Global Caccioppoli-Type and Poincar ´e Inequalities with Orlicz Norms

Volume 2010, Article ID 727954,27pages doi:10.1155/2010/727954

Research Article

Global Caccioppoli-Type and Poincar ´e Inequalities with Orlicz Norms

Ravi P. Agarwal

and Shusen Ding

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

2Department of Mathematics, Seattle University, Seattle, WA 98122, USA

Correspondence should be addressed to Shusen Ding,sding@seattleu.edu Received 16 July 2009; Revised 1 December 2009; Accepted 14 March 2010 Academic Editor: Jozsef Szabados

Copyrightq2010 R. P. Agarwal and S. Ding. 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.

We obtain global weighted Caccioppoli-type and Poincar´e inequalities in terms of Orlicz norms for solutions to the nonhomogeneousA-harmonic equationdAx, dω Bx, dω.

1. Introduction

The L^p-theory of solutions of the homogeneousA-harmonic equation dAx, dω 0 for differential forms has been very well developed in recent years. Many L^p-norm estimates and inequalities, including the Hardy-Littlewood inequalities, Poincar´e inequalities, Caccioppoli-type estimates, and Sobolev imbedding inequalities, for solutions of the homogeneousA-harmonic equation have been established; see1–11. Among these results, the Caccioppoli-type inequalities and the Poincar´e inequalities for differential forms have become more and more important tools in analysis and related fields, including partial differential equations and potential theory. However, the study of the nonhomogeneous A-harmonic equationdAx, dω Bx, dωjust began4,6. Roughly, the Caccioppoli-type inequalities or estimates provide upper bounds for the norms of∇uordu in terms of the corresponding norm ofuoru−c, whereuis a differential form or a function satisfying certain conditions. For example,umay be a solution of anA-harmonic equation or a minimizer of a functional, andcis some constant ifuis a function or a closed form ifuis a differential form.

Different versions of the Caccioppoli-type inequalities and the Poincar´e inequalities have been established during the past several decades. For instance, Sbordone proved in12the following version of the Caccioppoli-type inequality:

−

BR/2

A|du|dx≤C−

BR

A

|u−u_R| R

dx 1.1

for a quasiminimizeruof the functionalFΩ;v

ΩA|dv|dx, whereAis a continuous, convex, and strictly increasing function satisfying the so-calledΔ2-condition,B_Ris a ball with radiusR >0, andu_R⁻

BRu dx; see12. Using the above Caccioppoli-type inequality, Fusco and Sbordone obtained in13the higher integrability result

−

BR/2

A^r|du|dx≤C

−

BR

A|du|dx _r

1.2

for the gradient of minimizers of the functional IΩ;v, where r > 1 is some constant. In 14, Greco et al. studied the variational integrals whose integrand grows almost linearly with respect to the gradient and the related equation divAx, f∇u 0, whereAis slowly increasing to∞. For instance,At log^α1t, α >0, orAt log loget. They proved that the minimizerusubject to the Dirichlet datavsatisfies the estimate

Ω|∇u|A^1±ε|∇u|dx≤C

Ω|∇v|A^1±ε|∇v|dx 1.3

at least for some smallε >0. In15, Cianchi and Fusco investigated the higher integrability properties of the gradient of local minimizers of an integral functional of the formJu,Ω

Ωfx, u, dudx, whereΩ is an open subset ofRⁿ,n ≥ 2, andf is a Carathodory function defined in Ω × R^N × R^nNsatisfying some growth conditions. Using a new form of the Caccioppoli inequality and some other tools, such as the Sobolev inequality and a generalized version of the Gehring lemma, they proved that if u is a local minimizer of Ju,Ω, for Ω0⊂⊂Ωthere existsδ >0 such that

Ω0

A|du|

A|du|

|du|

_δ

dx <∞, 1.4

whereAsatisfies the so-calledΔ2-condition. However, all versions of the Caccioppoli-type inequality developed or used in12–15are about the minimizeruof some functional. In this paper, we will prove the Caccioppoli-type inequalities and the Poincar´e inequalities with the L^slogL^α-norm for differential forms satisfying the nonhomogeneousA-harmonic equation.

The method developed in this paper could be used to establish other L^slogL^α-norm inequalities for solutions of the homogeneousA-harmonic equation or the nonhomogeneous A-harmonic equation.

