Volume 2010, Article ID 680714,23pages doi:10.1155/2010/680714
Research Article
Optimal Interior Partial Regularity for
Nonlinear Elliptic Systems for the Case
1
< m <
2
under Natural Growth Condition
Shuhong Chen
1, 2and Zhong Tan
21Department of Information and Mathematics Sciences, China Jiliang University, Hangzhou,
Zhejiang 310018, China
2School of Mathematical Science, Xiamen University, Xiamen, Fujian 361005, China
Correspondence should be addressed to Shuhong Chen,[email protected]
Received 16 November 2009; Accepted 18 March 2010
Academic Editor: Shusen Ding
Copyrightq2010 S. Chen and Z. Tan. 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 consider the interior regularity for weak solutions of second-order nonlinear elliptic systems with subquadratic growth under natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly the regularity we obtained is optimal.
1. Introduction
In this paper we consider optimal interior partial regularity for the weak solutions of nonlinear elliptic systems with subquadratic growth under natural growth condition of the following type:
−n
α1
DαAαix, u, Du Bix, u, Du, i1, . . . , N inΩ, 1.1
whereΩis a bounded domain inRn,uandB
itaking values inRN, andAαi·,·,·has value inRnN.N > 1, u : Ω → RNDu {D
αui}, 1 ≤ α ≤ n, 1 ≤ i ≤ N stand for the div ofu and 1< m <2. To define weak solution to1.1, one needs to impose certain structural and regularity conditions onAαi and the inhomogeneityBi, as well as to restrictuto a particular class of functions as follows, for 1< m <2,
E1Aα
ix, u, pare differentiable functions inpand there existsL >0 such that
∂A
α i
x, u, p ∂pβj
E2Aαi is uniformly strongly elliptic, that is, for someλ >0, we have
∂Aα i
x, u, p ∂pjβ ξ
i αξ
j β≥λ
1p2m−2/2|ξ|2, ∀x∈Ω, u∈RN, p, ξ∈RnN, 1.3
E3There existsβ∈0,1andK:0,∞ → 0,∞monotone nondecreasing such that
Aαix, ξ, p−Aαix, ξ, p ≤K|ξ||x−x|mξ−ξmβ/m1pm/2 1.4
for allx,x∈Ω, ξ,ξ∈RN, andp∈RnN; without loss of generality, we takeK≥1. FurthermoreE1 allows us to deduce the existence of a functionωt, s : 0,∞× 0,∞→0,∞withωt,0 0 for alltsuch thatt→ωt, sis monotone nondecreasing for fixeds,s→ωt, sis concave and monotone nondecreasing for fixedt, and such that for all x, u∈Ω×RNandp, q∈RnN, we have
Aα
ipjβ
x, u, p−Aα
ipjβ
x, u, q≤C1p2q2m−2/2ωp,p−q, 1.5
E4there exist constantsaandb, such that
Bi
x, u, p≤apmb, 1.6
or E4
Bi
x, u, p≤Cpm−εb, ε >0. 1.7
Definition 1.1. By a weak solution of1.1with structure assumptionsE1–E4 orE4, we mean a vector valued functionu∈W1,mΩ, RN∩L∞Ω, RNsuch that
ΩA
α
ix, u, DuDαϕidx
ΩBix, u, Duϕ
idx, 1.8
for allϕ∈C∞0Ω, RN.
Even under reasonable assumptions onAα
In the class direct proofs, one “freezes the coefficients” with constant coefficients. The solution of the Dirichlet problem associated to these coefficients with boundary datauand the solution itself can then be compared. This procedure was first carried out by Giaquinta and Modica5 .
But the technique of harmonic approximation is to show that a function which is “approximately-harmonic” liesL2close to some harmonic function. This technique has its origins in Simon’s proof6 of the regularity theorem of Allard7 . Which also be used in8
to find a so-calledε-regularity theorem for energy minimizing harmonic maps. The technique of harmonic approximation allows the author to simplify the originalε-regularity theorem due to Schoen and Uhlenbeck9 .
In the remarkable proof whenm ≡ 2 given by Duzaar and Grotowski in 10, the key difference is that the solution is compared not to the solution of the Dirichlet problem for the system with frozen coefficients, but rather to an A-harmonic function which is close to w in L2, where w is a function corresponding from weak solutions. In particular, the optimal regularity result can be obtained. In 11, 12 , we deal with the optimal partial regularity of the weak solution to1.1for the case m > 2 by the method of A-harmonic approximation technique, which is advantage to the result of 13. The extension of A-harmonic approximation technique also can be found in14,15 .
