdx.

2.12

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Since Bx, u, du ∈ Λ l−1,so

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