Optimal partial regularity for nonlinear sub-elliptic systems with Dini continuous coefficients in Carnot groups

R E S E A R C H

Open Access

Optimal partial regularity for nonlinear

sub-elliptic systems with Dini continuous

coefficients in Carnot groups

Jialin Wang

_{and Dongni Liao}

*_{Correspondence:}

[email protected] School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi 341000, P.R. China

Abstract

This paper is concerned with the interior regularity for nonlinear sub-elliptic systems with Dini continuous coefficients under superquadratic controllable growth conditions in Carnot groups. We adapt the technique of theA-harmonic

approximation to the case of sub-elliptic systems in divergence form, and we show a partial regularity result for weak solutions. In particular, our result is optimal in the sense that in the case of Hölder continuous coefficients we obtain directly the optimal Hölder exponent for the horizontal gradient of weak solutions on its regular set.

Keywords: optimal partial regularity; Dini continuity coefficient; controllable growth condition; Carnot group; sub-elliptic system

1 Introduction and statements of main results

In this paper, we consider the following nonlinear sub-elliptic systems under superqua-dratic controllable growth (m> ) conditions in Carnot groupsGin divergence form:

–

k

i=

XiAαi(ξ,u,Xu) =B α

(ξ,u,Xu) in,α= , , . . . ,N, (.)

whereis a bounded domain inG,X={X, . . . ,Xk}withXi(i= , . . . ,k) to see the next

section (.) below,u= (u_{, . . . ,u}N_{) :→R}N_,Aα

i(ξ,u,p) :×RN ×Rk×N→Rk×N, and

Bα_{(ξ,u,p) :×R}N_×Rk×N_→RN_.

Under the coefficientsAα

i assumed to be Dini continuous, the purpose of this paper is

to establish optimal partial regularity to the system (.) under the superquadratic con-trollable growth conditions. Such an assumption is much weaker than the assumption of Hölder continuity; see [, ] for the case of sub-elliptic systems. More precisely, we assume for the continuity ofAα

i with respect to the variables (ξ,u) that

 +|p|–m_Aα

i(ξ,u,p) –Aαi(ξ˜,u˜,p)≤κ

|u|μd(ξ,ξ˜) +|u–u˜| (.)

for allξ,ξ˜∈,u,u˜ ∈RN_{, andp∈R}k×N_{, whereκ: (, +∞)→[, +∞) is nondecreasing,}

andμ: (, +∞)→[, +∞) is nondecreasing and concave withμ(+) = . We also require

thatr→r–γ_{μ(r) is nonincreasing for someγ∈(, ) and that}

M(r) =

ˆ r



μ(ρ)

ρ dρ<∞, for somer> . (.)

We employ the method of anA-harmonic approximation to investigate the sub-elliptic system (.) under this weaker assumption, and we establish optimal partial regular-ity results: Roughly speaking, assume additionally to the standard hypotheses (see pre-cisely (H), (H), and (H) below) that ( +|p|)–m_Aα

i(ξ,u,p) satisfies (.) and (.). Let

u∈HW,m_(,RN_{) be a weak solution of (.). Thenuis of classC}_{outside a closed} singu-lar setSingu⊂of Haar measure . Furthermore, forξ∈\Singu, the derivativeXuof uhas the modulus of continuityr→M(r) in a neighborhood ofξ. Our result is optimal in the sense that whenμ(ρ) =ργ,  <γ < , we haveM(r) =γ–rγ_{, and},γ _{regularity is}

known to be optimal in that case; see [].

As is well known, one, in general, cannot expect that weak solutions of the sub-elliptic system (.) will be classical,i.e. C_{-solutions, even under reasonable assumptions onA}α i

andBα_{. This was first shown by De Giorgi [] in the Euclidean space; we also refer the}

reader to Giaquinta [], and Chen and Wu [] for further discussion and additional ex-amples. Then the goal is to establish partial regularity theory. Moreover, a new method calledA-harmonic approximation technique is introduced by Duzaar and Steffen in [], and simplified by Duzaar and Grotowski in [], to study elliptic systems with quadratic growth case. Then similar results have been proved for more generalAα

i orBαin the

Eu-clidean setting; see [–] for Hölder continuous coefficients, and [–] for Dini con-tinuous coefficients.

However, turning to sub-elliptic equations and systems in Carnot groups, some new difficulties will arise due to non-commutativity of the horizontal vectorsXi. With regard

to cases of sub-elliptic equations, we refer the reader to Domokos [–], Capogna [, ], Manfredi and Mingione [], and Mingioneet al.[] for more interesting results and details. Several results were focused on sub-elliptic systems. Capogna and Garofalo [], and Shores [] considered the quadratic growth case, and Föglein [], Wang and Niu [], and Wang and Liao [] treated non-quadratic cases. We note that the assumption of Hölder continuity on the coefficientsAα

i is required for those regularity results of

sub-elliptic systems mentioned above.

When the assumption of Hölder continuity ofAα_i is weakened to Dini continuity, one may ask how to establish partial regularity for the nonlinear sub-elliptic systems under superquadratic controllable growth conditions in the Carnot groups. This paper is de-voted to this topic. To define weak solutions to (.), we assume the following structure conditions onAα

i andBα:

(H) The termAα

i(ξ,u,p) is differentiable inp, and there exists some constantLsuch

that

Aα

i,pj_β(ξ,u,p)≤L

 +|p|

m–

 _{, (ξ,u,p)}∈×_RN_×Rk×N_{,m> , (.)}

where we writeAα

i,pj_β(ξ,u,p) =

∂Aα_i(ξ,u,p)

∂pj_β .