The purpose of this paper is to establish the optimal partial regularity of weak solution to 1.1 under natural growth condition with subquadratic growth, that is, the case of 1 < m < 2, directly. Indeed the main difficulty in our setting is that the exponent of the integral function is negative−1/2<m−2/2<0, which means we cannot use the amplify technique as usually. Motivated by the technique used in16 , where the authors considered the minimizers of nonquadratic functional, we removed the hinder at last. And then with the help ofA-harmonic approximation technique, one can find a∂Aαi/∂pβjx0, ux0,ρ,Dux0,ρ -harmonic function, which is close to a functionw in sense of L2, the functionw is which we defined inLemma 4.2and which is a corresponding function from the weak solutionu. Thanks to the standard results of linear theory presented in Section 2and the elementary inequalities, we obtain the decay estimate of
Φx0, ρ, p0
−
Bρx0
VDu−Vp02dx 1.9
and the optimal regularity. Now we may state the main result.
Theorem 1.2. Letu ∈ W1,mΩ, RN∩L∞Ω, RN m ∈ 1,2be a weak solution of 1.1with
supΩ|u|M. Suppose that the natural growth conditions (E1)–(E4) (or (E4)) and2aM < λhold. Then there existsΩ0that is open inΩandu∈C1,βΩ0, RNforβis defined in (E3). Furthermore,
Ω\Ω0 Σ1Σ2, 1.10
where
Σ1
x0∈Ω: lim inf
ρ→0 − Bρx0
Du−Dux0,ρmdx >0
,
Σ2
x0∈Ω: lim sup
ρ→0
Du
x0,ρ
∞
.
1.11
2. The
A
-Harmonic Approximation Technique and
Preliminary Lemmas
In this section, we present the A-harmonic approximation lemma, the key ingredient in proving our regularity result, and some useful preliminaries will be need in later. At first, we introduce two new functions.
Throughout the paper we will use the functionsV Vp : Rn → Rn andW Wp :
Rn → Rndefined by
Vξ ξ
1|ξ|22−m/4
, Wξ ξ
1|ξ|2−m
, 2.1
for eachξ∈Rnand for anym >1. From the elementary inequality
x2/2−m≤ x1≤21−2−m/2x2/2−m, 2.2
applied to the vectorx 1,|ξ|2−m∈R2we deduce that
1|ξ|22−m/2≤1|ξ|2−m≤2m/21|ξ|22−m/2, 2.3
which immediately yields
|Wξ| ≤ |Vξ| ≤Cm|Wξ|. 2.4
The purpose of introducingWis the fact that in contrast to|V|2/m, the function|W|2/m is a convex function onRk. This can easily be shown as follows. Firstly a direct computation yields thatt → W2/mt t2/m1t2−m−1/m is convex and monotone increasing on0,∞ withW2/m0 0. Secondly we have
W
ξη
2
2/m Wξη 2
2/m
≤W
|ξ|η 2
2/m
≤ W|ξ|
2/mWη2/m
2
|Wξ|2/mWη2/m
2 ,
2.5
for anyξ, η∈Rn.
We use a number of properties ofV Vpwhich can be found in17, Lemma 2.1 .
Lemma 2.1. Letm ∈ 1,2andV, W : Rn → Rn be the functions defined in2.1. Then for any
ξ, η∈Rnandt >0there holds:
iii|Vξη| ≤cm|Vξ||Vη|;
iv m/2|ξ−η| ≤ |Vξ−Vη|/1|ξ|2|η|2m−2/4≤Ck, m|ξ−η|; v|Vξ−Vη| ≤ck, m|Vξ−η|;
vi|Vξ−η| ≤Cm, M|Vξ−Vη|,for allη with|η| ≤M.
The inequalitiesi–iiialso hold if we replaceV byW.
For later purposes we state the following two simple estimates which can easily be deduced fromLemma 2.1iandvi. Forξ, η ∈Rnwith|η| ≤Mwe have for|ξ−η| ≤1 the estimate
ξ−η2≤Cm, M
Vξ−Vη2; 2.6
as for|ξ−η|>1 we have
ξ−ηm≤Cm, MVξ−Vη2. 2.7
The next result we would state is theA-harmonic approximation lemma, which is prove in18 .
Lemma 2.2A-harmonic approximation lemma. Letκ, K be positive constants. Then for any
ε >0there existδδn, N, κ, K, ε∈0,1 with the following property. For any bilinear formAon
RnN which is elliptic in the sense of Legendre-Hadamard with ellipticity constantκand upper bound
K, for anyv∈W1,mB
ρx0, RNsatisfying
−
Bρx0
|WDv|2dx≤γ2≤1,
−
Bρx0
ADv, Dϕdx≤γδsup Bρx0
Dϕ, 2.8
for allϕ∈C10Bρx0, RN, there exists anA-harmonic functionhsatisfying
−
Bρx0
|WDh|2dx≤1, −
Bρx0
W
v−γh ρ
2dx≤γ2ε. 2.9
Definition 2.3. Here a functionhis calledA-harmonic if it satisfies
Bρx0
ADh, Dϕdx0, 2.10
for allϕ∈C10Bρ, RN.
