Volume 2010, Article ID 685046,18pages doi:10.1155/2010/685046
Research Article
Gradient Estimates for Weak Solutions of
A
-Harmonic Equations
Fengping Yao
Department of Mathematics, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Fengping Yao,[email protected]
Received 29 December 2009; Accepted 23 March 2010
Academic Editor: Wang Yong
Copyrightq2010 Fengping Yao. 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 gradient estimates in Orlicz spaces for weak solutions ofA-Harmonic Equations under the assumptions thatAsatisfies some proper conditions and the given function satisfies some moderate growth condition. As a corollary we obtainLp-type regularity for such equations.
1. Introduction
In this paper we consider the following general nonlinear elliptic problem:
divA∇u, x div|f|p−2f inΩ, 1.1
whereΩis an open bounded domain inRn,f f1, . . . , fnandAAξ, x:Rn×Rn → Rn
are two given vector fields, andAis measurable inxfor eachξand continuous inξfor almost everywherex. Moreover, for givenp∈1,∞the structural conditions on the functionAξ, x are given as follows:
Aξ, x− Aη, x·ξ−η≥C1ξ−ηp, 1.2
|Aξ, x| ≤C2
1|ξ|p−1, 1.3
Aξ, x·ξ≥C3|ξ|p−C4, 1.4
Aξ, x− Aξ, y≤C5wx−y1|ξ|p−1
for allξ, η∈Rn,x, y∈Ω,and some positive constantsC
i>0,i1,2,3,4,5. Here the modulus
of continuitywx:R → Ris nondecreasing and satisfies
wr−→0 asr −→0. 1.6
Especially whenAξ, x |ξ|p−2ξ,1.1is reduced to be quasilinear elliptic equations of p-Laplacian type
div|∇u|p−2∇udiv|f|p−2f inΩ. 1.7
As usual, the solutions of1.1are taken in a weak sense. We now state the definition of weak solutions.
Definition 1.1. A functionu∈Wloc1,pΩis a local weak solution of1.1if for anyϕ∈W01,pΩ, one has
ΩA∇u, x· ∇ϕ dx Ω|f|
p−2f· ∇ϕ dx. 1.8
DiBenedetto and Manfredi1and Iwaniec2obtainedLq,q≥p, gradient estimates
for weak solutions of1.7while Acerbi and Mingione3studied the case that p px. Moreover, the authors4,5obtained Lq,q ≥ p, gradient estimates for weak solutions of
quasilinear elliptic equation ofp-Laplacian type
divA∇u· ∇up−2/2A∇udiv|f|p−2f inΩ 1.9
under the different assumptions on the coefficients A and the domain Ω. Boccardo and Gallou¨et6,7obtained W1,q, q < np−1/n−1, p ≤ n, regularity for weak solutions
of the problem−divax, u, Du fwith some structural conditions.
Recently, Byun and Wang8obtainedW1,p, 2≤p <∞, regularity for weak solutions
of the general nonlinear elliptic problem
diva∇u, x divf inΩ, 1.10
withaξ, xsatisfyingδ, R-vanishing condition and the following structural conditions:
aξ, x−aη, x·ξ−η≥C1ξ−η2,
|aξ, x| ≤C21|ξ|, ∇ξaξ, x≥C
3.
The purpose of this paper is to extend theLp-type estimates in8to theLφ-type estimates in
Orlicz spaces for the more general problem1.1withAsatisfying1.2–1.5. In particular, we are interested in estimates like
Br
φ|∇u|pdx≤C
B2r
φ|f|pdxφ
B2r |u|pdx
1
, 1.12
whereCis a constant independent fromuandf. Indeed, ifφx |x|q/pwithq > p,1.12is
reduced to the classicalLqestimate.
Orlicz spaces have been studied as a generalization of Lp spaces since they were
introduced by Orlicz 9 see10–16. The theory of Orlicz spaces plays a crucial role in a very wide spectrum see 17. Here for the reader’s convenience, we will give some definitions on the general Orlicz spaces. We denote byΦthe function class that consists of all functionsφ:0,∞ → 0,∞which are increasing and convex.