(H) The termAα

i(ξ,u,p) is uniformly elliptic,i.e.for someλ>  we have

Aα

i,pj_β(ξ,u,p)η α iη

β j ≥λ

(H) There exist a modulus of continuityμ: (, +∞)→[, +∞) and a nondecreasing functionκ: [, +∞)→[, +∞) such that

 +|p|–

m _Aα

i(ξ,u,p) –A α

i(ξ˜,u˜,p)≤κ

|u|μd(ξ,ξ˜) +|u–u˜|. (.)

(H) The termBα_{satisfies the superquadratic controllable growth condition}

Bα(ξ,u,p)≤c +|u|r–+|p|m(r–)/r, (.)

wherecis a positive constant, andr=_QmQ_–m ifm<Q; orQ≤r< +∞ifm=Q. Without loss of generality we can assumeκ≥ and that

(μ) μis nondecreasing withμ(+) = ,μ() = ;

(μ) μis concave, in the proof of the regularity result we have to require thatr→r–γ_μ(r) is nonincreasing for some exponentγ ∈(, );

(μ) Dini’s conditionM(r) =´_rμ_ρ(ρ)dρ<∞for somer> .

Note that (H) one infers

Aα_i(ξ,u,p) –Aα_i(ξ,u,p˜)≤C(L) +|p|+|˜p|

m–

 |_p–p˜|_{, (.)}

and there exists a continuously nonnegative and bounded function ω(s,t) : [,∞)× [,∞)→[,∞), whereω(s, ) =  for alls, andω(s,t) is monotonously nondecreasing insfor fixedt;ω(s,t) is concave and monotonously nondecreasing intfor fixeds, such that for all (ξ,u,p), (ξ˜,u˜,p˜)∈×RN_×Rk×N_,

Aα

i,pj_β(ξ,u,p) –A α

i,pj_β(ξ,u,p˜)≤C

 +|p|+|˜p|

m–

 _ω|_p|_,|_p–p˜|_{. (.)}

Further (H) allows us to deduce the following inequality; see [] for more details:

Aα_i(ξ,u,p) –Aα_i(ξ,u,p˜)(p–p˜)≥λ

|p–p˜|+|p–p˜|m, (.)

with a positive constantλ.

To obtain partial regularity for weak solutions, the key point is to establish an excess-decay estimate for the functional

(ξ,ρ,p) =

Bρ(ξ)

|Xu–p|+|Xu–p|mdξ, (.)

here we writeffl_Bρ(ξ

)u dξ=|Br(ξ)|

–

G

´

Bρ(ξ)u dξ, which itself leads to the continuity of the

horizontal gradientXuof the weak solutionuvia the integral characterization of continu-ity by Campanato; see []. Since the non-commutative basic vector fields{Xi}of Lie

alge-bras corresponding to the Carnot group are more complicated than gradient vector fields

{ ∂

∂xi}in the Euclidean space, we have to find different auxiliary functions in treating

Theorem  Assume that coefficients Aα

i and Bα satisfy (H)-(H), (μ)-(μ). Let u∈

HW,m_(,RN_{)be a weak solution to the system(.),i.e.,}

ˆ

Aα_i(ξ,u,Xu)Xiϕαdξ=

ˆ

Bα(ξ,u,Xu)ϕαdξ ∀ϕ∈C_∞,RN. (.)

Then there exists a relatively closed setSingu⊂such that u∈C_(\Singu,RN_).

Fur-thermore,Singu⊂∪and Haar meas(\Singu) = ,where

=

ξ∈:sup

r>

|uξ,r|+(Xu)ξ,r=∞

,

=

ξ∈: lim

r→+inf Br(ξ)

Xu– (Xu)ξ,r



dξ>  .

In addition,forτ∈[γ, )andξ∈\Singu the horizontal derivative Xu has the modulus of continuity r→rτ_{+M(r)in a neighborhood ofξ}

.

2 Preliminaries

A Carnot groupGof stepris a simply connected, nilpotent Lie group whose Lie algebraG admits a stratification,i.e.,G=V_⊕V_{⊕ · · · ⊕V}r_{, such that [V}_,Vj_{] =V}j+_{,j= , . . . ,r– ,} and [V_,Vr_{] ={}. LetX}l

i denote a left-invariant basis vector field ofVl with ≤l≤r

and ≤i≤ml, wheremlis the dimension ofVl. For the sake of simplicity, we letXi=Xi,

k=m,X={X, . . . ,Xk}be the basis of left-invariant vector fields ofV, and denote by∇G=

(X, . . . ,Xk) the sub-elliptic gradient. We will say thatXi(i= , . . . ,k) are the horizontal

vector fields with the form

Xi=∂i+ n

j=i+

aij(ξ)∂j, Xi() =∂i, (.)

whereaij(ξ) is a polynomial inξ. For a vector valued functionu= (u, . . . ,uN) :G→RN,

we letXiuα(i= , . . . ,k,α= , . . . ,N) be a horizontal direction derivative, and say thatXu

is the horizontal Jacobian with entriesXiuα.

Denoting

ξ=ξ,ξ, . . . ,ξr=x_,x_, . . . ,x_m;x_, . . . ,x_m; . . . ;xr_, . . . ,xr_m

l

∈G,

the distance from the origin is defined by

d(ξ) =

_r

l=

_m l

i=

xl_i

r! lr!

. (.)

For anyξ,η∈G, we setd(ξ,η) =d(η–◦ξ), whereη–= –η= (–η, . . . , –ηr) is the inverse ofη, and◦is the multiplication rule inGdefined byξ◦ξ=ξ+ξ+P(ξ,ξ),ξ,ξ∈G, where

P:G×G→Ghas polynomial components.