Lemma 2.4. Leth∈W1,1B
ρx0, RNbe such that
Bρx0
ADh, Dϕdx0, 2.11
for anyϕ ∈ C1
0Bρx0, RN, where A ∈ RnN is elliptic in the sense of Legendre-Hadamard with ellipticity constantκand upper boundK. Thenh∈C∞Bρx0, RNand
ρ sup Bρ/2x0
D2h sup Bρ/2x0
|Dh| ≤Ca− Bρ
|Dh|dx, 2.12
where the constantCadepends only onn, N, κ, andK.
The next lemma is a more general version of 17, Lemma 2.7 , which itself is an extension of 3, Lemma 3.1, Chapter V . The proof in which can easily be adapted to the present situation by replacing the condition of homogeneity byLemma 2.1ii.
Lemma 2.5. Let0 ≤ ν < 1, a, b ≥ 0, v ∈ LpB
ρx0, andg be a nonnegative bounded function satisfying
gt≤νgs a
Bρx0
V
v s−t
2dxb, 2.13
for allρ/2≤t < s≤ρ. Then there exists a constantCCνsuch that
gρ 2
≤Cν
a
Bρx0
V
v ρ
2dxb
. 2.14
And then we state a Poincare type inequality involving the functionV, which have been found in17 and, in a sharp way, in18 .
Lemma 2.6Poincare-type inequality. Letm∈1,2andu∈W1,mB
ρ, RN, Bρ⊂Ω, then
−
Bρ
V
u−u
x0,ρ ρ
mdx
1/m
≤CPn, N, m
−
Bρ
|VDu|2dx
1/2
, 2.15
wherem 2n/n−m. In particular, the previous inequality is valid withm replaced by2.
We conclude the section with an algebraic fact can be retrieved again from 16 ,
Lemma 2.7. For everyt∈−1/2,0andμ≥0, one has
1≤
1 0
μ2AsA−A2
t
ds
μ2|A|2A2
t ≤ 8
2t1, 2.16
for anyA,A∈RnN, not both zero ifμ0.
3. A Caccioppoli Second Inequality
Forx0 ∈Ω, u0 ∈RN, p0 ∈RnN, we defineP {pix}, i1,· · ·, N, pi ui0pi0αxα−x0α and we simply writeP u0p0x−x0.
In order to prove the main result, our first aim is to establish a suitable Caccioppoli inequality.
Lemma 3.1Caccioppoli second inequality. Letu∈W1,mΩ, RN∩L∞Ω, RN 1 < m <2
be a weak solution of 1.1withsupΩ|u|Mand2aM < λhold under natural growth conditions (E1)–(E4) (or (E4)). Then for everyx0 ∈ Ω, u0 ∈ RN, p0 ∈ RnN, and arbitraryρwith0 < ρ < min{1,distx0, ∂Ω}, one has
−
Bρ/2x0
V
Du−p02dx≤Cc
−
Bρx0
V
u−u0−p0x−x0 ρ
2dxG
, 3.1
for
GK|u0|p01p0m/2
σ
ρ2βmaxap0mb
2
, ap0mb
m/m−1
ρ2. 3.2
whereσmax{2m/m−2β, m2β/m−1}>2and the constantCcCcn, N, m, L, λ, M.
Proof. Let Bρx0 ⊂ Ω. Choose ρ/2 ≤ t < s ≤ ρ and a standard cut off function η ∈
C1
0Bρx0,0,1 with η ≡ 1 onBtx0, which satisfies |∇η| ≤ 1/s−t. Foru0 ∈ RN and p0∈RnN, let
vu−u0−p0x−x0 3.3
and define
ϕηv, ψ1−ηv. 3.4
Then
and further there holds
Dϕm≤Cm|
Dv|m v s−t
m; Dψm≤Cm
|Dv|m v s−t
m. 3.6
Using hypothesisE2, fromLemma 2.7, and as the elementary inequality
1 3
1|b|2|a|2≤1|a|2|b−a|2≤31|a|2|b|2, 3.7
we can get
Bsx0
Aαix, u, p0Dϕ
−Aαix, u, p0
Dαϕidx
Bsx0 1
0 ∂Aα
i
x, u, p0θDϕ
∂Pβj dθDβϕ
jD αϕidx
≥λ
Bsx0 1
0
1p0θDϕ2
m−2/2
dθDϕ2dx
≥3m−2/2λ
Bsx0
1p02Dϕ2
m−2/2
Dϕ2dx,
3.8
A simple calculation yields
3m−2/2λ
Bsx0
1p02Dϕ2m−2/2Dϕ2dx
≤ −
Bsx0 1
0 ∂Aα
i
x, u, Du−θDψ ∂Pβj dθDβψ
jD αϕidx
−
Bsx0
Aαix, u, p0
−Aαix, P, p0
Dαϕidx
−
Bsx0
Aαix, P, p0−Aiαx0, u0, p0Dαϕidx
Bsx0
Bix, u, Duϕidx
≤IIIIIIIV.