Throughout this paper, we always assume thatΩis an open subset ofRⁿ,n≥2. Then- dimensional Lebesgue measure of a setE⊆Rⁿis denoted by|E|. We say thatwis a weight if w∈L¹_locRⁿandw >0 a.e. For 0< p <∞, we denote the weightedL^p-norm of a measurable functionfoverEby f _p,E,wα

E|fx|^pw^αxdx^1/p,whereαis a real number. We write f _p,E f _p,E,wα ifw 1. A continuously increasing functionϕ : 0,∞ → 0,∞with ϕ0 0 andϕ∞ ∞is called an Orlicz function. The Orlicz spaceL^ϕΩconsists of all measurable functionsfonΩsuch that

Ωϕf k

dx <∞ 1.5

for somekkf>0.L^ϕΩis equipped with the nonlinear Luxemburg functional f

ϕ inf

k >0 : 1

|Ω|

Ωϕf k

dx≤1

. 1.6

A convex Orlicz functionϕis often called a Young function. Ifϕis a Young function, then · ϕ defines a norm inL^ϕΩ, which is called the Luxemburg norm or Orlicz norm. The Orlicz spaceL^ψΩ with ψt t^plog^αet/c will be denoted byL^plogL^αΩ and the corresponding norm will be denoted by f L^plogL^αΩ, where 1 ≤ p < ∞,α ≥ 0, andc > 0 are constants. The spacesL^plogL⁰ΩandL¹logL¹Ωare usually referred asL^pΩand LlogLΩ, respectively. From16, we have the equivalence

f

L^plogL^αΩ≈

Ω

f^plog^α

e f f

p,Ω

dx

1/p

. 1.7

Similarly, we have f

L^plogL^αΩ,μ≈

Ω

f^plog^α

e f f

p,Ω

dμ

_1/p

, 1.8

whereμis a measure defined bydμwxdxandwxis a weight. In this paper, we simply write

f

L^plogL^αE,w^α

E

|f|^plog^α

e f f

p,E

w^αdx

1/p

, 1.9

and f L^plogL^αE f L^plogL^αE,1, wherewis a weight.

We keep using the traditional notations related to differential forms in this paper. Let Λ Λ Rⁿbe the linear space of the -covectors onRⁿ, 1,2, . . . , n. It is a normed space of dimension_n

. A differential -formωonΩis a Schwartz distribution onΩwith values inΛ Rⁿ. We writeDΩ,Λ for the space of all differential -forms andL^pΩ,Λ for all -formsωx

IωIxdxI

ωi1i2···i xdxi1∧dxi2∧ · · · ∧dxi withωI ∈L^pΩ,Rfor all ordered -tuplesI. Thus,L^pΩ,Λ is a Banach space with norm

ω _p,Ω

Ω|ωx|^pdx 1/p

⎛

⎝

Ω

I

|ωIx|² p/2

dx

⎞

⎠

1/p

. 1.10

We useL^plogL^αΩ,Λ to denote the space of all differential -formsuonΩwith

u _L^p_logL^α_Ω

Ω|u|^plog^α

e |u|

u _p,Ω

dx 1/p

<∞. 1.11

We use d : DΩ,Λ → DΩ,Λ ¹ to denote the differential operator and d : DΩ,Λ ¹ → DΩ,Λ to denote the Hodge codifferential operator given byd −1^nl1 donDΩ,∧^l1, 0,1, . . . , n. Hereis the well-known Hodge star operator. We useBto denote a ball andσB,σ >0, is the ball with the same center asBand with diameterσdiamB.

A differential formuis called closed ifdu 0 and a differential formvis called coclosed if dv0.

Definition 1.1. LetA:Ω×Λ Rⁿ → Λ RⁿandB:Ω×Λ Rⁿ → Λ ⁻¹Rⁿbe two operators satisfying the conditions:

|Ax, ξ| ≤a|ξ|^p−1, Ax, ξ·ξ≥ |ξ|^p, |Bx, ξ| ≤b|ξ|^p−1 1.12

for almost everyx∈Ωand allξ∈Λ Rⁿ. Then the nonlinear elliptic equation

dAx, dω Bx, dω 1.13

is called the nonhomogeneousA-harmonic equation for differential forms. Herea, b >0 are constants and 1< p <∞is a fixed exponent associated with1.13.

We should notice that if the operatorBequals 0 in1.13, then1.13reduces to the following homogeneousA-harmonic equation, or theA-harmonic equation:

dAx, dω 0, 1.14

which has received much investigation during the recent years; see3,5,7–11. A solution to 1.13is an element of the Sobolev spaceW_loc^1,pΩ,Λ ⁻¹such that

ΩAx, dω·dϕBx, dω·ϕ0 1.15

for allϕ ∈ W_loc^1,pΩ,Λ ⁻¹with compact support. The solutions of theA-harmonic equation are calledA-harmonic tensors. For any differential formωdefined in a bounded and convex domainD, there is a decomposition

ωdTω Tdω. 1.16

Using the operatorT, we can define thel-formω_D∈DD,Λ by

ω_D|D|⁻¹

D

ω y

dy, 0, andω_DdTω, 1,2, . . . , n, 1.17

for allω∈L^pD,Λ , 1≤p <∞. It is known thatuDis a closed form. Hence,u−uDis still a solution of1.13wheneveruis a solution of1.13.