We denote by ωG =|B(ξ, )|G the volume of unit pseudo-ball. Then the volume of

the pseudo-ball BR(ξ) ={η∈G|d(ξ,ξ) <R} is given by |B(ξ,R)|G =ωGRQ, whereQ= r

The horizontal Sobolev spaceHW,p_{() (≤p<∞) is defined as}

HW,p() =u∈Lp() :Xiu∈Lp(),i= , , . . . ,k

.

ThenHW,p() is a Banach space with the norm

uHW,p₍₎=u_Lp₍₎+ k

i=

XiuLp₍₎. (.)

Lu [] showed the following Poincaré type inequality related to Hörmander’s vector fields foru∈HW,q_(B

R(ξ)),  <q<Q, ≤p≤_QqQ_–q:

BR(ξ)|

u–uξ,R|pdξ 

p

≤CpR

BR(ξ)| Xu|qdξ

 q

, (.)

whereuξ,R=

ffl

Br(ξ)u dξ=|Br(ξ)|

–

G

´

Br(ξ)u dξ. Note the fact that the horizontal vectors

Xidefined in (.) fit the Hörmander’s vector fields, and that (.) is valid forp=q=m

(≥).

Let  <m,q<∞,f ∈Lm()∩Lq(). By Hölder’s inequality, it follows for anyrwith m<r<q

fr≤ fm–αfαq, (.)

where  <α<  with _r=–_mα+α_q.

Following [], for technical convenience, lettingη(t) =μ(√t), we have the correspond-ing properties forη: (η)ηis continuous, nondecreasing andη() = ; (η)ηis concave, andr→r–γ_{η(r) is nonincreasing for some exponentγ ∈(, ); (η)H(r) := M}₍√_{r) =} [´_r

√

η(ρ)

ρ dρ]

_{<∞for somer> . Changingκby a constant, but keepingκ≥, we can} assume that (η)η() = , implyingη(t)≥tfort∈[, ].

From the fact thatηis nondecreasing we concludesη(t)≤sη(t) for all ≤t≤s. We use the nonincreasing property ofr→η(_rr) andη()≤. Combining both cases we get

sη(t)≤sη(s) +t, s∈[, ],t> . (.)

From (η), we deduce forθ∈(, ),t> ,j∈N∪ {}, 

γ

 –θγη/θjt=

ˆ θj_t

θ(j+)t τγ–η

/_(θj_t)

(θj_t)γ/ dτ≤

ˆ θj_t

θ(j+)t η/_(τ)

τ dτ,

which implies

∞

j=

η/θjt≤ γ ( –θγ₎

ˆ t



η/_(τ)

τ dτ = γ

( –θγ₎H

/_(t).

It yields particularly thatη(t)≤γ_H(t) for allt≤, andt→t–γ_{H(t) is also}

In the sequel, we let ρ(s,t) = ( +s+t)–κ(s+t)–, andK(s,t) = ( +t)mκ(s+t) for s,t≥. The constantCmay vary from line to line. Note thatρ≤ and thats→ρ(s,t), t→ρ(s,t) are nonincreasing functions.

3 Caccioppoli type inequality

In this section, we begin by introducing theA-harmonic approximation lemma, and the proof is similar to [], also see Föglein [] withp=  in the Heisenberg group. We mainly prove the Caccioppoli type inequality for weak solutions of the systems (.) with control-lable growth conditions.

Lemma  Letλand L be fixed positive numbers,and k,N∈Nwith k≥.Assume for any givenε> ,there existsδ=δ(k,N,λ,ε)∈(, ]with the following properties:

(I) for anyA∈Bil(Rk×N_)satisfying

A(ν,ν)≥λ|ν| and A(ν,ν¯)≤L|ν||¯ν|, ν,ν¯∈Rk×N, (.)

(II) for anyw∈HW,(Bρ(ξ),RN)satisfying

Bρ(ξ)|

Xw|dξ≤ and (.)

_Bρ(ξ)A(Xw,Xϕ)dξ≤δ sup

Bρ(ξ)|

Xϕ|, ∀ϕ∈C_Bρ(ξ),RN

. (.)

Then there exists anA-harmonic function h such that

Bρ(ξ)|

Xh|dξ≤ and ρ– Bρ(ξ)|

h–w|dξ≤ε. (.)

Shores in [] established the following prior estimate for the weak solutionuto the sub-elliptic systems with constant coefficients in Carnot groups:

sup

Bρ/(ξ)

|u|+ρ|Xu|+ρXu≤Cρ Bρ(ξ)

|Xu|dξ. (.)

Lemma  Let u∈HW,m_(,RN_{)be a weak solution to the system(.)under the conditions}

(H)-(H)and(μ)-(μ).Then for everyξ= (ξ,ξ, . . . ,ξr)∈,u∈RN,p∈Rk×N and  <ρ<R<ρ_m/(|u|,|p|)≤such that BR(ξ)⊂⊂,the inequality

ˆ

Bρ(ξ)

|Xu–p|+|Xu–p|mdξ

≤ Cc

(R–ρ)

ˆ

BR(ξ)

u–u–ξ–ξ_pdξ

+ Cc

(R–ρ)m

ˆ

BR(ξ)

u–u–ξ–ξ_pmdξ

+CcωGRQK

|u|,|p|ηR+Cc ˆ

BR(ξ)

 +|u|r+|Xu|mdξ m(r–)

r(m–)

(.)

holds,whereξ_{= (ξ}

Proof We test the sub-elliptic system (.) with the testing functionϕ=φ_vwithv=u– u– (ξ–ξ)p, whereφ∈C∞(BR(ξ),RN) is a cut-off function satisfying ≤φ≤,|Xφ| ≤

C

R–ρ, andφ≡ onBρ(ξ). This yields

ˆ

BR(ξ)

Aα_i(ξ,u,Xu)φ(Xu–p)dξ= –

ˆ

BR(ξ)

φXφAα_i(ξ,u,Xu)v dξ

+

ˆ

BR(ξ)

Bα_(ξ,u,Xu)ϕα_dξ.