Definition 1.2. A functionφ∈Φis said to satisfy the globalΔ2condition, denoted byφ∈Δ2,
if there exists a positive constantKsuch that for everyt >0,
φ2t≤Kφt. 1.13
Moreover, a functionφ ∈Φis said to satisfy the global∇2 condition, denoted byφ ∈ ∇2, if there exists a numbera >1 such that for everyt >0,
φt≤ φat
2a . 1.14
Remark 1.3. 1 We remark that the global Δ2 ∩ ∇2 condition makes the functions grow
moderately. For example, φt |t|α1 |log|t ∈ Δ
2∩ ∇2 forα > 1. Examples such as tlog1tare ruled out by∇2, and those such as expt2are ruled out byΔ2.
2In fact, ifφ∈Δ2∩ ∇2, thenφsatisfies for 0< θ2≤1≤θ1<∞,
φθ1t≤Kθ1α1φt, φθ2t≤2aθ2α2φt, 1.15
whereα1log2Kandα2loga21.
3Under condition1.15, it is easy to check thatφ∈Φsatisfiesφ0 0 and
lim
t→0 φt
t tlim→∞
t
Definition 1.4. Letφ ∈Φ. Then the Orlicz classKφΩis the set of all measurable functions
g:Ω → Rsatisfying
Ωφ
gdx <∞. 1.17
The Orlicz spaceLφΩis the linear hull ofKφΩ.
Remark 1.5. We remark that Orlicz spaces generalizeLq spaces in the sense that if we take
φt tq,t≥0, thenφ∈Δ
2∩ ∇2, so for this special case,
LφΩ LqΩ. 1.18
Moreover, we give the following lemma.
Lemma 1.6see10,12,15. Assume thatφ∈Δ2∩ ∇2andg ∈LφΩ. Then
1Kφ LφandC∞
0 is dense inLφ,
2Lα1Ω⊂LφΩ⊂Lα2Ω⊂L1Ω, whereα
1andα2are defined in1.15,
3
Ωφ
gdx ∞
0
x∈Ω:g> λdφλ, 1.19
4
∞
0 1
μ {x∈Ω:|g|>aμ}gdx d
φbμ≤C Ωφ
gdx, 1.20
for anya, b >0, whereCCa, b, φ.
Now we are set to state the main result.
Theorem 1.7. Assume thatφ ∈ Δ2∩ ∇2 and|f|p ∈ LφlocΩ. Ifuis a local weak solution of 1.1
withAsatisfying1.2–1.5, then one has
|∇u|p∈LφlocΩ 1.21
with the estimate1.12, that is,
Br
φ|∇u|pdx≤C
B2r
φ|f|pdxφ
B2r |u|pdx
1
, 1.22
Remark 1.8. We remark that the globalΔ2∩ ∇2 condition is optimal. Actually, the authors in
15have proved that ifuis a solution of the Poisson equation−ΔufinRn, then
Rn
φD2udx≤C
Rn
φfdx 1.23
holds if and only ifφ∈Δ2∩ ∇2.
Our approach is based on the paper18. Recently Acerbi and Mingione18obtained localLq,q ≥ p, gradient estimates for the degenerate parabolic p-Laplacian systems which
are not homogeneous ifp /2. There, they invented a new iteration-covering approach, which is completely free from harmonic analysis, in order to avoid the use of the maximal function operator.
This paper will be organized as follows. InSection 2, we give a new normalization method and the iteration-covering procedure, which are very important to obtain the main result. We finish the proof ofTheorem 1.7inSection 3.
2. Preliminary Materials
2.1. New Normalization
In this paper we will use a new normalization method, which is much influenced by8,19, so that the highly nonlinear problem considered here is invariant.
For eachλ≥1, we define
uλx
ux
λ , fλx
fx
λ , 2.1
Aλξ, x A
λξ, x
λp−1 . 2.2
Lemma 2.1new normalization. Ifu∈Wloc1,pΩis a local weak solution of 1.1andAsatisfies
1.2–1.5, then
1Aλsatisfies1.2–1.5with the same constantsCi 1≤i≤4,
2uλis a local weak solution of
divAλ∇uλ, x
1
λp−1divA∇u, x div
|fλ|p−2fλ in Ω. 2.3
Proof. We first prove thatAλsatisfies1.2–1.5with the same constantsCi 1≤i≤4. From
1.2and2.2we find that
Aλξ, x− Aλ
η, x·ξ−η 1 λp
Aλξ, x− Aλη, x·λξ−λη
≥C1 1
λpλξ−λη p
C1ξ−ηp
for allξ, η∈Rn. That is to say,A
λsatisfies1.2. Moreover,Aλsatisfies1.3-1.4since
|Aλξ, x| |A
λξ, x| λp−1 ≤C2
1 λp−1 |ξ|
p−1
≤C2
1|ξ|p−1,
Aλξ, x·ξ 1
λpAλξ, x·λξ≥
1 λp
C3|λξ|p−C4
≥C3|ξ|p−C4
2.5
for allξ, η∈Rnandλ≥1. Furthermore,
Aλξ, x− Aλ
ξ, y 1
λp−1Aλξ, x− A
λξ, y
≤ C5
λp−1wx−y1|λξ|
p−1
≤C5wx−y1|ξ|p−1
2.6
for allξ, η∈Rnandλ≥1.