ByE1,Lemma 2.7and3.7, there holds
I≤C
Bsx0
1|Du|2Du−Dψ2m−2/2DψDϕdx. 3.10
Noting that suppDψ ⊂ Bt\Bs and−1/2 < m−2/2 < 0, one can take the domain
Bsx0intoBsx0∩{|Dψ|>1}∩{|Dϕ|>1}, Bsx0∩{|Dψ|>1}∩{|Dϕ| ≤1}, Bsx0∩{|Dψ| ≤ 1}∩{|Dϕ|>1}, andBsx0∩{|Dψ| ≤1}∩{|Dϕ| ≤1},four parts, and then by Young inequality and the estimations2.6and2.7, thus there is
I ≤C1
Bsx0\Btx0
VDu−p02dx
Bsx0
V
v ρ
2dx
. 3.11
From the structure conditionE3yields
II ≤
Bsx0
K|u0|p01p0m/2|v|βDϕdx. 3.12
Similar toI, we split the domain of integration into four parts as follows. And on the partBsx0∩ {|v/s|>1} ∩ {|Dϕ| ≤1}, we see
K|u0|p01p0m/2|v|βDϕ
≤εDϕ2CεK|u0|p01p0m/2|v|β
2
≤εDϕ2Cεv
s
mCεK|u0|p01p0m/2sβ
2m/m−2β
≤εC|VDv|2CεV
v s
2CεK|u0|p01p0m/2
2m/m−2β s2β,
3.13
as on the setBsx0∩ {|v/s| ≤1} ∩ {|Dϕ|>1}, there are
K|u0|p01p0m/2|v|βDϕ
≤K|u0|p01p0m/2sβDϕ
≤εDϕmCεK|u0|p01p0m/2sβm/m−1
≤εC|VDv|2CεV
v s
2CεK|u0|p01p0m/2
m/m−1 s2β,
and on the caseBsx0∩ {|v/s| ≤1} ∩ {|Dϕ| ≤1}, one can get
K|u0|p01p0m/2|v|βDϕ
≤K|u0|p01p0m/2sβDϕ
≤εDϕ2CεK|u0|p01p0m/2sβ
2
≤εC|VDv|2CεV
v s
2CεK|u0|p01p0m/22s2β,
3.15
Finally, noting that supΩ|u| M, then for the caseBsx0∩ {|v/s|> 1} ∩ {|Dϕ| > 1}, there exists a constant 0<2m−1/m2β <1 such that
K|u0|p01p0m/2|v|βDϕ
≤K|u0|p01p0m/2|v|β2m−1/m2β2Mp0s2−m2β/m2βDϕ
≤εDϕmCεK|u0|p01p0m/2m/m−1
× |v|2mβ/m2β2Mp0s2−m2β/m2β·m/m−1 ≤εDϕ2Cεv s
m
CεK|u0|p01p0m/2m2β/m−1s2β2Mp0s2−m2β/m−1
≤ε|VDv|2CεV
v s
2
CεK|u0|p01p0m/2
m2β/m−1
2Mp0s
2−m2β/m−1s2β.
3.16
Combining these estimations onII, we have
II ≤Cε
Bsx0
|VDv|2dxCε
Bsx0
V
v s
2dx
Cε12Mp0s
2−m2β/m−1K|u
0|p01p0m/2
σ
αnsn2β,
3.17
forσmax{2, m/m−1, m2β/m−1, 2m/m−2β}max{m2β/m−1, 2m/m− 2β}.