Adding this to the equations

–

ˆ

BR(ξ)

Aα_i(ξ,u,p)φ(Xu–p)dξ= 

ˆ

BR(ξ)

φXφAα_i(ξ,u,p)v dξ

–

ˆ

BR(ξ)

Aα

i(ξ,u,p)Xϕαdξ

and

 =

ˆ

BR(ξ)

Aα_i(ξ,u,p)Xϕα, (.)

we have

ˆ

BR(ξ)

Aα_i(ξ,u,Xu) –Aα_i(ξ,u,p)φ(Xu–p)dξ

= 

ˆ

BR(ξ)

Aα

i(ξ,u,p) –Aαi(ξ,u,Xu)

φvXφdξ

+

ˆ

BR(ξ)

Aα_iξ,u+ξ–ξ_p,p–Aα_i(ξ,u,p)Xϕαdξ

+

ˆ

BR(ξ)

Aα_i(ξ,u,p)–Aαi

ξ,u+ξ–ξ_p,pXϕαdξ

+

ˆ

BR(ξ)

Bα_(ξ,u,Xu)ϕα_dξ

=:I+II+III+IV+V, (.)

with the obvious labeling forI-V.

The left hand side of (.) can be estimated via the uniformly elliptic condition (H) (also see (.)),

λ

ˆ

BR(ξ)

|Xu–p|+|Xu–p|mφdξ

≤

ˆ

BR(ξ)

Forε>  to be fixed later we have, using the version (.) of (H), and Young’s inequality,

I≤C(L)

ˆ

BR(ξ)

 +|Xu|+|p|m– |_Xu–p||_φ||_v||_Xφ|_dξ

≤Cε

ˆ

BR(ξ)

|Xu–p||φ|dξ+ C

ε(R–ρ)

ˆ

BR(ξ)

|v|dξ

+Cε

ˆ

BR(ξ)

|Xu–p|m|φ|m–m _dξ+ C εm–_(R–ρ)m

ˆ

BR(ξ)

|v|mdξ. (.)

Using Jensen’s inequality, (.), and the fact thatη(ts)≤tη(s) fort≥, we arrive at

 +|p|

m κ(·)

ˆ

BR(ξ)

η|v|_dξ

≤ωGRQ–

 +|p|

m

κ(·)Rη

BR(ξ)

|v|dξ

≤ωGRQ–

BR(ξ)

|v|dξ+ +|p|

m

κ(·)Rη +|p|

m

κ(·)R

≤R–

ˆ

BR(ξ)|

v|dξ+ωGRQ

 +|p|mκ(·)ηR, (.)

whereκ(·) is abbreviation of the functionκ(|u|+|p|). Also note that (.) in the second last inequality is applied with the assumptionR<ρ_m/(|u|,|p|)≤.

Using the Dini continuity condition (H), Young’s inequality, and (.) inII, we obtain

II≤ +|p|

m

_κ|_u|_+R|_p|

ˆ

BR(ξ)

η|_v||_Xu–p|_φ_dξ,

≤ε

ˆ

BR(ξ)

|Xu–p||φ|dξ+ε– +|p|mκ(·)

ˆ

BR(ξ)

η|v|dξ

≤ε

ˆ

BR(ξ)

|Xu–p||φ|dξ+ 

ε(R–ρ)

ˆ

BR(ξ)

|v|dξ

+ε–ωGRQ

 +|p|mκ(·)ηR. (.)

Similarly,

III≤ +|p|

m

_κ|_u|_+R|_p|

ˆ

BR(ξ)

η|_v||_v||_Xφ||_φ|_dξ,

≤ C

(R–ρ)

ˆ

BR(ξ)

|v|dξ+ +|p|

m κ(·)

ˆ

BR(ξ)

η|v|dξ

≤ C

(R–ρ)

ˆ

BR(ξ)

|v|dξ+ωGRQ

 +|p|mκ(·)ηR (.)

and

IV≤ +|p|m_κ|_u|_+R|_p|

×

ˆ

BR(ξ)

≤ε

ˆ

BR(ξ)

|Xu–p||φ|dξ+ Cε (R–ρ)

ˆ

BR(ξ)

|v|dξ

+ ε–ωGRQ

 +|p|mκ(·)η

BR(ξ)

R +|p|dξ

≤ε

ˆ

BR(ξ)

|Xu–p||φ|dξ+ Cε (R–ρ)

ˆ

BR(ξ)

|v|dξ

+ ε–ωGRQ

 +|p|

m+

κ(·)ηR, (.)

where we have usedκ≥ in the last inequality.

Finally, the termV can be estimated by using the controllable growth condition (H), Hölder’s inequality, and Young’s inequality. This yields

V≤c

ˆ

BR(ξ)

 +|u|r–+|Xu|m(–/r)ϕdξ

≤c

ˆ

BR(ξ)

|ϕ|rdξ /rˆ

BR(ξ)

 +|u|r+|Xu|mdξ (r–)/r

≤c

ˆ

BR(ξ)| Xϕ|mdξ

/mˆ

BR(ξ)

 +|u|r+|Xu|mdξ (r–)/r

≤ε

ˆ

BR(ξ)

|Xϕ|mdξ+C(ε)

ˆ

BR(ξ)

 +|u|r+|Xu|mdξ

m(r–)/r(m–)

≤m–ε

ˆ

BR(ξ)

|Xu–p|m|φ|mdξ+

m–_Cε (R–ρ)m

ˆ

BR(ξ)

|v|mdξ

+C(ε)

ˆ

BR(ξ)

 +|u|r+|Xu|mdξ

m(r–)/r(m–)

, (.)

where we have used the fact|Xϕ|m≤m–(φm|Xu–p|m+_(R–C_ρ)m|v|m).