Finally we prove2. Indeed, sinceuis a local weak solution of1.1, it follows from
Definition 1.1,2.1, and2.2that
ΩAλ∇uλ, x· ∇ϕ dx 1 λp−1
ΩA∇u, x· ∇ϕ dx
1
λp−1 Ω|f|
p−2f· ∇ϕ dx Ω|fλ|
p−2fλ· ∇ϕ dx.
2.7
Thus we complete the proof.
2.2. The Iteration-Covering Procedure
In this subsection we give one important lemmathe iteration-covering procedure, which is much motivated by18. To start with, letube a local weak solution of the problem1.1. By a scaling argument we may as well assume thatr1 inTheorem 1.7. We write
λ0
−
B2
|∇u|pdx1 −B2|f|
pdx
1/p
, 2.8
where >0 is going to be chosen later in3.47. Moreover, for anyx∈Ωandρ >0, we write
Jλ
Bρx
−
Bρx
|∇uλ|pdy1
−Bρx|fλ|
pdy,
2.9
Eλ1
x∈B1:|∇uλ|p>1
From1.6, we can choose a proper constantR0 R0∈0,1such that
wR0≤. 2.11
Lemma 2.2. Givenλ≥λ∗:10/R0n/pλ01, there exists a family of disjoint balls{Bρixi}, xi∈ Eλ1such that0< ρiρxi≤R0/10and
Jλ
Bρixi
1, Jλ
Bρxi
<1 for anyρ > ρi. 2.12
Moreover, one has
Eλ1⊂
i∈N
B5ρixi∪negligible set, 2.13
Bρixi≤3
{x∈Bρixi:|∇uλ|p>1/3}
|∇uλ|pdx
1
{x∈Bρixi:|fλ|p>/3} |fλ|pdx
. 2.14
Proof. 1We first claim that
sup
w∈B1 sup
R0/10≤ρ≤R0 Jλ
Bρw
≤1. 2.15
To prove this, fix anyw ∈B1 andR0/10 ≤ρ ≤ R0. Letλ ≥ λ∗ 10/R0n/pλ01. Then we have
−
Bρw
|∇uλ|pdx≤ |
B1|
Bρw
−
B1
|∇uλ|pdx
≤ 10 R0 n − B1
|∇uλ|pdx
1 λp 10 R0 n − B1
|∇u|pdx.
2.16
Similarly,
−
Bρw
|fλ|pdx≤
10 R0 n − B1
|fλ|pdx 1
λp 10 R0 n − B1
|f|pdx. 2.17
Consequently, combining the two inequalities above,2.8and2.9, we know that
Jλ
Bρw
≤1 2.18
2Now for a.e. w ∈ Eλ1, a version of Lebesgue’s differentiation theorem implies
that
lim
ρ→0Jλ
Bρw
>1, 2.19
which implies that there exists someρ >0 satisfying
Jλ
Bρw
>1. 2.20
Therefore from2.15we can select a radiusρw∈0, R0/10such that
ρw: max
ρ|Jλ
Bρw
1,0< ρ≤ R0 10
. 2.21
Then we observe that
Jλ
Bρww
1 2.22
and that forρw< ρ≤R0,
Jλ
Bρw
<1. 2.23
From the argument above we know that for a.e. w ∈ Eλ1 there exists a ball Bρww constructed as above. Therefore, applying Vitali’s covering lemma, we can find a family of disjoint balls{Bρixi}i∈Nwith xi ∈ Eλ1and ρi ρxi ∈ 0, R0/10,so that 2.12and
2.13hold truely.
