AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 676215 10.1155/2013/676215 676215 Research Article Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition Chen Shuhong 1 Tan Zhong 2 Eloe Paul 1 Department of Mathematics and Information Science Zhangzhou Normal University Zhangzhou, Fujian 363000 China fjzs.edu.cn 2 School of Mathematical Science Xiamen University Xiamen, Fujian 361005 China xmu.edu.cn 2013 28 4 2013 2013 31 12 2012 02 04 2013 2013 Copyright © 2013 Shuhong Chen and Zhong 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 boundary regularity for weak solutions of second-order quasilinear elliptic systems under natural growth condition with super quadratic growth and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

1. Introduction

This paper considers boundary regularity for weak solutions of quasilinear elliptic systems (1)-Dα(Aijαβ(x,u)Dβuj)=Bi(x,u,Du),xΩ, where Ω is a bounded domain in Rn with boundary of class C1,n2 and u takes value in RN,N>1. Each Aijαβ maps Ω×RN into R, and each Bi maps Ω×RN×RnN into R. A partial regularity theory of (1) must have a priori existence weak solutions. Here we assume that weak solutions exist and consider partial regularity of weak solutions directly. We further impose certain structural conditions on Aijαβ and Bi with m>2 as follows.

There exists L>0 such that (2)Aijαβ(x,ξ)(ν,ν~)L(1+|ξ|2)(m-2)/2|ν||ν~|Aijαβforall(x,ξ)Ω¯×RN,ν,ν~RnN.

Aijαβ(x,ξ) is uniformly strongly elliptic; that is, for some λ>0 we have (3)Aijαβ(x,ξ)(ν,ν)λ(1+|ξ|2)(m-2)/2|ν|2Aijαβforall(x,ξ)Ω¯×RN,νRnN.

Assume that AijαβC0(Ω×RN,RnN) and further that Aijαβ is uniformly continuous on sets of the form Ω¯×{ξ:|ξ|M}, for any fixed M,0<M<.

(Natural growth condition). There exist constants a and b, with a possibly depending on M>0, such that (4)|Bi(x,ξ,ν)|a(M)|ν|m+b

for all xΩ¯,ξRN with |ξ|M and νRnN.

Further hypothesis (H3) deduces, writing ω(·) for ω(M,·), the existence of a monotone nondecreasing concave function ω:[0,)[0,) with ω(0)=0, continuous at 0, such that (5)|Aijαβ(x,u)-Aijαβ(y,v)|ω(|x-y|m+|u-v|m), for all x,yΩ¯,u,vRN with |u|,|v|M .

There exist s with s>n and a function gH1,s(Ω,RN), such that (6)u|Ω=g|Ω.

Note that we trivially have gH1,2(Ω,RN). Further, by the Sobolev embedding theorem we have gC0,κ(Ω,RN) for any κ[0,1-(n/s)]. If g|Ω0, we will take g0 on Ω.

If the domain we consider is an upper half unit ball B+, the boundary condition becomes as follows.

There exist s with s>n and a function gH1,s(B+,RN), such that (7)u|D=g|D.

Here we write Bρ(x0)={xRn:|x-x0|<ρ}, and further Bρ=Bρ(0), B=B1. Similarly we denote upper half balls as follows: for x0Rn-1×{0}, we write Bρ+(x0) for {xRn:xn>0,|x-x0|<ρ} and set Bρ+=Bρ+(0), B+=B1+. For x0Rn-1×{0} we further write Dρ(x0) for {xRn:xn=0,|x-x0|<ρ} and set Dρ=Dρ(0),D=D1.

Definition 1.

By a weak solution of (1) one means a vector valued function uW1,m(Ω¯,RN)L(Ω¯,RN) such that (8)ΩAijαβ(x,u)(Dβuj,Dαφi)dx=ΩBi(x,u,Du)·φidx holds for all test-functions φC0(Ω¯,RN) and, by approximation, for all φW01,m(Ω¯,RN)L(Ω¯,RN).

Under such assumptions, even the boundary data is smooth, one cannot expect full regularity of (1) at the boundary . Then, our goal is to establish partial boundary regularity.

After the partial regularity results of the type in this paper were proved by Giusti and Miranda in , there are some previous partial regularity results for quasilinear systems. For example, regularity up to boundary for nonlinear and quasilinear systems  has been studied by Arkhipova. Wiegner  established boundary regularity for systems in diagonal form first, and the proof was generalized and extended by Hildebrandt and Widman . Jost and Meier  deduced full regularity in a neighborhood of the boundary for minima of functionals with the form ΩA(x,u)|Du|2dx. Furthermore, Duzaar et al. obtained the boundary Hausdorff dimension on the singular sets of solutions to even more general systems in [10, 11] recently. Further discussion for regularity theory can be seen in [12, 13] and their references.

Inspired by , in this paper, we would establish boundary regularity for quasilinear systems under natural growth condition by the method of A-harmonic approximation.

The technique of A-harmonic approximation  is a natural extension of the harmonic approximation technique, which originated from Simon's proof of Allard's  ε-regularity theorem. In this context, using the A-harmonic approximation technique, we obtain the following regularity results.