Applying these estimates to (.), we obtain

ˆ

Bρ(ξ)

|Xu–p|+|Xu–p|mφdξ

≤(C+ )(ε+ε–)

(R–ρ)

ˆ

BR(ξ)

|v|dξ+C(ε

–m_{+ }m–_ε)

(R–ρ)m

ˆ

BR(ξ)

|v|mdξ

+ε–+ ωGRQ

 +|p|mκ(·)ηR +C(ε)

ˆ

BR(ξ)

 +|u|r+|Xu|mdξ

m(r–)/r(m–) ,

where=λ– (C+  + m–)ε. The conclusion follows by choosing suitableεsuch that

> . We refer to [] for a similar and detailed argument.

4 Proof of main theorem

Lemma  Let Bρ(ξ)⊂⊂ withρ ≤ρ

m 

 (|u|,|p|) andϕ∈C∞ (Bρ(ξ),RN)satisfying

|ϕ| ≤ρandsup_Bρ(ξ

)|Xϕ| ≤.Then there exists a constant C≥,such that

Bρ(ξ)

Aα

i,pj_β(ξ,u,p)(Xu–p)Xϕ α

dξ

≤C

(ξ,ρ,p) +ω

 m|_p

|,(ξ,ρ,p)

_(ξ

,ρ,p)

+F|u|,|p|η_ρ _sup Bρ(ξ)

|Xϕ|, (.)

where F(s,t) =K(s,t) + ( +s+t)r–_{and C= }m– _{(C(L) + c+  + Cp) > .}

Proof We first write

Aα_i(ξ,u,Xu) –Aαi(ξ,u,p)=

ˆ 

 Aα

i,pj_β

ξ,u,θXu+ ( –θ)p

(Xu–p)dθ.

Noting that´_Bρ(ξ

)A α

i(ξ,u,p)Xϕαdξ=  and using the weak form ofu, we conclude

Bρ(ξ)

Aα

i,pj_β(ξ,u,p)(Xu–p)Xϕ α

dξ

≤

Bρ(ξ) ˆ 



Aα

i,pj_β(ξ,u,p) –A α i,pj_β

ξ,u,θXu+ ( –θ)p

(Xu–p)dθ

dξ

× sup

Bρ(ξ)

|Xϕ|

+

Bρ(ξ)

Aα_i(ξ,u,Xu) –Aαi

ξ,u+p(ξ–ξ),Xu

sup

Bρ(ξ)

|Xϕ|

+

Bρ(ξ)

Aα_iξ,u+p(ξ–ξ),Xu

–Aα_i(ξ,u,Xu) sup

Bρ(ξ)

|Xϕ|

+

Bρ(ξ)

Bα_(ξ,u,Xu)ϕα_dξ

:=I+II+III+IV, (.)

with the obvious meaning ofI-IV. In order to estimate the termI, we use (.) and (.) to first get

Aα

i,pj_β(ξ,u,p) –A α i,pj_β

ξ,u,θXu+ ( –θ)p

 m+(–m)

≤C +|p|+θ(Xu–p) +pm– _ω|_p|_{,θ(Xu–p)}m

×L +|p|

m–

 _{+L +θ(Xu–p}

) +p

m–_ –_m

≤C(L) +|p|

m–  |_Xu–p

|m–ω

 m|_p

|,|Xu–p|

Using (.), Hölder’s inequality, the fact thatt→ω_{(s,t) is concave, and Jensen’s} inequal-ity, we have

I≤C(L)

Bρ(ξ) ωm|_p

|,|Xu–p|

|Xu–p|m– sup

Bρ(ξ)

|Xϕ|dξ

≤C(L) sup

Bρ(ξ)

|Xϕ|

Bρ(ξ)

ω|p|,|Xu–p|dξ 

m

Bρ(ξ)

|Xu–p|mdξ m–

m

≤C(L) sup

Bρ(ξ)| Xϕ|ωm

|p|,

Bρ(ξ)|

Xu–p|dξ

Bρ(ξ)|

Xu–p|mdξ m–

m

≤C(L)ωm|_p

|,(ξ,ρ,p)

_(ξ

,ρ,p) sup

Bρ(ξ)

|Xϕ|, (.)

where we have used the fact _mm_–<  and the assumption(ξ,ρ,p)≤.

The termIIcan be estimated using the Dini continuity condition (.) and the fact that

η(ts)≤tη(s) fort≥. We have

II≤ sup

Bρ(ξ)

|Xϕ|κ(·)μρ +|p|

Bρ(ξ)

 +|Xu|

m  _dξ

≤ sup

Bρ(ξ)|

Xϕ|κ(·) +|p|η_ρ Bρ(ξ)

 +|p|+|Xu–p|m _dξ

≤m– _sup Bρ(ξ)|

Xϕ|κ(·) +|p|

+m_

η_ρ_+κ₍·_{) +}|_p

|



ηρ

+

Bρ(ξ)

|Xu–p|mdξ

≤m– _{(ξ,ρ,p) + κ}₍·_{) +}|_p|+ m

_η_ρ _sup

Bρ(ξ)

|Xϕ|, (.)

where we have used the fact thatη(ρ)≤η(_ρ), which follows from the nondecreasing

property of the functionη(t), (η), and our assumptionρ≤ρ≤.