2. Preliminaries

The purpose of this section is to establish some preliminary results that will be used in the proof of our main theorems. In 6, the weighted Poincar´e inequality for solutions of the nonhomogeneousA-harmonic equation was established. From 7, we have the following local Poincar´e inequality.

Lemma 2.1. Letu ∈DΩ,∧ be a differential form in a domainΩ ⊂ Rⁿanddu ∈ L^sΩ,∧ ¹, 0,1, . . . , n. Assume that 1< s <∞. Then

u−u_{B s,B}≤C|B|^1/n du _s,σB 2.1

for all ballsBwithσB⊂Ω. HereCis a constant independent ofuandσ >1 is some constant.

From7, we have the following local Caccioppoli-type inequality.

Lemma 2.2. Letu ∈ DΩ,Λ^l,l 0,1, . . . , n, be a solution of the nonhomogeneousA-harmonic equation1.13in a domainΩ⊂Rⁿand letρ >1 be some constant. Assume that 1< s <∞is a fixed exponent associated with theA-harmonic equation1.13. Then there exists a constantC, independent ofu, such that

du _s,B ≤C|B|^−1/n u−c _s,ρB 2.2

for all ballsBwithρB⊂Ωand all closed formsc.

The following weak reverse H ¨older inequality appears in7.

Lemma 2.3. Letube a solution of 1.13inΩand 0< s,t < ∞. Then there exists a constantC, independent ofu, such that

u _s,B≤C|B|^t−s/st u _t,σB 2.3

for all balls or cubesBwithσB⊂Ωfor someσ >1.

Now, we prove the following local Orlicz norm estimates.

Proposition 2.4. Letube a solution of 1.13inΩ,α >0, σ >1, and 1< p <∞. Then there exists a constantC, independent ofu, such that

uB L^plogL^αB≤C u _L^p_logL^α_σB, 2.4 u−uB L^plogL^αB ≤C u−c _LplogL^ασB 2.5 for all ballsBwithσB⊂Ωand diamB≥d₀. Hered₀ >0 is a constant andcis any closed form.

Proof. LetB⊂Ωbe a ball with diamB≥d0 >0. Chooseε >0 small enough and a constant Mlarge enough such that|B|^−ε/p2 ≤M. FromLemma 2.3, we have

uB pε,B≤C₁|B|^p−pε/ppε uB p,σB 2.6 for some σ > 1. Similar to 3.4 in the proof of Theorem 3.1, we may assume that

|uB|/ uB p,B≥1 onB. For aboveε >0, there existsC2>0 such that

log^α

e |uB| uB p,B

≤C2

|uB| uB p,σB

ε

. 2.7

From2.6and2.7, it follows that

uB L^plogL^αB

B

|uB|^plog^α

e |uB| uB p,B

dx

_1/p

≤C3

1 uB ε

p,σB

B

|uB|^pεdx _1/p

≤ C3

uB ε/p p,σB

B

|uB|^pεdx

_1/pεpε/p

≤ C₄ uB ε/p

p,σB

|B|^p−pε/ppε uB p,σB

_pε/p

≤C5|B|^−ε/p2 uB p,σB.

2.8

From17, we know that

uB p,σB≤C₆ u _p,σB. 2.9

Putting2.9into2.8and noting that

log^α

e |u|

u _p,σB

≥1 2.10

forα >0, we obtain

uB L^plogL^αB≤C₇ u _p,σB≤C₈ u _L^p_logL^α_σB. 2.11 This ends the proof of inequality2.4. Ifcis a closed differential form, from1.16and1.17, we find that

cdTc Tdc dTc cB. 2.12

Applying triangle inequality and2.4, we conclude that

u−u_{B L}plogL^αB u−c−uB−c_{B L}plogL^αB

u−c−u−c_{B L}^p_logL^α_B

≤ u−c _L^p_logL^α_B u−c_{B L}plogL^αB

≤ u−c _L^p_logL^α_BC₉ u−c _L^p_logL^α_σB

≤C10 u−c _L^p_logL^α_σB

2.13

for any closed formc. The proof ofProposition 2.4has been completed.

Next, extend the weak reverse H ¨older inequality above to the case of Orlicz norms.

Lemma 2.5. Letube a solution of 1.13inΩ,σ >1, and 0< s, t <∞. Then there exists a constant C, independent ofu, such that

u _LslogL^αB≤C|B|^t−s/st u _LtlogL^βσB 2.14

for any constantsα >0 andβ >0, and all ballsBwithσB⊂Ωand diamB≥d₀>0, whered₀is a fixed constant.

The proof of Lemma 2.5is similar to that of Proposition 2.4. For completeness, we proveLemma 2.5as follows.

