Some Caccioppoli Estimates for Differential Forms

Journal of Inequalities and Applications Volume 2009, Article ID 734528,11pages 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,[email protected]

Received 31 March 2009; Accepted 26 June 2009

Recommended by Shusen Ding

We prove the global Caccioppoli estimate for the solution to the nonhomogeneousA-harmonic equation d∗Ax, 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.

Copyrightq2009 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

d∗ Ax, u, du Bx, u, du. 1.1

Whenuis a 0-form, that is,uis a function,1.1is equivalent to

divAx, u,∇u Bx, u,∇u. 1.2

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

−divAx,∇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 thep-harmonic tensors and the relations between quasiregular mappings and the exterior algebraor differential formsin3. In 1993, Iwaniec and Lutoborski discussed the Poincar´e inequality for differential forms when 1 < p < nin

Ding proved the Caccioppli estimates for the solution to the nonhomogeneousA-harmonic equationd∗Ax, du Bx, du in10, where the operatorB satisfies|Bx, ξ| ≤ |ξ|p−1. In 2004, D’Onofrio and Iwaniec introduced thep-harmonic type system in11, which is an important extension of the conjugateA-harmonic equation. Lots of work on the solution to thep-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 operatorsA:Ω×Λl×Λl1 → Λl1 andB:Ω×Λl_×Λl1 _{→ Λ}l_{satisfy the following conditions on a bounded convex domainΩ:}

|Ax, u, ξ| ≤a|ξ|p−1bx|u−u_Ω|p−1ex,

|Bx, u, ξ| ≤cx|ξ|p−1dx|u|p−1fx, ξ, Ax, u, du ≥ |ξ|p−dx|u−u_Ω| −gx

1.3

for almost everyx∈Ω, alll-differential formsuandl1-differential formsξ. Whereais a positive constant andbxthroughgxare measurable functions onΩsatisfying:

b, e∈LmΩ, c∈Ln/1−ε, d, f, g∈LtΩ 1.4

with some 0 < ε ≤ 1, 1/m 1−1/p−p−1/χp,1/t 1−ε/p−p−ε/χp,andχis the Poincar´e constant.

Now we introduce some notations and operations about exterior forms. Let

e1, e2, . . . , en denote the standard orthogonal basis ofRn. Forl 0,1, . . . , n, we denote the

linear space of alll-vectors byΛl _Λl_Rn_{, spanned by the exterior producte}

I ei1∧ei2 ∧

· · · ∧eil, corresponding to all orderedl-tuplesI i1, i2, . . . , il, 1≤i1 < i2 <· · · < il ≤n. The

Grassmann algebraΛ ⊕Λl _{is a graded algebra with respect to the exterior products. For}

ααIeI∈ΛandββIeI ∈Λ, then its inner product is obtained by

α, βαIβI 1.5

with the summation over allI i1, i2, . . . , iland all integersl 0,1, . . . , n. The Hodge star

operator∗:Λ → Λis defined by the rule

∗1ei1∧ei2∧ · · · ∧ein,

α∧ ∗ββ∧ ∗αα, β∗1 1.6

for allα, β∈Λ. Hence the norm ofα∈Λcan be given by

|α|2_{α, α∗α∧ ∗α∈Λ}

0R. 1.7

Throughout this paper, Ω ⊂ Rn _{is an open subset. For any constant σ > 1, Q denotes a}

αIofαis Schwartz distribution onΩ. We denote the space spanned by differentiall-form on ΩbyDΩ,Λl_{. We writeL}p_Ω,Λl_{for thel-formαα}

IdxI onΩwithαI ∈LpΩfor all

orderedl-tupleI. ThusLp_Ω,Λl_{is a Banach space with the norm}

αp,Ω

Ω|α|

p

1/p

⎛

⎝

Ω

I

|αI|2

p/2⎞

⎠

1/p

. 1.8

Similarly,Wk,pΩ,Λldenotes thosel-forms onΩwhich all coefficients belong toWk,pΩ. The following definition can be found in3, page 596.