And noting thatK≥1, and thatm/m−1≥2, and similarly asII, we see
III≤Cε
Bsx0
|VDv|2dxCε
Bsx0
V
v s
2dx
CεK|u0|p01p0m/2β
m/m−1
αnsn2β,
and forμpositive to be fixed later, we have
IV
Bsx0
a|Du|mu−u0−p0x−x0ηdx
Bsx0
vsbsηdx. 3.19
On the part{Bsx0} ∩ {|Du−p0| ≥1} ∩ {|v/s| ≤1}, argue anginous asIIandIII, by Young’s inequality and2.6and2.7, we have
a|Du|mu−u0−p0x−x0ηv s
bsη
≤a1μDu−p0m
11 μ
p0m
u−u0−p0x−x0η
εb2s2η2Cεv s
2
≤a1μ2Mp0s
|VDv|2a
1 1 μ
p0m|v|ηεb2s2η2Cε
V
v
s
2
≤a1μ2Mp0s|VDv|2εa2
1 1 μ
2
p02m
s2η2εb2s2η2CεV
v s
2. 3.20
Similarly, on the part{Bsx0} ∩ {|Du−p0| ≥1} ∩ {|v/s| ≤1}, we see
a|Du|mu−u0−p0x−x0η
vsbsη
≤a1μ2Mp0s|VDv|2εsm/m−1ηm/m−1
×
bm/m−1am/m−1
1 1 μ
m/m−1 p0m
2/m−1
CεV
v s
2,
3.21
and on the part{Bsx0} ∩ {|Du−p0| ≥1} ∩ {|v/s| ≤1},
a|Du|mu−u0−p0x−x0η
vsbsη
≤a1μDu−p0m|v|η
a
1 1 μ
p0mb
≤εs2Du−p02Cε2Mp0s2m−1/2−ma1μ2/2−mv s 2 ε a
1 1 μ
p0mb
2
s2η2Cεv s
2
≤ε|VDv|2Cε2Mp0s
2m−1/2−ma1μ2/2−mVv s 2 ε a
1 1 μ
p0mb
2
s2η2CεV
v s 2, 3.22
and on the part{Bsx0} ∩ {|Du−p0| ≥1} ∩ {|v/s| ≥1},
a|Du|mu−u0−p0x−x0ηv s
bsη
≤εs2Du−p02Cε
2Mp0s
2−2mm2/2−m
a1μ2/2−mv s m ε a
1 1 μ
p0mb
m/m−1
sm/m−1ηm/m−1Cεv s
m
≤ε|VDv|2Cε2Mp0s
2−2mm2/2−m
a1μ2/2−mV
v s 2 ε a
1 1 μ
p0mb
m/m−1
sm/m−1ηm/m−1CεV
v s 2. 3.23
Combining these estimates inIV, and noting thatm/m−1>2 ands≤1, η≤1, we have
IV ≤Cε, M, a
Bsx0
|VDv|2dxCε, M, m, a
Bsx0
V v s 2dx max a
1 1 μ
p0mb
2 ,
a
1 1 μ
p0mb
m/m−1 αnsn2.
3.24
Finally, onBtx0we useLemma 2.1ivandvito bound the integrand of the left-hand side of3.9from below:
λ1p02Dϕ2
m−2/2
Dϕ2λ1p02|Dv|2
m−2/2
|Dv|2
≥Cmλ1p02|Du|2m−2/2Du−p02
≥Cn, N, m, λVDu−Vp02
≥Cn, N, m, λ, M|VDv|2.
Using this in3.9together with the estimatesI,II,III, andIV we finally arrive at
C1
Btx0
|VDv|2dx≤C2
Bsx0\Btx0
|VDv|2V
v s−t
2
dx
C3
Bsx0
|VDv|2V
v s−t
2
dxC4
K|u0|p01p0m/2
σ
αnsn2β
C5max
a
1 1 μ
p0mb
2 ,
a
1 1 μ
p0mb
m/m−1 αnsn2.
3.26
The proof is now completed by applyingLemma 2.5.
4. The Proof of the Main Theorem
In this section we proceed to the proof of the partial regularity result and hence consider u∈W1,mΩ, RN∩L∞Ω, RN 1 < m <2to be a weak solution of1.1. Then we have the following.
Lemma 4.1. Considerρ <1andϕ ∈C∞0 Bρx0, RNwithsupBρx0|Dϕ| ≤ 1. Furthermore fixed
p0 in RnN and set u0 u x0,ρ −
Bρx0udxand Φx0, ρ, p0 ≤ 1. Then for the weak solutionu ∈ W1,mΩ, RN∩L∞Ω, RN1 < m <2to systems1.1withsup
Ω|u| Mand2aM < λbeing
hold, there holds
Bρx0
⎡ ⎣∂Aαi
∂pjβ
x0, u0, p0
Du−p0
⎤⎦
·Dαϕidx
≤Ceαnρn
ω1/2p0, Φ1/2x0, ρ, p0Φ1/2x0, ρ, p0 Φx0, ρ, p0ρβHp0 4.1
forCeCeCp, N, n, Land where one defines
Φx0, ρ, p0−
Bρx0
VDu−V
p02dx,
Ht KM t1Mtm/2σ,
4.2
Proof. We assume initially that supBρx
0|Dϕ| ≤ 1. ApplyingLemma 2.7and noting that the definition of the weak solution of1.1, for 0≤t≤1, we deduce
Bρx0
⎡ ⎣ 1
0 ∂Aα
i
∂pjβ
x0, u0, p0tDu−p0dtDu−p0
⎤
⎦·Dαϕidx
Bρx0
Aαix0, u0, Du−Aαix0, u0, p0·Dαϕidx
Bρx0
Aαix0, u0, Du−Aαix, u, Du
·Dαϕidx Bρx0
Bi·, u, Du·ϕidx.