Proof. For any ballB ⊂Ωwith diamB≥d₀ >0, we may chooseε >0 small enough and a constantC1such that

|B|^−ε/st≤C₁. 2.15

FromLemma 2.3, we have

u _sε,B≤C₂|B|^t−sε/tsε u _t,σB 2.16

for someσ >1. Similar to3.5in the proof ofTheorem 3.1, we may assume that|u|/ u t,B≥1 onB. For aboveε >0, there existsC₃>0 such that

log^α

e |u|

u _s,B

≤C3

|u|

u _t,σB ε

. 2.17

From2.16and2.17, we have

u _LslogL^αB

B

|u|^slog^α

e |u|

u _s,B

dx _1/s

≤C4

1 u ^ε_t,σB

B

|u|^sεdx _1/s

≤ C₄ u ^ε/s_t,σB

B

|u|^sεdx

_1/sεsε/s

≤ C5

u ^ε/s_t,σB

|B|^t−sε/tsε u _t,σBsε/s

≤C₆|B|^{t−s−ε/st} u _t,σB.

2.18

From2.15and2.18and using log^βe|u|/ u t,σB≥1,β >0, we obtain

u _LslogL^αB≤C6|B|^{t−s−ε/st} u _t,σB

≤C₇|B|^{t−s−ε/st}

σB

|u|^tlog^β

e |u|

u _t,σB

dx _1/t

≤C7|B|^{t−s−ε/st} u _LtlogL^βσB

≤C₈|B|^t−s/st u _LtlogL^βσB.

2.19

This ends the proof ofLemma 2.5.

Using a similar method developed in the proof ofLemma 2.5and from Lemma 2.9 in 6, we can prove the following version of the weak reverse H ¨older inequality with Orlicz norms. Note that the following version of the weak reverse H ¨older inequality cannot be obtained by replacingubyduinLemma 2.5sincedumay not be a solution of1.13.

Lemma 2.6. Letube a solution of1.13inΩ,σ >1, and 0< s,t <∞. Then there exists a constant C, independent ofu, such that

du _LslogL^αB≤C|B|^t−s/st du _LtlogL^βσB 2.20

for all ballsBwithσB⊂Ωand diamB≥d0>0. Hered0is a fixed constant, andα >0 andβ >0 are any constants.

It is easy to see that for any constantk, there exist constantsm >0 andM >0, such that

mloget≤log

e t k

≤Mloget, t >0. 2.21

From the weak reverse H ¨older inequalityLemma 2.3, we know that the norms u s,Band u t,B are comparable when 0 < d1 ≤ diamB ≤ d2 < ∞. Hence, we may assume that 0 <

m1≤ u s,B ≤M1<∞and 0< m2≤ u t,B≤M2 <∞for some constantsmiandMi, i1,2.

Thus, we have

C₁loge|u|≤log

e |u|

u _s,B

≤C₂loge|u|,

C₃loge|u|≤log

e |u|

u _t,B

≤C₄loge|u|

2.22

for anys >0 andt >0, whereCiis a constant,i1,2,3,4. Using2.22, we obtain

C5

B

|u|^slog^α

e |u|

u _t,B

dx 1/s

≤ u _L^s_logL^α_B≤C6

B

|u|^slog^α

e |u|

u _t,B

dx 1/s

, 2.23 C7 u _L^t_logL^α_B ≤

B

|u|^tlog^α

e |u|

u _s,B

dx _1/t

≤C8 u _L^t_logL^α_B 2.24

for any ballBand anys >0, t >0,andα >0. Consequently, we see that u L^slogL^αB<∞if and only if

B

|u|^slog^α

e |u|

u _t,B

dx _1/s

<∞. 2.25

We recall the Muckenhoupt weights as follows. More properties and applications of Muckenhoupt weights can be found in1.

Definition 2.7. A weight wxis called an A_rEweight in a set E ⊂ Rⁿ for r > 1, write w∈ArE, if

sup

B

1

|B|

B

w dx 1

|B|

B

1 w

_1/r−1 dx

_r−1

<∞ 2.26

for any ballB⊂E.

We will need the following reverse H ¨older inequality forArE-weights.

Lemma 2.8. Ifw ∈A_rE,r >1, then there exist constantsk > 1 andC, independent ofw, such that

w _k,Q≤Cq^1−k/k w _1,Q 2.27

for all balls or cubesQ⊂E.

3. Caccioppoli-Type Estimates

In recent years different versions of Caccioppoli-type estimates have been established; see 1,2,4,12–15,17–19. The Caccioppoli-type estimates have become powerful tools in analysis and related fields. The purpose of this section is to prove the following Caccioppoli-type estimates withL^plogL^α-norms for solutions to the nonhomogeneousA-harmonic equation.