Definition 1.13. We denote the exterior derivative by

d:DΩ,Λl−→DΩ,Λl1, 1.9

and its formal adjointthe Hodge co-differentialis the operator

d∗:DΩ,Λl−→DΩ,Λl−1. 1.10

The operatorsdandd∗are given by the formulas

dα

I

dαI∧dxI, d∗ −1nl1∗d∗. 1.11

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

W_loc1,pΩ,Λl−1such that

Ω

Ax, u, du, dϕBx, u, du, ϕ≡0 1.12

for allϕ∈W₀1,pΩ,Λl−1_{with compact support.}

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

whenB0 andAsatisfies

|Ax, ξ| ≤K|ξ|p−1, Ax, ξ, ξ ≥ |ξ|p. 1.13

We notice that the nonhomogeneousA-harmonic equationd∗Ax, du Bx, duand the

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 2.11. Letαbe a positive exponent, and letαi,βi,i1,2, . . . , N, be two sets ofNreal

numbers such that0 < αi < ∞and 0 ≤ βi < α. Suppose thatzis a positive number satisfying the

inequality

zα≤αizβi 2.1

then

z≤Cαiγi_{, 2.2}

whereCdepends only onN, α, βi,and whereγi α−βi−1.

By the inequalities2.13and3.28in5, One has the following lemma.

Lemma 2.25. LetΩbe a bounded convex domain inRn_{, then for any differential formu, one has}

|d|u−uΩ|| ≤Cn, p|du|. 2.3

Lemma 2.35. Iff, g≥0and for any nonnegativeη∈C∞₀ Ω, one has

Ωηf dx≤

Ωg dx, 2.4

then for anyh≥0, one has

Ωηfh dx≤

Ωgh dx. 2.5

Theorem 2.4. Suppose thatΩis a bounded convex domain inRn_{, anduis a solution to1.1which}

satisfies1.3, andp >1, then for anyη∈C∞₀Ω, there exist constantsCandk, such that

ηdu_p,Ω ≤CdiamΩsp−11u−u_Ωdη_p,Ω diamΩsp/ε−1ηu−u_Ωp,Ω

kdiamΩsp−11dη_p,ΩkdiamΩsp/ε,

2.6

Proof. We assume that Bx, u, du _IωIdxI. For any nonnegative η ∈ C∞₀ Ω, we let

ϕ1 −IηsignωIdxI, then we have dϕ1 −IsignωIdη ∧dxI. By usingϕ ϕ1 in

the equation1.12, we can obtain

Ω

Bx, u, du,

I

ηsignωIdxI

dx

Ω

Ax, u, du,−

I

signωIdη∧dxI

dx, 2.7

that is,

Ω

I

η|ωI|dx

Ω

Ax, u, du,−

I

signωIdη∧dxI

dx. 2.8

By the elementary inequality

_n

i1

ai2

1/2

≤n

i1

|ai|, 2.9

2.8becomes

Ωη|Bx, u, du|dx Ωη I ω2 I

1/2

dx ≤ Ωη I

|ωI|dx

Ω

Ax, u, du,−

I

signωIdη∧dxI

dx ≤ Ω

Ax, u, du,−

I

signωIdη∧dxI

dx.

2.10

Using the inequality

|a, b| ≤ |a| |b|, 2.11

then2.10becomes

Ωη|Bx, u, du|dx≤

Ω|Ax, u, du|

I

signωIdη∧dxI

dx

≤

Ω|Ax, u, du| ·

I

signωIdη∧dxIdx

Ω|Ax, u, du|

I

dηdx.

SinceBx, u, du∈Λl−1_{,so we can deduce}

Ωη|Bx, u, du|dx≤C

l−1

n

Ω|Ax, u, du|

dηdx. 2.13

Now we letϕ2−u−uΩηp, thendϕ2−pηp−1dη∧u−uΩ−ηpdu. We useϕϕ2in1.12,

then we can obtain

−

Ω

Ax, u, du, pηp−1dη∧uηpdudx−

Ω

Bx, u, du, ηpudx≡0. 2.14

So we have

Ω

Ax, u, du, ηpdudx−

Ω

Ax, u, du, pηp−1dη∧u−uΩ dx − Ω

Bx, u, du, ηpudx.