3From2.12we see that
Jλ
Bρixi
:−
Bρixi
|∇uλ|pdx1
−B ρixi
|fλ|pdx1.
2.24
That is to say,
Bρ
ixi
Bρixi
|∇uλ|pdx1
B ρixi
|fλ|pdx.
2.25
Therefore, by splitting the right-side two integrals in2.25as follows we have
Bρixi≤
{x∈Bρixi:|∇uλ|p>1/3}
|∇uλ|pdx
Bρixi 3
1
{x∈B
ρixi:|fλ|>/3}
|fλ|pdxBρixi 3 .
2.26
3. Proof of Main Result
In the following it is sufficient to consider the proof ofTheorem 1.7as an a priori estimate, therefore assuming a priori that|∇u|p∈L∞
locΩ⊂L
φ
locΩ. This assumption can be removed in a standard way via an approximation argument as the one in12,15,18.
We first give the following localLpestimates for problem1.1.
Lemma 3.1. Suppose that|f|p∈LφB
2R,B2R ⊂Ω,and letu∈Wloc1,pΩbe a local weak solution of
1.1withAsatisfying1.2–1.5. Then one has
BR
|∇u|pdx≤C
1 Rp
B2R
|u|pdx
B2R
|f|pdx1
. 3.1
Proof. We may choose the test functionϕ ζpu ∈ W1,p
0 B2R in Definition 1.1, whereζ ∈ C∞0Rnis a cutofffunction satisfying
0≤ζ≤1, ζ≡1 inBR, ζ≡0 in R n
B2R
, |∇ζ| ≤C
R. 3.2
Then we have
B2R
A∇u, x· ∇ζpudx
B2R
|f|p−2f· ∇ζpudx 3.3
and then write the resulting expression as
I1I2I3I4, 3.4
where
I1
B2R
ζpA∇u, x· ∇u dx,
I2−
B2R
pζp−1uA∇u, x· ∇ζ dx,
I3
B2R
ζp|f|p−2f· ∇u dx,
I4
B2R
pζp−1u|f|p−2f· ∇ζ dx.
3.5
Estimate ofI1.Using1.4, we find that
I1≥C
B2R
Estimate ofI2.From Young’s inequality withτ,1.3, and3.2we have
I2≤
B2R
pζp−1|A∇u, x||u||∇ζ|dx
≤C
B2R
pζp−11|∇u|p−1|u||∇ζ|dx
≤τ
B2R
ζp|∇u|p1dxCτ
B2R
|∇ζ|p|u|pdx
≤τ
B2R
ζp|∇u|p1dxCτ Rp
B2R |u|pdx.
3.7
Estimate ofI3.From Young’s inequality withτwe have
I3≤τ
B2R
ζp|∇u|pdxCτ B2R
|f|pdx.
3.8
Estimate ofI4.From Young’s inequality and3.2we have
I4≤C
B2R
|∇ζ|p|u|pdx
B2R |f|pdx
≤C
1 Rp
B2R
|u|pdx
B2R |f|pdx
.
3.9
Combining the estimates ofIi 1≤i≤4, we deduce that
C
B2R
ζp|∇u|pdx≤2τ
B2R
ζp|∇u|pdxCτ
1 Rp
B2R
|u|pdx
B2R
|f|pdx1
3.10
and then finish the proof by choosingτsmall enough.
Letvbe the weak solution of the following reference equation:
divA∇v, x∗ 0 inBr,
vu on∂Br,
3.11
We first state the definition of the global weak solutions.
Definition 3.2. Assume thatv∈W1,pB
r. One says thatv∈W1,pBrwithv−u∈W01,pBris
the weak solution of3.11inBr if one has
Br
A∇v, x∗· ∇ϕ dx0 3.12
for anyϕ∈W01,pBr.
From the definition above we can easily obtain the following lemma.