4.3
Rearranging this, we find
Bρx0
⎡ ⎣∂Aαi
∂pjβ
x0, u0, p0Du−p0
⎤
⎦·Dαϕidx
Bρx0
⎡ ⎣ 1
0 ∂Aαi
∂pjβ
x0, u0, p0
dtDu−p0
⎤⎦
·Dαϕidx
Bρx0
⎡ ⎣ 1
0
⎛ ⎝∂Aαi
∂pjβ
x0, u0, p0−∂A
α i
∂pjβ
x0, u0, p0tDu−p0
⎞
⎠dtDu−p0 ⎤ ⎦
·Dαϕidx Bρx0
Aαix0, u0, Du−Aαix, u0p0x−x0, Du·Dαϕidx
Bρx0
Aαix, u0p0x−x0, Du
−Aαix, u, Du·Dαϕidx
Bρx0
Bi·, u, Du·ϕidx
IIIIIIIV.
4.4
Using the structure conditionE1and the estimate1.5for the modulus of continuity of∂Aα
i/∂p j
β, byLemma 2.7and let
s1Bρx0∩%Du−p0≤1
&
, s2Bρx0∩%Du−p0>1
&
we can derive
I
Bρx0 1
0
L1p02m−2/2L1p0tDu−p02m−2/2
1/2
·
L1p02p0tDu−p02m−2/2ωp0,tDu−p0
1/2
dtDu−p0dx
≤C
s1
ω1/2p0,Du−p0Du−p0dxC
s2
ω1/2p0,Du−p0Du−p0m/2dx. 4.6
Noting that the estimates 2.6 and 2.7, using first H ¨older’s inequality and then Jensen’s inequality:
I ≤CαnρnΦ1/2
x0, ρ, p0
ω1/2p0,Φ1/2
x0, ρ, p0
, 4.7
here we have usedΦ1/mx
0, ρ, p0≤Φ1/2x0, ρ, p0forΦx0, ρ, p0≤1.
By E3, Young inequality, 2.6, 2.7, and noting the function K monotone nondecreasing andKM|p0|≥1 and thatρ≤1, we can estimateIIas follows
II≤
Bρx0
K|u0|p0ρβ1p0β1|Du|m/2dx
≤KMp0ρβ
1p0βm/2αnρnβ
KMp01p0β
2 αnρn2β
Φx0, ρ, p0
αnρn
KMp01p0β
4/4−m
αnρn4β/4−m
≤Φx0, ρ, p0
αnρn2
KMp01p0βm/2
σ
αnρnβ,
4.8
for 1<4/4−m≤2< σ.
Similar to 3.11, to estimate III, one can divide the domain Bρx0 as previously mentioned. On the setBρx0∩{|v/ρ|>1}∩{|Du−p0| ≤1}, form/m−β<2m/m−2β≤σ,
K|u0|p01|Du|m/2|v|β
≤KMp01p0m/2|v|βKMp0|v|β
≤2εv ρ
mCεKMp01p0m/2
m/m−β
ρmβ/m−β
CεKMp0m/m−βρmβ/m−β
≤2εCV
v ρ
22CεKMp01p0m/2
σ
ρβ,
while on the partBsx0∩ {|v/ρ| ≤1} ∩ {|Du−p0|>1}and noting that 1<2/2−β<2< σ,
K|u0|p01|Du|m/2|v|β
≤KMp01p0m/2ρβ
vρβKMp0ρβDu−p0m/2
≤εv ρ
2CεKMp01p0m/2
2/2−β
ρ2β/2−β
εDu−p0mCε
KMp02ρ2β
≤εCV
v ρ
2εC|VDv|2CεKMp01p0m/2
σ
ρβ.
4.10
OnBsx0∩ {|v/ρ| ≤1} ∩ {|Du−p0| ≤1}
K|u0|p01|Du|m/2|v|β
≤KMp01p0m/2|v|βK
Mp0|v|β
≤εv ρ
2CεKMp01p0m/2
2/2−β
ρ2β/2−β
≤εCV
v ρ
2CεKMp01p0m/2
σ
ρβ.
4.11
Finally, on the case Bsx0∩ {|v/ρ| > 1} ∩ {|Du−p0| > 1}, there exists a constant 0< m/mβ<1 such that
K|u0|p01|Du|m/2|v|β
≤KMp01p0m/2
|v|βm/mβ2Mp0ρ
β2/mβ
KMp0Du−p0m/2
|v|βm/mβ2Mp0ρ
β2/mβ
≤Cεv
ρ
mCεDu−p0mCε
KMp01p0m/2
mβ/m
2Mp0ρ
β2/m ρβ
CεKMp02mβ/m−β
2Mp0ρ
β2/m−β
ρ2mβ/m−β
≤CεV
v ρ
2Cε|VDv|2Cε, n, NKMp01Mp0m/2
σ
ρβ,
4.12
Whereas,Lemma 2.1yields
III≤Cε
Bρx0
VDu−Vp02
dxCε
BSx0
V
v ρ
2dx
Cε, n, NKMp01Mp0m/2
σ
αnρnβ,
4.13
whereσis defined inLemma 3.1. Noting that supBρx
0|ϕ| ≤ρ≤1, and by Young’s inequality, we see
IV ≤C
Bρx0
a|Du|mbϕdx
≤C
Bρx0
aDu−p0mϕdxC
Bρx0
bp0mρdx.