2.15

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

0≤

Ωη

p_|du|p_dx≤

Ω

Ax, u, du, ηpudx

Ω

|dx| |u−uΩ|pgdx

≤

Ω

Ax, u, du, pηp−1dη∧u−u_Ωdx

Ω

Bx, u, du, ηpu−u_Ωdx

Ωη

p_{|dx| |u−u}

Ω|pgdx

≤

Ω|Ax, u, du|pη

p−1_dη|u−u

Ω|dx

Ω|Bx, u, du|η

p_|u−u

Ω|dx

Ωη

p_{|dx| |u|}p_gdx

≤Cl−1

n p

Ω|Ax, u, du|η

p−1_dη|u−u

Ω|dx

Ωη

p_{|dx| |u−u}

Ω|pgdx

≤C1

Ωη

p−1_dη|u−u

Ω||du|p−1dx

Ωη

p−1_dη|u−u

Ω|

|bx| |u−uΩ|p−1|e|

dx

Ωη

p_{|dx| |u−u}

Ω|pgdx

,

2.16

We suppose thatu−uΩ IuIdxI,k e1/p−1 g1/p and let u1 IuI

ksignuIdxI,then we havedu1duand

|u−u_Ω|k≤ |u1|

I

|uI|k2

1/2

≤ |u−u_Ω|Cl_n−1k. 2.17

Combining2.16and2.17, we have

Ωη

p_|du

1|pdx

≤C1

Ωη

p−1_dη|u−u

Ω| |du|p−1dx

Ωη

p−1_dη|u−u

Ω|

|bx| |u−u_Ω|p−1|e|dx

Ωη

p_{|dx| |u−u}

Ω|pgdx

≤C1

Ωη

p−1_dη|u

1| |du1|p−1dx

Ωη

p−1_dη|u 1|

|bx|k1−p|e||u−uΩ|p−1kp−1

dx

Ωη

p_|dx|k−p_g|u−u

Ω|pkpdx

≤C2

Ωη

p−1_dη|u

1| |du1|p−1dx

Ωη

p−1_dη|u 1|

|bx|k1−p|e||u−uΩ|kp−1dx

Ωη

p_|dx|k−p_g|u−u

Ω|kpdx

≤C2

Ωη

p−1_dη|u

1||du1|p−1dx

Ωη

p−1_b

1xdη|u1|pdx

Ωη

p_d

1x|u1|pdx

,

2.18

whereC2C12p−1,b1x |bx|k1−p|e|andd1x |dx|k−p|g|.By simple computations,

we getb1x ≤ bx1 andd1x ≤ dx1.

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−1_dη|u

1| |du1|p−1dx≤ u1dη_p,Ωηdu1p_p,−_Ω1, 2.19

Ωη

p−1_b

1xdη|u1|pdx

Ωη

p−1_b

1xdη|u1||u1|p−1dx

≤ b1xm,Ω

Ω

ηp−1dη|u1||u1|p−1

m/m−1

dx

≤ b1xm,Ωu1dη_p,Ωu1ηp_χp,−1_Ω

≤C3b1xm,ΩdiamΩsp−1u1dηp,Ωd|u1η|p_p,−_Ω1

≤C4diamΩsp−1u1dη_p,Ω

u1dη_p,Ωd|u1|η_p,Ω

p−1

≤C5diamΩsp−1u1dη_p,Ω

u1dηp_p,−_Ω1η|du|p_p,−_Ω1

C5diamΩsp−1u1dη_p,Ω

u1dηp_p,−_Ω1η|du1|p_p,−_Ω1

C5diamΩsp−1

u1dηp_p,Ωu1dη_p,Ωηdu1p_p,−_Ω1

.

2.20

By the similar computation, we can obtain

Ωη

p_d

1x|u1|pdx

Ωη

p_d

1x|u1|p−ε|u1|εdx

≤C6diamΩsp−εu1ηε_p,Ω

u1dηp_p,−_Ωεηdu1p_p,−_Ωε

.