Theorem 3.1. Letu∈L^plogL^αΩ,Λ , 0,1, . . . , n−1, be a solution to the nonhomogeneous A-harmonic equation1.13inΩ⊂Rⁿ. Then, there exists a constantC, independent ofu, such that

du _L^p_logL^α_B≤C|B|^−1/n u−c _L^p_logL^α_σB 3.1

for some constantσ >1 and all ballsBwithσB ⊂ Ωand diamB≥ d0 >0. Hered0, 1< p < ∞ andα >0 are constants, andc∈L^plogL^αΩ,Λ is any closed form.

Proof. LetB⊂Ωbe a ball with diamB≥d₀ >0. Letε >0 be small enough and a constant C1large enough such that

|B|^−ε/p2≤C₁. 3.2

Applying Lemma 2.9 in6, we have

du _pε,B≤C₂|B|^p−pε/ppε du _p,σB 3.3

for someσ > 1. We may assume that|du|/ du p,B ≥ 1 onB. Otherwise, settingB1 {x ∈ B:|du|/ du p,B ≥1},B₂ {x∈B :|du|/ du p,B <1}, and using the elementary inequality

|ab|^s≤2^s|a|^s|b|^s, wheres >0 is any constant, we have

du _L^p_logL^α_B

B

|du|^plog^α

e |du|

du _p,B

dx _1/p

B1|du|^plog^α

e |du|

du _p,B

dx

B2|du|^plog^α

e |du|

du _p,B

dx _1/p

≤2^1/p

⎛

⎝

B1

|du|^plog^α

e |du|

du _p,B

dx 1/p

B2

|du|^plog^α

e |du|

du _p,B

dx 1/p⎞

⎠. 3.4

First, we estimate the first term on the right. Since |du|/ du p,B > 1 onB₁, then forε > 0 appeared in3.2, there existsC₃>0 such that

log^α

e |du|

du _p,B

≤C3

|du|

du _p,σ1B ε

. 3.5

Combining3.2,3.3, and3.5, we obtain

B1

|du|^plog^α

e |du|

du _p,B

dx _1/p

≤C₄ 1

du ^ε_p,σ1B

B1

|du|^pεdx _1/p

≤C₄ 1

du ^ε_p,σ1B

B

|du|^pεdx _1/p

C₄ du ^ε/p_p,σ1B

B

|du|^pεdx

_1/pεpε/p

≤ C5

du ^ε/p_p,σ1B

|B|^p−pε/ppε du _p,σ1Bpε/p

≤C6 du _p,σ1B,

3.6 whereσ1 >1 is a constant. Since

log^α

e |du|

du _p,B

≤M1log^αe1≤M2, x∈B2, 3.7

we can estimate the second term similarly

B2

|du|^plog^α

e |du|

du _p,B

dx 1/p

≤C₇ du _p,σ2B, 3.6

whereσ₂ >1 is a constant. From3.4,3.6, and3.6, we have

du _L^p_logL^α_B≤C₈ du _p,σ3B, 3.8

whereσ3 max{σ1, σ2}. ByLemma 2.2, we obtain

du _p,σ3B≤C9|B|^−1/n u−c _p,σ4B 3.9 for someσ4> σ3and all closed formsc. Note that

log^α

e |u−c|

u−c _p,σ2B

≥1, α >0. 3.5

Combining last three inequalities, we obtain

du _L^p_logL^α_B≤C10|B|^−1/n u−c _p,σ4B≤C10|B|^−1/n u−c _L^p_logL^α_σ4B. 3.10

The proof ofTheorem 3.1has been completed.

If we revise3.5and3.5in the proof ofTheorem 3.1, we obtain the following version of Caccioppoli-type estimate.

Corollary 3.2. Letu∈L^plogL^αΩ,Λ , 0,1, . . . , n−1, be a solution to the nonhomogeneous A-harmonic equation1.13inΩ⊂Rⁿ. Then, there exists a constantC, independent ofu, such that

B

|du|^plog^α

e |du|

du _p,Ω

dx _1/p

≤ C

diamB

σB

|u−c|^plog^α

e |u−c|

u−c _p,Ω

dx _1/p

3.11 for some constantσ >1 and all ballsBwithσB ⊂ Ωand diamB≥ d₀ >0. Hered₀, 1< p < ∞ andα >0 are constants, andc∈L^plogL^αΩ,Λ is any closed form.

Theorem 3.3. Letu∈L^plogL^αΩ,Λ , 0,1, . . . , n−1, be a solution to the nonhomogeneous A-harmonic equation1.13in a bounded domainΩ⊂Rⁿandwx∈A_rΩfor somer >1. Then, there exists a constantC, independent ofu, such that

du _L^p_logL^α_B,w≤C|B|^−1/n u−c _L^p_logL^α_σB,w 3.12 for any closed formc, some constantσ >1 and all ballsBwithσB⊂Ωand diamB≥d0>0. Here d₀, 1< p <∞andα >0 are constants, andc∈L^plogL^αΩ,Λ is any closed form.