Similarly, it follows that by using (.), (.), and the Poincaré inequality (.) in the special casep=q= 

III≤ sup

Bρ(ξ)

|Xϕ| Bρ(ξ)

κ(·)η|_v|_{ +}|_Xu| m  _dξ

≤m– _sup Bρ(ξ)|

Xϕ|

Bρ(ξ)|

Xu–p|mdξ+κ(·)

Bρ(ξ)

η|v|dξ

+κ(·) +|p|

m 

Bρ(ξ)

η|_v|_dξ

≤m– _sup Bρ(ξ)

|Xϕ|

(ξ,ρ,p)

+ ρ– Bρ(ξ)

|v|dξ+κ(·)ηρ+κ(·) +|p|

m

η_ρ

≤m– _sup Bρ(ξ)

|Xϕ|

+ Cp Bρ(ξ)|

Xu–p|dξ+ κ(·) +|p|mη_ρ

≤m– _{( + Cp)(ξ,ρ,p) + κ}₍·_{) +}|_p|m_η_ρ _sup Bρ(ξ)

|Xϕ|. (.)

Note that by the assumptionsup_Bρ(ξ)|ϕ| ≤ρ≤ and (η), we have

IV≤c

Bρ(ξ)

 +|u|r–+|Xu|m(–/r)|ϕ|dξ

≤c

Bρ(ξ)

|Xu|m(–/r)|ϕ|dξ+c

Bρ(ξ)

u–u–pξ–ξ_r–|ϕ|dξ

+cρ +|u|+ρ|p|r–

≤c

Bρ(ξ)

|Xu–p|mdξ r–

r

Bρ(ξ)

|ϕ|rdξ 

r

+c

Bρ(ξ)

|p|mdξ r–

r

Bρ(ξ)

|ϕ|rdξ 

r

+c

Bρ(ξ)

|Xu–p|mdξ

r– m

Bρ(ξ)

|ϕ|rdξ 

r

+cρ +|u|+|p|

r–

≤c

Bρ(ξ)|

Xu–p|mdξ r–

r

Bρ(ξ)|

ϕ|rdξ 

r

+cρ +|u|+|p|r–+|p|m(–r)

≤c(ξ,ρ,p) + cη



_ρ_{ +}|_u

|+|p|

r–

, (.)

where we have usedr– ≥m( – /r) andη(s)≤ fors∈(, ].

Combining the estimates (.)-(.) with (.), we obtain the conclusion with C =

m– _{(C(L) + c+  + Cp).}

The following lemma is to establish the excess improvement of the functionalas in (.). The strategy of the proof is to approximate the given solution byA-harmonic func-tions, for which suitable decay estimates are available from the classical theory.

Lemma  Assume that the conditions of Lemmaand the following smallness conditions

hold:

ωm|_uξ,ρ|_{+(Xu)ξ,ρ,ξ,ρ, (Xu)ξ,ρ+}/_{ξ,ρ, (Xu)ξ,ρ}≤δ

, (.)

CF|uξ,ρ|,(Xu)ξ,ρη

ρ≤δ, (.)

with C= CC,together with the radius condition

Then we have the following excess improvement estimate withτ∈[γ, ):

(ξ,θρ)≤θτ(ξ,ρ) +K∗

|uξ,ρ|,(Xu)ξ,ρη

ρm–m _,

where we have abbreviated(ξ,r) =(ξ,r, (Xu)ξ,r),and K∗(s,t) =CF( +s,  +t).

Proof We definew= [u–uξ,ρ– (Xu)ξ,ρ(ξ–ξ)]σ–, where

σ=C

(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

ρ _(.)

with C >  in Lemma . Then we have Xw=σ–(Xu– (Xu)ξ,ρ). Now we consider

Bρ(ξ)⊂⊂such thatρ≤ρ

m 

 (|uξ,ρ|,|(Xu)ξ,ρ|)≤. It yields

Bρ(ξ)

|Xw|dξ=σ–(ξ,ρ)≤ 

C_ ≤. (.)

Applying Lemma  onBρ(ξ) tou, we have, for anyϕ∈C∞(Bρ(ξ),RN),

Bρ(ξ)

Aα

i,pj_β

ξ,uξ,ρ, (Xu)ξ,ρ

XwXϕdξ

≤

/(ξ,ρ) +ω



m_{(Xu)ξ,ρ,(ξ,ρ)+}δ



sup

Bρ(ξ)

|Xϕ|. (.)

In consideration of the smallness condition (.), we see that (.) and (.) imply the conditions (.) and (.) in Lemma , respectively. Also note that assumptions (H) and (H) with u=uξ,ρ, and p= (Xu)ξ,ρ, imply the conditions (.). So there exists a

Aα

i,pj_β(ξ,uξ,ρ, (Xu)ξ,ρ)-harmonic functionh∈HW

,_(B

ρ(ξ),RN) such that

Bρ(ξ)

|Xh|dξ≤ (.)

and

ρ– Bρ(ξ)

|w–h|dξ≤ε. (.)