Lemma 3.3. Ifv∈W1,pB
10ρixiis the weak solution of 3.11inB10ρixi, wherexi∈Eλ1and ρiare defined inLemma 2.2, then one has
−
B10ρixi
|∇v|pdx≤C
−
B10ρixi
|∇u|pdx1
. 3.13
Proof. Choosing the test functionϕu−v∈W01,pB10ρixi, fromDefinition 3.2, we find that
B10ρixi
A∇v, x∗· ∇u−vdx0. 3.14
That is to say,
B10ρixi
A∇v, x∗· ∇v dx
B10ρixi
A∇v, x∗· ∇u dx. 3.15
From1.4, we conclude that
B10ρixi
A∇v, x∗· ∇v dx≥C
B10ρixi
|∇v|pdx−B10ρi
. 3.16
Moreover, from1.3and Young’s inequality withτwe have
B10ρixi
A∇v, x∗· ∇u dx≤C
B10ρixi
1|∇v|p−1|∇u|dx
≤τ
B10ρixi
|∇v|pdxCτ
B10ρixi
1|∇u|pdx.
Combining the estimates of Ii i 1,2 and selecting a small enough constant τ > 0, we
deduce that
−
B10ρixi
|∇v|pdx≤C
−
B10ρixi
|∇u|pdx1
3.18
and then finish the proof.
Lemma 3.4. Suppose thatv ∈ W1,pB
10ρixiis the weak solution of 3.11in B10ρixiwithA
satisfying1.2–1.5. If
−
B10ρixi
|∇u|pdx≤1, −
B10ρixi
|f|pdx≤,
3.19
then there existsN0>1such that
sup
B5ρixi
|∇v| ≤N0, 3.20
−
B10ρixi
|∇u−v|pdx≤. 3.21
Proof. If the conclusion3.21is true, then the conclusion3.20can follow from20, Lemma
5.1.
Next we are set to prove 3.21. We may choose the test function ϕ u − v ∈ W01,pB10ρixiin Definitions1.1and3.2to find that
B10ρixi
A∇u, x· ∇u−vdx
B10ρixi
|f|p−2f· ∇u−vdx,
B10ρixi
A∇v, x∗· ∇u−vdx0,
3.22
wherex∗∈B10ρixiis a fixed point. Then a direct calculation shows the resulting expression as
where
I1
B10ρixi
A∇u, x− A∇v, x· ∇u−vdx,
I2−
B10ρixi
A∇v, x− A∇v, x∗· ∇u−vdx,
I3
B10ρixi
|f|p−2f· ∇u−vdx.
3.24
Estimate ofI1.Equation1.2implies that
I1 ≥C
B10ρixi
|∇u−v|pdx. 3.25
Estimate ofI2.From1.5and the fact thatρi∈0, R0/10we obtain
I2≤Cw
10ρi
B10ρixi
1|∇v|p−1|∇u||∇v|dx
≤CwR0
B10ρixi
1|∇v|p1|∇v|p−1|∇u|dx,
3.26
then it follows from2.11, Young’s inequality, andLemma 3.3that
I2≤C
B10ρixi
1|∇v|pdx
B10ρixi
|∇u|pdx
≤C
B10ρixi
1|∇u|pdx.
3.27
Furthermore, using3.19we can obtain
I2 ≤CB10ρixi. 3.28
Estimate ofI3.Using Young’s inequality withτ, we have
I3≤
B10ρixi
|f|p−1|∇u−v|dx
≤τ
B10ρixi
|∇u−v|pdxCτ
B10ρixi |f|pdx.
Combing all the estimates ofIi 1 ≤i≤ 3and selecting a small enough constantτ >0, we
obtain
B10ρixi
|∇u−v|pdx≤CB10ρixiC
B10ρixi |f|pdx,
3.30
then it follows from3.19that
−
B10ρixi
|∇u−v|pdx≤Cε. 3.31
This completes our proof.
In view ofLemma 2.2, givenλ≥λ∗:10/R0n/pλ01, we can construct the disjoint family of balls{Bρixi}i∈N, wherexi ∈Eλ1. Fix anyi∈N. It follows fromLemma 2.2that
−
Bρxi
|∇uλ|pdx≤1, − Bρxi
|fλ|pdx≤ for any ρ > ρ
i. 3.32
Furthermore, from the new normalization inLemma 2.1, we can easily obtain the following corollary ofLemma 3.4.
Corollary 3.5. Suppose thatvλ∈W1,pB10ρixiis the weak solution of
divAλ∇vλ, x∗ 0 in Qr,
vλuλ on∂pQr, 3.33
withx∗∈B10ρixiandAλsatisfying1.2–1.5. Then there existsN0>1such that
sup
B5ρixi
|∇vλ| ≤N0,
−
B10ρixi
|∇uλ−vλ|pdx≤C.