Proof. LetBbe a ball withσB ⊂ Ωand diamB ≥ d0 > 0. SinceΩis bounded, thend0 ≤ diamB ≤ diamΩ < ∞. Thus, 0 < v₁ ≤ |B| ≤ v₂ < ∞for some constants v₁ and v₂. By Lemma 2.3, we have

m₁ u _s,ρ1B≤ u _t,B ≤m₂ u _s,ρ2B 3.13

for any solutionuof1.13and any constantss, t >0, where 0 < ρ1 <1, ρ2 >1, 0< m1 <1, andm₂ >1 are some constants. ByLemma 2.8, there exist constantsk >1 andC₀ >0, such that

w _k,B≤C₀|B|^1−k/k w _1,B. 3.14

Choosespk/k−1, then 1< p < sandks/s−p. We know thatu∈L^plogL^αΩ,Λ impliesu∈L^pΩ,Λ . Then, for any closed formc∈L^plogL^αΩ,Λ , it follows thatu−c∈ L^plogL^αΩ,Λ . Thus, u−c ∈ L^pΩ,Λ . By Caccioppoli inequality with L^p-norms, we know thatdu ∈ L^pΩ,Λ which gives du p,Ω N < ∞. If du p,B 0, thendu 0 a.e.

on B and Theorem 3.3follows. Thus, we may assume that 0 < m₁ ≤ du s,B < M₁ and 0 < m₂ ≤ du p,B < M₂ by3.13. Since 1/p 1/s s−p/ps, by the H ¨older inequality, 3.14and2.23, we have

du _L^p_logL^α_B,w

B

|du|^plog^α

e |du|

du _p,B

w dx 1/p

B

|du|log^α/p

e |du|

du _p,B

w^1/p _p

dx 1/p

≤

B

|du|^slog^αs/p

e |du|

du _p,B

dx 1/s

B

w^s/s−pdx

_s−p/sp

≤C1

B

|du|^slog^αs/p

e |du|

du _s,B

dx

_1/s

B

w^kdx

1/k_1/p

≤C₂|B|^1−k/kp w ^1/p_1,B

B

|du|^slog^αs/p

e |du|

du _s,B

dx _1/s

.

3.15

ApplyingTheorem 3.1yields

B

|du|^slog^αs/p

e |du|

du _s,B

dx 1/s

≤C3|B|^−1/n u−c _LslogL^αs/pσ1B. 3.16

Herecis any closed form. Next, chooset p/r. Using3.16andLemma 2.5withβ α/r, we obtain

B

|du|^slog^αs/p

e |du|

du _p,B

dx _1/s

≤C4|B|^−1/n|B|^t−s/st u−c _LtlogL^βσ2B 3.17

for someσ2> σ1. Using the H ¨older inequality again with 1/t1/p p−t/pt, we obtain

u−c _LtlogL^βσ2B

σ2B

|u−c|^tlog^β

e |u−c|

u−c _t,σ2B

dx 1/t

⎛

⎝

σ2B

|u−c|log^β/t

e |u−c|

u−c _t,σ2B

w^1/pw^−1/p t

dx

⎞

⎠

1/t

≤

σ2B

|u−c|^plog^βp/t

e |u−c|

u−c _t,σ2B

w dx

1/p

σ2B

1 w

_t/p−t dx

_p−t/pt

≤ u−c _L^p_logL^α_σ2B,w

σ2B

1 w

_1/r−1 dx

_r−1/p .

3.18

Combining3.15,3.17, and3.18, we conclude that

du _L^p_logL^α_B,w

≤C₅|B|^−r/p−1/n u−c _L^p_logL^α_σ2B,w

⎛

⎝

B

w dx

σ2B

1 w

_1/r−1 dx

_r−1⎞

⎠

1/p

≤C5|B|^−r/p−1/n u−c _L^p_logL^α_σ2B,w

w _1,σ2B· 1

w

1/r−1,σ2B

1/p

.

3.19

Sincew∈ArΩ, then

w _1,σ2B· 1

w

1/r−1,σ2B

_1/p

⎛

⎝

σ2B

w dx

σ2B

1 w

_1/r−1 dx

_r−1⎞

⎠

1/p

⎛

⎝|σ2B|^r 1

|σ2B|

σ2B

w dx 1

|σ2B|

σ2B

1 w

_1/r−1 dx

_r−1⎞

⎠

1/p

≤C6|B|^r/p.

3.20

Substituting the last inequality into3.19it follows obviously that

du _L^p_logL^α_B,w≤C₇|B|^−1/n u−c _L^p_logL^α_σ2B,w. 3.21

This ends the proof ofTheorem 3.3.

Letα1 inTheorem 3.3; we obtain the following corollary.