4.14
OnD1{Bρx0∩ {|Du−p0|>1}}, by2.7and Young inequality, we have
4.14≤C
D1
aVDu−Vp02dxCαnρn1
bp0m
. 4.15
On the other hand, onD2 {Bρx0∩ {|Du−p0| ≤1}}, using2.6and Young inequality, we have
Du−p0m≤
Du−p021≤VDu−Vp021. 4.16
Thus
4.14≤C
D2
aVDu−Vp02dxCαnρn1
abp0m
. 4.17
Combining these estimates and noting that definition ofHt, we derive
IV ≤C
Bρx0
VDu−Vp02
dxCαnρn1Hp0. 4.18
ByLemma 2.6, there is
III ≤Cε, CP, n, N Bρx0
VDu−Vp02dxCεHp0αnρnβ. 4.19
We next establish an initial excess-improvement estimate, assuming that the excess Φρis initially sufficient small. We also defineΓρ
Φρ 4δ−2H2ρ2β, wx ux−
ux0,ρ−γhx0−Dux0,ρx−x0, andγC6CeΓρ, whereC6stands for the constantsCm, M formLemma 2.1vi. The precise statement is the following.
Lemma 4.2excess-improvement. Consider weak solutionu∈W1,mΩ, RN∩L∞Ω, RN 1<
m < 2 satisfying the conditions of Theorem 1.2 and β fixed in (E3). Then we can find positive constantsCi, Ck, andδ,andθ∈0,1/4 (withCidepends only onn,N,m,λ,andLand withCk,
δandθdepending only on these quantities as well asβ) such that the smallness conditionρ∈0, ρ :
2√2Caγ≤1,
ω1/2Dux0,ρ,Φ1/2ρ Φ1/2ρ≤ δ 2,
Dux0,ρ
≤M1, for given constant0≤M1 <∞
2CiρβH
1Dux0,ρ≤ δ 2,
CeCaΦ
ρ≤1,
4.20
together imply the growth condition
Φθρ≤θ2βΦρCkρ2βH2
1Dux0,ρ. 4.21
Here one uses the abbreviateΦρ Φx0, ρ,Dux0,ρ.
Proof. Forε >0 to be determined later, we takeδδn, N, λ,Λ, ε∈0,1to be corresponding constant from theA-harmonic approximation lemma, that is,Lemma 2.2, and set
wx ux−ux0,ρ−γhx0
−Dux0,ρx−x0,
Γρ
Φρ4δ−2H2ρ2β, γC
6CeΓ
ρ.
4.22
whereC6stands for the constantCm, MfromLemma 2.1vi. Then, from2.4andLemma 2.1vi, we have
−
Bρx0
|WDw|2dx≤ −
Bρx0
|VDw|2dx≤C6Φ2ρ≤γ2. 4.23
And byLemma 4.1and the smallness condition
ω1/2Dux0,ρ
,Φ1/2ρ Φ1/2ρ≤ δ
we can deduce
−Bρx0
⎡ ⎣∂Aαi
∂pβj
x0, ux0,ρ,Dux0,ρ
Dw
⎤
⎦·Dαϕidx
≤γ
ω1/2Du
x0,ρ
,Φ1/2ρΦ1/2ρ ΦρρβHDu x0,ρ
C1Γρ Bsupρx0
Dϕ
≤γ
ω1/2Dux0,ρ,Φ1/2ρ Φ1/2ρ δ 2
sup Bρx0
Dϕ
≤γδsup
Bρx0
Dϕ.
4.25
Inequalities 4.23 and 4.25 fulfill the condition of A-harmonic approximation lemma, which allow us to apply Lemma 2.2. Therefore we can find a function h ∈ W1,mB
ρx0, RNwhich is∂Aαi/∂pβjx0, ux0,ρ,Dux0,ρ-harmonic such that
−
Bρx0
|WDh|2dx≤1, −
Bρx0
W
w−γh
ρ
2dx≤γ2ε. 4.26
With the help ofLemma 2.1iiiandv, we have
Φ2θρ−
Bθρx0
VDu−VDux0,θρ2dx
≤C−
Bθρx0
VDu−Dux0,θρ2dx
≤C−
Bθρx0
VDu−Dux0,ρ−γDhx02dx
CVDux0,θρ−Dux0,ρ−γDhx02,
4.27
where the constantCdepends only onn, N, andm.