3.34
Proof. FromCorollary 3.5, for anyλ≥λ∗:10/R0n/pλ01 we have
x∈B5ρixi:|∇u|>2N0λx∈B5ρixi:|∇uλ|>2N0 ≤x∈B5ρixi:|∇uλ−vλ|> N0
x∈B5ρixi:|∇vλ|> N0
x∈B5ρixi:|∇uλ−vλ|> N0
≤ 1
N0p B5ρixi
|∇uλ−vλ|pdx
≤CBρixi,
3.35
then it follows from2.14inLemma 2.2that
x∈B5ρ
ixi:|∇u|>2N0λ
≤C
{x∈Bρixi:|∇uλ|p>1/3}
|∇uλ|pdx 1
{x∈B
ρixi:|fλ|p>/3} |fλ|pdx
, 3.36
whereCCn, p. Recalling the fact that the balls{Bρixi}are disjoint and
i∈N
B5ρixi⊃Eλ1 {x∈B1:|∇uλ|>1} 3.37
for anyλ≥λ∗:10/R0n/pλ01 and then summing up oni∈Nin the inequality above, we have
x∈B1:|∇u|p>2N0pλp|{x∈B1:|∇u|>2N0λ}|
≤
i
x∈B5ρ
ixi:|∇u|>2N0λ
≤C
{x∈B2:|∇uλ|p>1/3}
|∇uλ|pdx 1
x∈B2:|fλ|p>/3|fλ|
pdx
for anyλ≥λ∗:10/R0n/pλ01. RecallingLemma 1.63, we compute
B1
φ|∇u|pdx ∞
0
x∈B1:|∇u|p> μdφμ
2N0
pλp ∗
0
x∈B1:|∇u|p> μdφμ
∞
2N0pλp∗
x∈B1:|∇u|p> μdφμ
2N0
pλp ∗
0
x∈B1:|∇u|p> μdφμ
∞
λ∗
x∈B1:|∇u|p>2N0pλpd
2N0pλp
:J1J2.
3.39
Estimate ofJ1.From the definition ofλ0in2.8we deduce that
λp∗ ≤Cλp01≤C
−
B1
u|∇|pdx1 −B1|f|
pdx1
, 3.40
then it follows fromLemma 3.1that
λp∗≤C
−
B2
|u|pdx−
B2
|f|pdx1
−B1|f|
pdx1
≤C
−
B2
|u|pdx−
B2
|f|pdx1
,
3.41
whereCCn, p, . Therefore, by1.15and Jensen’s inequality, we conclude that
J1≤φ
2N0pλp∗
|B1|
≤C φ − B2 |u|pdx
φ
−
B2 |f|pdx
1 ≤C φ B2 |u|pdx
B2
φ|f|pdx1
,
3.42
Estimate ofJ2.From3.38we deduce that
J2≤C ∞
0 {x∈B2:|∇uλ|p>1/3}
|∇uλ|pdx d
2N0pλp
C ∞
0 {x∈B2:|fλ|p>/3}|fλ|
pdx d2N
0pλp
.
3.43
Setμλp. The above inequality and2.1imply that
J2≤C ∞
0 1
μ {x∈B2:|∇u|p>μ/3}|∇
u|pdx d2N0pμ
C ∞
0 1
μ {x∈B2:|f|p>μ/3}|f|
pdx d2N
0pμ
,
3.44
then it follows fromLemma 1.64that
J2≤C1
B2
φ|∇u|pdxC2
B2
φ|f|pdx, 3.45
whereC1Cn, p, φandC2Cn, p, φ, .
Combining the estimates ofJ1andJ2, we obtain
B1
φ|∇u|pdx≤C1
B2
φ|∇u|pdxC3
B2
φ|f|pdxC4φ
B2 |u|pdx
1, 3.46
whereC3C3n, p, φ, andC4C4n, p, φ, . Selecting suitablesuch that
C1 1
2 3.47
and reabsorbing at the right-side first integral in the inequality above by a covering and iteration argumentsee21, Lemma 4.1, Chapter 2, or22, Lemma 2.1, Chapter 3, we have
B1
φ|∇u|pdx≤C
B2
φ|f|pdxφ
B2 |u|pdx
1
. 3.48
Acknowledgments
This work is supported in part by Tianyuan Foundation10926084and Research Fund for the Doctoral Program of Higher Education of China20093108120003. Moreover, the author wishes to the department of mathematics at Shanghai university which was supported by the Shanghai Leading Academic Discipline ProjectJ50101and Key Disciplines of Shanghai MunicipalityS30104.