Using Lemma  on the ballBθρ(ξ) with u=uξ,θρ,θ ∈(, /], and replacingpby

(Xu)ξ,ρ+σ(Xh)ξ,θρ, we obtain

ˆ

Bθρ(ξ)

Xu– (Xu)ξ,ρ–σ(Xh)ξ,θρ



+Xu– (Xu)ξ,ρ–σ(Xh)ξ,θρ m

dξ

≤ Cc

(θρ)

ˆ

Bθρ(ξ)

u–uξ,θρ–

(Xu)ξ,ρ+σ(Xh)ξ,θρ

ξ–ξ_dξ

+ Cc

(θρ)m

ˆ

Bθρ(ξ)

u–uξ,θρ–

(Xu)ξ,ρ+σ(Xh)ξ,θρ

ξ–ξ_mdξ

+CcωG(θρ)Q

K|uξ,θρ|,(Xu)ξ,ρ+σ(Xh)ξ,θρη

+Cc ˆ

Bθρ(ξ)

 +|u|r+|Xu|mdξ

m(r–)/r(m–)

:=I+II+III+IV. (.)

Note that the smallness conditions (.)-(.) implyσ_C(=C

C+CFηδ–)≤ with C=max{C, (θ)–Q}, where we have assumed _CCδ≤, which is no restriction. Then it follows by applying the prior estimate (.) for theA-harmonic functionh

σ(Xh)ξ,θρ≤σ sup Bθρ(ξ)

|Xh| ≤σC

Bρ(ξ)

|Xh|dξ 



≤σC≤. (.)

Furthermore, it follows by Poincaré inequality (.)

|uξ,θρ| ≤ |uξ,ρ|+|uξ,θρ–uξ,ρ|

≤ |uξ,ρ|+ (θ)

–Q/

Bρ(ξ)

u– (Xu)ξ,ρ

ξ–ξ_–uξ,ρ

 dξ

/

≤ |uξ,ρ|+ (θ)

–Q/_ρC

p/(ξ,ρ)

≤ |uξ,ρ|+ σCp

C(θ)

Q 

≤ |uξ,ρ|+σ

C≤ |uξ,ρ|+ , (.)

where we have used the definition ofσ (.) and the factC>Cp.

Recall thatg(τ) =´_Bθρ(ξ

)(u–τ)

_{dξ has a minimal value atτ=u}

ξ,θρ. Noting thatu–

((Xu)ξ,ρ+σ(Xh)ξ,θρ)(ξ

_–ξ

) has mean valueuξ,θρ on the ballBθρ(ξ), and using the

definition ofw, and Poincaré inequality (.), we have

 (θρ)

Bθρ(ξ)

u–uξ,θρ–

(Xu)ξ,ρ+σ(Xh)ξ,θρ

ξ–ξ_dξ

≤ σ

(θρ)

Bθρ(ξ)

|w–h|dξ+

Bθρ(ξ)

h–hξ,θρ– (Xh)ξ,θρ

ξ–ξ_dξ

≤σ

(θ)–Q–ε+C_p(θρ)

Bθρ(ξ)

Xhdξ

≤Cθ–Q–ε+θ(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

ρ, (.)

whereC=C

(–Q+ CpC)≥, and in the last inequality we have used the definition of σ (.) withu=uξ,θρ, and the fact that

Bθρ(ξ)

Xhdξ≤ sup

Bρ(ξ)

Xh≤Cρ– Bρ(ξ)

|Xh|dξ≤Cρ–.

Then it follows

I≤CCcωG(θρ)Q

θ–Q–ε+θ(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

For  <m<Q, we haveQ_Qm–m=_m∗ <_m <_. Then there existst∈(, ) such that _m =_( – t) +_m∗t. Using (.), (.), Young’s inequality, and (.) in turn, we have

Bθρ(ξ)

u–uξ,θρ–

(Xu)ξ,ρ+σ(Xh)ξ,θρ

ξ–ξ_mdξ

≤

Bθρ(ξ)

u–uξ,θρ–

(Xu)ξ,ρ+σ(Xh)ξ,θρ

ξ–ξ_dξ (–t)m_

×

Bθρ(ξ)

u–uξ,θρ–

(Xu)ξ,ρ+σ(Xh)ξ,θρ

ξ–ξ_m∗dξ tm

m∗

≤(θρ)Cθ–Q–ε+θ(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

ρ(–t) m 

×

C_pm(θρ)m

Bθρ(ξ)

Xu–(Xu)ξ,ρ+σ(Xh)ξ,θρ m

dξ t

≤ε–t(Cp)mt 

–t_(θρ)_Cθ–Q–_ε+θ

×(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

ρ m 

+ε(θρ)m

Bθρ(ξ)

Xu–(Xu)ξ,ρ+σ(Xh)ξ,θρ m

dξ

≤ε–t(Cp)mt 

–t_(C)m_(θρ)m_θ–Q–_ε+θ

×(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

ρ

+ε(θρ)m

Bθρ(ξ)

Xu–(Xu)ξ,ρ+σ(Xh)ξ,θρ m

dξ, (.)

here we have used the assumptionσC≤. Furthermore, we obtain the estimate for the termII,

II≤CcωG(θρ)Q

ε–t(Cp)mt 

–t_(C)m_θ–Q–_ε+θ

×(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

ρ

+εCc

ˆ

Bθρ(ξ)

Xu–(Xu)ξ,ρ+σ(Xh)ξ,θρ m

dξ. (.)

Also, (.) and (.) yield

III≤CcωG(θρ)Q

K

 +|uξ,ρ|,  +(Xu)ξ,ρη

(θρ)

≤CcωG(θρ)QF

 +|uξ,ρ|,  +(Xu)ξ,ρη

(θρ).