Corollary 3.4. Letu∈L^plogLΩ,Λ , 0,1, . . . , n−1, be a solution to the nonhomogeneous A-harmonic equation1.13in a bounded domainΩ⊂Rⁿandwx∈A_rΩfor somer >1. Then, there exists a constantC, independent ofu, such that

du _L^p_logLB,w≤C|B|^−1/n u−c _L^p_logLσB,w 3.22 for any closed formc, some constantσ >1 and all ballsBwithσB⊂Ωand diamB≥d0>0. Here d₀and 1< p <∞are constants, andc∈L^plogLΩ,Λ is any closed form.

We know that ifw ∈ArEand 0 < λ≤ 1, thenw^λ ∈ ArE. Thus, under the same conditions ofTheorem 3.3, we also have the following estimate:

du _L^p_logL^α_B,w^λ ≤C|B|^−1/n u−c _L^p_logL^α_σB,w^λ, 3.23 wherec is any closed form, and 0 < λ ≤ 1 andα > 0 are any constants. Chooseλ 1/p, 1< p <∞, in3.23. Then, for closed formcand any constantα >0, we have

du _L^p_logL^α_B,w^1/p≤C|M|^−1/n u−c _L^p_logL^α_σB,w^1/p. 3.24 We have proved Caccioppoli-type inequalities withL^plogL^α-norms for solutions to the nonhomogeneousA-harmonic equation. Using the same method developed in12, we can obtain the more general version of the Caccioppoli-type inequality for differential forms satisfying certain conditions. A special useful Young functionψ : 0,∞ → 0,∞, termed

anN-function, is a continuous Young function such thatψx 0 if and only ifx 0 and lim_x→0ψx/x0, lim_{x→ ∞}ψx/x ∞.We say that a differential formu∈W_loc^1,1Ω,Λ is a k-quasiminimizer for the functional

IΩ;v

Ωψ|dv|dx 3.25

if and only if, for everyφ∈W_loc^1,1Ω,Λ with compact support, I

suppφ;u

≤k·I

suppφ;uφ

, 3.26

wherek >1 is a constant. We say thatψsatisfies the so-calledΔ2-condition if there exists a constantp >1 such thatψ2t≤pψtfor allt >0, from which it follows that

ψλt≤λ^pψt 3.27

for anyt >0 andλ≥1; see12.

We will need the following lemma which can be found in19or12.

Lemma 3.5. Letftbe a nonnegative function defined on the intervala, bwitha≥ 0. Suppose that fors, t∈a, bwitht < s,

ft≤ A

s−t^α Bθfs 3.28 holds, where A, B, α, and θ are nonnegative constants with θ < 1. Then, there exists a constant CCα, θsuch that

f ρ

≤C A

R−ρ_α B

3.29

for anyρ, R∈a, bwithρ < R.

Theorem 3.6. Letube ak-quasiminimizer for the functional3.25and letψ be a Young function satisfying theΔ2-condition. Then, for any ball BR ⊂ Ω with radius R, there exists a constant C, independent ofu, such that

BR/2

ψ|du|dx≤C

BR

ψ

|u−c|

R

dx, 3.30

wherecis any closed form.

The proof ofTheorem 3.6is the same as that of Theorem 6.1 developed in12. For the complete purpose, we include the proof ofTheorem 3.6as follows.

Proof. LetBR Bx0, R ⊂ be a ball with radiusR and center x0, R/2 < t < s < R. Set ηx g|x−x₀|, where

gτ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1, 0≤τ≤t, affine, τ < t < s, 0, τ≥s.

3.31

Then,η∈W₀^1,∞Bs,ηx 1 onB_t, and dηx

⎧⎨

⎩

s−t⁻¹, t≤ |x−x0| ≤s,

0, otherwise. 3.32

Letvx ux ηx^pc−ux. We find that

dv 1−η^p

duη^ppdη

η c−ux. 3.33

Sinceψis an increasing convex function satisfying theΔ2-condition, we obtain

ψ|dv|≤ 1−η^p

ψ|du| η^pψ

pdη

η |c−ux|

. 3.34

Using the definition of thek-quasiminimizer and3.27, it follows that

Bs

ψ|du|dx≤k

Bs

ψ|dv|dx

≤k

Bs\Bt

1−η^p

ψ|du|dx

Bs

η^pψ

pdη

η |c−ux|

dx

≤k

Bs\Bt

ψ|du|dxp^p

Bs

ψdη|u−c|

dx

.

3.35

Applying3.35,3.32, and3.27, we have

Bt

ψ|du|dx≤

Bs

ψ|du|dx

≤k

Bs\Bt

ψ|du|dxp^p

Bs

ψ

2R|u−c|

s−tR

dx

≤k

Bs\Bt

ψ|du|dx

2pR_p s−t^p

Bs

ψ

|u−c|

R

dx

.