References
1 E. DiBenedetto and J. Manfredi, “On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems,”American Journal of Mathematics, vol. 115, no. 5, pp. 1107–1134, 1993.
2 T. Iwaniec, “Projections onto gradient fields andLp-estimates for degenerated elliptic operators,”
Studia Mathematica, vol. 75, no. 3, pp. 293–312, 1983.
3 E. Acerbi and G. Mingione, “Gradient estimates for thepx-Laplacean system,”Journal fur die Reine und Angewandte Mathematik, vol. 584, pp. 117–148, 2005.
4 S.-S. Byun and L. Wang, “Quasilinear elliptic equations with BMO coefficients in Lipschitz domains,”
Transactions of the American Mathematical Society, vol. 359, no. 12, pp. 5899–5913, 2007.
5 J. Kinnunen and S. Zhou, “A local estimate for nonlinear equations with discontinuous coefficients,”
Communications in Partial Differential Equations, vol. 24, no. 11-12, pp. 2043–2068, 1999.
6 L. Boccardo and T. Gallou¨et, “Nonlinear elliptic and parabolic equations involving measure data,”
Journal of Functional Analysis, vol. 87, no. 1, pp. 149–169, 1989.
7 L. Boccardo and T. Gallou¨et, “Nonlinear elliptic equations with right-hand side measures,”
Communications in Partial Differential Equations, vol. 17, no. 3-4, pp. 641–655, 1992.
8 S.-S. Byun and L. Wang, “Lp-estimates for general nonlinear elliptic equations,”Indiana University
Mathematics Journal, vol. 56, no. 6, pp. 3193–3221, 2007.
9 W. Orlicz, “ ¨Ueber eine gewisse Klasse von R¨aumen vom TypusB,”Bulletin International de l’Acad´emie Polonaise S´erie A, vol. 8, pp. 207–220, 1932.
10 R. A. Adams and J. J. F. Fournier,Sobolev Spaces, vol. 140 ofPure and Applied Mathematics, Academic Press, New York, NY, USA, 2nd edition, 2003.
11 A. Benkirane and A. Elmahi, “An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 36, no. 1, pp. 11–24, 1999.
12 S.-S. Byun, F. Yao, and S. Zhou, “Gradient estimates in Orlicz space for nonlinear elliptic equations,”
Journal of Functional Analysis, vol. 255, no. 8, pp. 1851–1873, 2008.
13 A. Cianchi, “Hardy inequalities in Orlicz spaces,”Transactions of the American Mathematical Society, vol. 351, no. 6, pp. 2459–2478, 1999.
14 V. Kokilashvili and M. Krbec,Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, River Edge, NJ, USA, 1991.
15 L. Wang, F. Yao, S. Zhou, and H. Jia, “Optimal regularity for the Poisson equation,”Proceedings of the American Mathematical Society, vol. 137, no. 6, pp. 2037–2047, 2009.
16 M. Weber, “Stochastic processes with value in exponential type Orlicz spaces,” The Annals of Probability, vol. 16, no. 3, pp. 1365–1371, 1988.
17 M. M. Rao and Z. D. Ren,Applications of Orlicz Spaces, vol. 250 ofMonographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2002.
18 E. Acerbi and G. Mingione, “Gradient estimates for a class of parabolic systems,”Duke Mathematical Journal, vol. 136, no. 2, pp. 285–320, 2007.
19 G. Mingione, “The Calder ´on-Zygmund theory for elliptic problems with measure data,”Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, vol. 6, no. 2, pp. 195–261, 2007.
20 G. M. Lieberman, “The natural generalization of the natural conditions of Ladyzhenskaya and uralltseva tseva for elliptic equations,”Communications in Partial Differential Equations, vol. 16, no. 2-3, pp. 311–361, 1991.
21 Y. Chen and L. Wu,Second Order Elliptic Partial Differential Equations and Elliptic Systems, American Mathematical Society, Providence, RI, USA, 1998.
22 M. Giaquinta,Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol. 105 of