Using the Poincaré type inequality, we have

Bθρ(ξ)

|Xu|m+|u|r+ dξ m(r–)

r(m–)

≤

m–

Bθρ(ξ)

Xu– (Xu)ξ,ρ m

dξ m(r–)

r(m–)

+

Bθρ(ξ)

r–u–uξ,ρ– (Xu)ξ,ρ

ξ–ξ_rdξ m(r–)

r(m–)

+ + r–uξ,ρ+ (Xu)ξ,ρ

ξ–ξ_r m(r–) r(m–)

≤C

(θ)–Q

Bρ(ξ)

Xu– (Xu)ξ,ρ m

dξ m(r–)

r(m–)

+m–(Xu)ξ,ρ mm(r–)_r(m–)

+C

(θ)–Q

Bρ(ξ)

Xu– (Xu)ξ,ρ m

dξ r–

m–

+ + r–uξ,ρ+ (Xu)ξ,ρ rm(r–)

r(m–)

≤C(θ)–Q(ξ,ρ)

m(r–)

r(m–)_{+C +}|_u

ξ,ρ|+(Xu)ξ,ρ m(r–) r(m–)

≤C +|uξ,ρ|+(Xu)ξ,ρ m(r–)

r(m–)_{, (.)}

where in the last inequality we have used the fact (θ)–Q(ξ,ρ)≤, implied by the as-sumptionσ_{C≤ withC=max{C, (θ)}–Q_{}. In view of}m(r–)

m– +

m m–( –



r) <

m(r–)

m– , we have

IV≤CωG(θρ)QF m

r(m–)_{ +}|_u

ξ,ρ|,(Xu)ξ,ρη

(θρ)m–m _{, (.)}

where we have used Qm_r(m(r_–)–)≤Q+_mm_–and (θρ)m–m ≤_η((_θρ)m–m ).

Joining the estimates ofI,II,III, andIVwith (.), we obtain

ˆ

Bθρ(ξ)

Xu– (Xu)ξ,ρ–σ(Xh)ξ,θρ

 dξ

+ ( –Ccε)

ˆ

Bθρ(ξ)

Xu– (Xu)ξ,ρ–σ(Xh)ξ,θρ m

dξ

≤Cc(C) m _ω

G(θρ)Q

(Cp)mtε–t 

–t _{+ θ}–Q–_ε+θ

×(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

ρ

+CcωG(θρ)QF

|uξ,ρ|,  +(Xu)ξ,ρη

(θρ)

+CωG(θρ)QF m

r(m–)_{ +}|_u

ξ,ρ|,(Xu)ξ,ρη

(θρ)m–m

≤CωG(θρ)Q

θ–Q–ε+θ(ξ,ρ) + δ–F

|uξ,ρ|,(Xu)ξ,ρη

ρ

+CωG(θρ)QF

 +|uξ,ρ|,  +(Xu)ξ,ρη

ρm–m _,

whereC=Cc(C)

m

[((C_p)mt_ε–t)–t + ] > .

We setP= (Xu)ξ,ρ+σ(Xh)ξ,θρ. Choosing a suitable smallε>  such that ( –Ccε) > ,

and considering the smallness condition (.) (it impliesρ≤ρ(|uξ,θρ|,|P|), see (.)

and (.) above), we deduce that

ξ,θρ, (Xu)ξ,θρ

=

Bθρ(ξ)

Xu– (Xu)ξ,θρ



+Xu– (Xu)ξ,θρ m

dξ

≤

Bθρ(ξ)

≤Cθ–Q–ε+θξ,ρ, (Xu)ξ,ρ

+ δ–ηρF|uξ,ρ|,(Xu)ξ,ρ

+CF +|uξ,ρ|,  +(Xu)ξ,ρη

ρm–m _{, (.)}

whereC= C

Q

[–Ccε]> .

For a givenτ∈[γ, ), we now specifyε=θQ+,θ∈(, /] such that Cθ≤θτ_{. Then}

we have

(ξ,θρ)≤θτ(ξ,ρ) +

Cθδ–+CF +|uξ,ρ|,  +(Xu)ξ,ρη

ρm–m

≤θτ(ξ,ρ) +CF

 +|uξ,ρ|,  +(Xu)ξ,ρη

ρm–m

:=θτ(ξ,ρ) +K∗

|uξ,ρ|,(Xu)ξ,ρη

ρm–m _{, (.)}

whereC= Cθδ–+C>  andK∗(s,t) =CF( +s,  +t). Then the proof of Lemma 

is complete.

ForT> , we find(T) >  (depending onQ,N,λ,L,τ, andω) such that

ωm_{T, }

(T)

+ 

 

(T)≤ 

δ and (.)

( +Cp)

(T)≤θQ/

 –θτT. (.)

With(T) from (.) and (.), we chooseρ(T)∈(, ] (depending onQ,N,λ,L,τ,

ω,η, andκ) such that

ρ(T)≤ρm/( + T,  + T), (.)

CF(T, T)η

ρ(T)

≤δ, (.)

K(T)ηρ(T)

≤θγ–θτ(T) and (.)

( +Cp)K(T)H

ρ(T)

≤θQ –θγθγ–θτT, (.)

whereK(T) :=K∗(T, T).

By the proof method of Lemma . in [] and conditions (.)-(.), Lemma  can be proved. As we know, it is sufficient to complete the proof of Theorem  once we obtain Lemma .

Lemma  Assume that for some T> and Bρ(ξ)⊂⊂we have

() |uξ,ρ|+|(Xu)ξ,ρ| ≤T; () ρ≤ρ(T);

() (ξ,ρ)≤(T).

Then the smallness conditions(.)-(.)are satisfied on the balls Bθj_ρ(ξ_)for j∈N∪ {}.

Moreover,the limitξ=limj→∞(Xu)ξ,θjρexists,and the estimate

Bρ(ξ)

|Xu–ξ|

_dξ≤C

r

ρ τ

(ξ,ρ) +H

r