3.36

Addingk

Btψ|du|dxto both sides of inequality3.36yields

Bt

ψ|du|dx≤ k k1

Bs

ψ|du|dx

2pRp

s−t^p

Bs

ψ

|u−c|

R

dx

. 3.37

Next, write ft

Btψ|du|dx,fs

Bsψ|du|dx,A 2pR^p

Bsψ|u−c|/Rdx, and B 0. From 3.37, we find that the conditions ofLemma 3.5 are satisfied. Hence, using Lemma 3.5withρR/2 andαp, we obtain3.30immediately. The proof ofTheorem 3.6 has been completed.

It should be noticed thatc ∈ L^plogL^αΩ,Λ is any closed form on the right side of each version of the Caccioppoli-type inequality. Hence, we may choosec 0 in each of the above Caccioppoli-type inequalities. For example, if we choose c 0 in Theorem 3.1 andTheorem 3.6, we obtain the following Corollaries3.7and3.8, respectively, which can be considered as the special version of the Caccioppoli-type inequality.

Corollary 3.7. Letu∈L^plogL^αΩ,Λ , 0,1, . . . , n−1, be a solution to the nonhomogeneous A-harmonic equation1.13inΩ⊂Rⁿ. Then, there exists a constantC, independent ofu, such that

du _L^p_logL^α_B≤C|B|^−1/n u _L^p_logL^α_σB 3.38 for some constantσ >1 and all ballsBwithσB ⊂ Ωand diamB≥ d0 >0. Hered0, 1< p < ∞ andα >0 are constants.

Corollary 3.8. Letu be ak-quasiminimizer for the functional 3.25 and ψ be a Young function satisfying theΔ2-condition. Then, for any ball BR ⊂ Ω with radius R, there exists a constant C, independent ofu, such that

BR/2

ψ|du|dx≤C

BR

ψ |u|

R

dx. 3.39

4. Poincar ´e Inequalities

In this section, we focus our attention on the local and global Poincar´e inequalities with L^plogL^α-norms. The main result for this section is Theorem 4.2, the global Poincar´e inequality for solutions of the nonhomogeneous A-harmonic equation. The following definition ofL^ϕμ-domains can be found in1.

Definition 4.1. Letϕbe a Young function on0,∞withϕ0 0. We call a proper subdomain Ω⊂RⁿanL^ϕμ-domain, if there exists a constantCsuch that

Ωϕσ|u−u_Ω|dμ≤Csup

B⊂Ω

B

ϕστ|u−uB|dμ 4.1

for allusuch thatϕ|u|∈L¹_locΩ;μ, where the measureμis defined bydμwxdx,wx is a weight andτ, σare constants with 0< τ ≤1, 0< σ≤1, and the supremum is over all balls B⊂Ω.

Theorem 4.2. Assume thatΩ⊂Rⁿis a boundedL^ϕμ-domain withϕt t^plog^αet/k, where k u−u_{B0 p,Ω}, 1 < p < ∞, andB₀ ⊂ Ωis a fixed ball. Letu ∈ DΩ,Λ⁰be a solution of the nonhomogeneousA-harmonic equation inΩand du ∈ L^pΩ,Λ¹as well asw ∈ A_rΩfor some r >1. Then, there is a constantC, independent ofu, such that

u−u_{Ω L}plogL^αΩ,w≤C|Ω|^1/n du _L^p_logL^α_Ω,w 4.2

for any constantα >0.

To proveTheorem 4.2, we need the following local Poincar´e inequalities, Theorems4.3 and4.4, with Orlicz norms.

Theorem 4.3. Letu ∈ DΩ,∧ be a solution of the nonhomogeneousA-harmonic equation in a domainΩ ⊂ Rⁿand du ∈L^pΩ,∧ ¹, 0,1, . . . , n. Assume that 1< p < ∞. Then, there is a constantC, independent ofu, such that

u−u_{B L}plogL^αB ≤C|B|^1/n du _L^p_logL^α_ρB 4.3

for all ballsBwithρB⊂Ωand diamB≥d0. Hereα >0 is any constant andρ >1 andd0 >0 are some constants.

Proof. LetB⊂Ωbe a ball with diamB≥d₀ >0. Chooseε >0 small enough and a constant C1such that

|B|^−ε/p2≤C1. 4.4

FromLemma 2.3, we have

u−uB pε,B ≤C2|B|^p−pε/ppε u−uB p,ρ1B 4.5

for someρ₁>1. Similar to the proof ofTheorem 3.1, we may assume that|u−uB|/ u−u_{B p,B} ≥ 1. Hence, for aboveε >0, there existsC3>0 such that

log^α

e |u−uB| u−u_{B p,B}

≤C3

|u−uB| u−u_{B p,ρ1B}

ε

. 4.6