2.21

We insert the three previous estimates 2.19,2.20 and2.21 into the right-hand side of

2.15, and set

z ηdu1p,Ω u1dη_p,Ω

, ζ ηu1p,Ω u1dη_p,Ω

, 2.22

the result can be written

zp≤C2zp−1C5diamΩsp−1

1zp−1C6diamΩsp−εζε

1zp−ε

≤C7

diamΩsp−1 11zp−1 diamΩsp−εζε1zp−ε

. 2.23

ApplyingLemma 2.1and simplifying the result, we obtain

z≤C7

diamΩsp−1 1 diamΩsp/ε−1ζ, 2.24

or in terms of the original quantities

ηdu1p,Ω ≤C7

diamΩsp−1 1u1dηp,Ω diamΩsp/ε−1ηu1p,Ω

Combining2.17and2.25, we can obtain

ηdu_p,Ω ≤C7

diamΩsp−1 1u−uΩdηp,ΩkdiamΩsp/ε

diamΩsp/ε−1ηu−uΩp,Ωk

diamΩsp−1 1dη_p,Ω.

2.26

If 1< p < ninTheorem 2.4, we can obtain the following.

Corollary 2.5. Suppose thatΩis a bounded convex domain inRn_{, anduis a solution to1.1which}

satisfies1.3, and1< p < n, then for anyη∈C∞₀ Ω, there exist constantsCandk, such that

ηdu_p,Ω≤Cu−uΩdηp,Ωηu−uΩp,Ωkdηp,Ωk|Ω|

, 2.27

whereCCn, p, l, a,b,d, εandke1/p−1g1/p.

Whenuis a 0-differential form, that is,uis a function, we have|d|u|| ≤ |du|. Now we useuin place ofu−uΩ in1.3, then1.1satisfying1.3is equivalent to5which satisfies

6in1, we can obtain the following result which is the improving result of1, Theorem 2.

Corollary 2.6. Letube a solution to the equationdivAx, u,∇u Bx, u,∇uin a domainΩ.

For any1< p < n, one denotesχn/n−p. Suppose that the following conditions hold

i|Ax, u, ξ| ≤a|ξ|p−1b|u|p−1e, wherea >0is a constant,b, e∈Lq_{such that 2003p−}

1/pχ1/p1/q1;

ii|Bx, u, ξ| ≤c|ξ|p−1d|u|p−1f,

iiiξ·Ax, u, ξ≥ |ξ|p−d|u|p−g,

whereb∈Ln/p−1_;c∈Ln/1−ε_{;d, f, g ∈L}n/p−ε_{with for someε∈0,1.Then for anyσ > 1and}

any cubes or ballsQsuch thatQ⊂σQ⊂Ω, one has

∇u_p,Q≤Cr−11u_p,σQkrn/p, 2.28

whereCandkare constants depending only on the above conditions andr is the diameter ofQ. One can write them

CCp, n, σ, ε;a,b,d,

ke1/p−1 g1/p.

If we letη∈C₀∞σQandηis a bump function, then we have the following.

Corollary 2.7. Suppose thatΩis a bounded convex domain inRn_{, anduis a solution to1.1which}

satisfies1.3, andp >1, then for anyσ >1and any cubes or ballsQsuch thatQ⊂σQ⊂Ω, there exist constantsCandk, such that

dup,Q≤C

u−uσQ_p,σQk

, 2.30

whereCCn, p, l, a,b,d, ε,diamQ,ke1/p−1g1/p, andχis the Poincar´e constant.

3. Some Examples

Example 3.1. The Sobolev inequality cannot be deduced to differential forms. For anyη ∈ C∞₀B,we only let

uηdx

B ∂η ∂ydx dy B ∂η ∂zdx

dz, 3.1

thenu∈C∞₀ B,Λ1_,and

du ∂η ∂xdx ∂η ∂ydy ∂η ∂zdz ∧dx ∂η ∂ydx B

∂2_η

∂y2dx

dy

B

∂2_η

∂y∂zdx dz ∧dy ∂η ∂zdx B

∂2_η

∂y∂zdx

dy

B

∂2_η

∂z2dx

dz ∧dz 0. 3.2

So we cannot obtain

1

|B|

B

|u|pχ_dx1/pχ_≤CdiamB 1

|B|

B

|du|p_dx1/p_{. 3.3}

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

Lemma 3.35. Letu∈DD,Λl_{, anddu∈L}p_D,Λl1_{, thenu−u}

Dis inLχpD,Λland

1

|D|

D|u−uD|

pχ_dx1/pχ_{≤Cn, p, ldiamD} 1

|D|

D|du|

p_dx1/p_{, 3.4}

Acknowledgment

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

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