We proceed to estimate the right-hand side of4.27. DecomposingBθρx0into the set with|Du−Dux0,ρ−γDhx0| ≤ 1 and that with|Du−Dux0,ρ−γDhx0|> 1, that using
Lemma 2.1iand H ¨older inequality, we obtain
Dux0,θρ−Dux0,ρ−γDhx0
≤ −
Bθρx0
Du−Dux0,ρ−γDhx0dx√2I1/2I1/m,
where we have abbreviated
I−
Bθρx0
VDu−Dux0,ρ−γDhx02dx. 4.29
Now, since|VA| V|A|andt → Vtis monotone increasing, we deduce from 4.27, also by usingLemma 2.1iandii, that there holds
Φ2θρ≤CIV2I1/2I1/m≤CII2/m, 4.30
whereCdepends only onn, N, andm. Therefore it remains for us to estimate the quantity I. By considering the cases|Dh| ≤1 and|Dh|>1 seperately and keeping in mind4.26, we haveusingLemma 2.1i:
−
Bρx0
|Dh|dx≤2√2. 4.31
Using the assumption|Dux0,ρ| ≤M1andLemma 2.4, this shows
Dux0,ργ|Dhx0| ≤M1γ|Dhx0|
≤M1γCa−
Bρx0
|Dh|dx
≤M12
√
2γCa
≤M11.
4.32
Lemma 3.1applied onBθρx0with ux0,ρ, respectivelyDux0,ργDhx0, instead of u0, respectively,p0; note that the constantCcdepends only onn, N, m, L, λ, M:
I≤Cc
⎡ ⎢ ⎣−
B2θρx0
V
⎛ ⎜
⎝u−ux0,ρ−
Dux0,ργDhx0x−x0 2θρ
⎞ ⎟ ⎠
2
dxG
⎤ ⎥
⎦, 4.33
for
G
Kux0,ρ
Dux0,ργDhx0
1Dux0,ργDhx0m/2
σ 2θρ2β
Lemma 2.1iiiyields
−
B2θρx0
V ⎛ ⎜
⎝u−ux0,ρ−
Dux0,ργDhx0
x−x0
2θρ ⎞ ⎟ ⎠ 2 dx ≤ −
B2θρx0
V
u−ux0,ρ−γhx0
−Dux0,ρx−x0−γhx γhx
2θρ
−γhx0−γDhx0x−x0 2θρ 2dx ≤C −
B2θρx0
V
w−γhx
2θρ
2V
γh−h0−Dhx0x−x0 2θρ 2 dx , 4.35
where the constantCis given byCmCc. To estimate the right-hand side of4.33we use
2.4,Lemma 2.1ii note that 1/2θ≥1and4.26to infer
−
B2θρx0
V
w−γh
2θρ
2dx≤Cm−
B2θρx0
W
w−γh
2θρ
2dx
≤Cm2θ−n−
Bρx0
W
w−γh 2θρ
2dx
≤Cm2θ−n−2−
Bρx0
W
w−γh
ρ
2dx
≤Cm2−n−2θ−n−2γ2ε.
4.36
UsingLemma 2.1i, Taylor’s theorem applied tohonB2θρx0,Lemma 2.4and4.31, we obtain
−
B2θρx0
V
γh−h0−Dhx0x−x0 2θρ
2dx
≤γ2−
B2θρx0
h−h0−Dhx0x−x0 2θρ
2dx
≤ γ2
4θ2ρ2 sup
B2θρx0
|hx−hx0−Dhx0x−x0|2
≤8C2aθ2γ2.
Using the smallness condition 2√2Caγ ≤1 and4.32together with the definition of
Hyields
Kux0,ρ
Dux0,ργDhx01Dux0,ργDhx0m/2
σ 2θρ2β
≤
KMDux0,ρ12Dux0,ρm/2 σ
2θρ2β
≤H1Dux0,ρ
2θρ2β,
max+aDux0,ργDhx0
mb2,aDux0,ργDhx0
mbm/m−1,2θρ2
≤H1Dux0,ρ2θρ2β.
4.38
Combining all the above estimates with4.33, and letεθn4forθ∈0,1/4 , we get
I≤Ci
θ2γ2H1Dux0,ρ2θρ2β, 4.39
where the constantCidepends only onn, N, L, m, λ, M, andθthe dependency fromθoccurs due to the fact thatδdepends onθ. Chooseθ ∈0,1/4suitable such thatCiθ2 ≤θ2β, and inserting this into4.30we easily findrecalling also thatΦρ≤1:
Φθρ≤θ2βΦρCkH2
1Dux0,ρρ2β, 4.40
where the constantCkhas the same dependencies asCi.
The regularity result then follows from the fact that this excess-decay estimate for any xin a neighborhood ofx0. From this estimate we concludeby Campanato’s characterization of H ¨older continuous functions 19, 20 that VDu has the modulus of continuity ρ → Φx0, ρby a constant timesρ2β. ByLemma 2.1ivthis modulus of continuity carries over to Du.
Acknowledgments
This work was supported by NCETXMU and the National Natural Science Foundation of China-NSAFno: 10976026.
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