Theorem 2.

Consider a bounded domain Ω in RN, with boundary of class C1. Let u be a bounded weak solution of (1) satisfying the boundary condition (H5), and uLM< with 2a(M)M<λ, where the structure conditions (H1)–(H3) hold for Aijαβ and (H4) holds for Bi. Consider a fixed γ(0,σ]. Then there exist positive R1 and ε0 (depending only on n,N,λ,L,b,M,a(M),ω(·),m, and γ) with the property that (9)BR(x0)Ω|u-ux0,R|2dx+gH1,s2R2(1-(n/s))+R2ε02 for some R(0,R1] for a given x0Ω implies uC0,γ(B¯R/2(x0)Ω¯,RN).

Note in particular that the boundary condition (H5) means that ux0,R makes sense: in fact, we have ux0,R=gx0,R. For νL1(Ω), x0Ω, we set νx0,R=ΩB¯R(x0)νdHn-1. In particular, for νL1(Dρ(x0)), x0D, we write νx0,ρ=Dρ(x0)νdHn-1.

Combining this result with the analogous interior  and a standard covering argument allows us to obtain the following bound on the size of the singular set.

Corollary 3.

Under the assumptions of Theorem 2 the singular set of the weak solution u has (n-2)-dimensional Hausdorff measure zero in Ω¯.

If the domain of the main step in proving Theorem 2 is a half ball, the result then is the following.

Theorem 4.

Consider a bounded weak solution of (1) on the upper half unit ball B+ which satisfies the boundary condition (H5) and uLM< with 2a(M)M<λ, where the structure conditions (H1)–(H3) hold for Aijαβ and (H4) holds for Bi. Then there exist positive R0 and ε0 (depending only on n,N,λ,L,b,M,a(M),M,ω(·),m, and γ) with the property that(10)BR+(x0)|u-ux0,R|2dx+gH1,s2R2(1-(n/s))+R2ε02, for some R(0,R0] for a given x0D, implies that there holds: uC0,σ(B¯R/2+(x0),RN).

Note that analogous to the above, the boundary condition (H5) ensures that ux0,R exists, and we have indeed ux0,R=gx0,R.

2. The A-Harmonic Approximation Technique

In this section we present the A-harmonic approximation lemma  and some standard results due to Companato .

Lemma 5 (A-harmonic approximation lemma).

Consider fixed positive λ and L, and n,NN with n2. Then for any given ε>0 there exists δ=δ(n,N,λ,L,ε)(0,1] with the following property: for any ABil(RnN) satisfying (11)A(ν,ν)λ|ν|2forallνRnN,|A(ν,ν-)|L|ν||ν-|forallν,ν-RnN for any wH1,2(Bρ+(x0),RN)   (for some ρ>0,x0Rn) satisfying (12)ρ2-nBρ+(x0)|Dw|2dx1,|ρ2-nBρ+(x0)A(Dw,Dφ)dx|δρsupBρ+(x0)|Dφ|,w|Dρ(x0)=0 for all φC01(Bρ+(x0),RN), there exists an A-harmonic function (13)vH~={·|ρ2-nBρ+(x0)|Dw~|2dx1,w~|Dρ(x0)w~H1,2(Bρ+(x0),RN)·|ρ2-nBρ+(x0)|Dw~|2dx1,w~|Dρ(x0)0} with (14)ρ-nBρ+(x0)|v-w|2dxε.

Next we recall a slight modification of a characterization of Hölder continuous functions originally due to Campanato .

Lemma 6.

Consider nN,n2, and x0Rn-1×{0}. Suppose that there are positive constants κ and α, with α(0,1] such that, for some νL2(B6R+(x0)), there holds the following: (15)infμR{ρ-nBρ+(y)|ν-μ|2dx}κ2(ρR)2α, for all yD2R(x0) and ρ4R; and (16)infμRρ-n{Bρ(y)|ν-μ|2dx}κ2(ρR)2α, for all yB2R+(x0) and Bρ(y)B2R+(x0).

Then there exists a Hölder continuous representative of the L2-class of ν on B¯R+(x0), and for this representative ν¯ there holds (17)|ν¯(x)-ν¯(z)|Cκ(|x-z|R)α, for all x,zB¯R+(x0), for a constant Cκ depending only on n and α.

We close this section by a standard estimate for the solutions to homogeneous second-order elliptic systems with constant coefficients .

Lemma 7.

Consider fixed positive λ and L, and n,NN with n2. Then there exists C0 depending only on n, N, λ, and L   (without loss of generality we take C01) such that, for ABil(RnN) satisfying (11), any A-harmonic function h on Bρ+(x0) with h|Dρ(x0)0 satisfies (18)ρ2supBρ/2+(x0)|Dh|2C0ρ2-nBρ+(x0)|Dh|2dx.

3. The Caccioppoli Inequality

In this section we would prove a suitable Caccioppoli inequality. First of all we recall two useful inequalities. The first is the Sobolev embedding theorem which yields the existence of a constant Cs depending only on s, n, and N such that for x0D,ρ1-|x0| there holds (19)supBρ+(x0)|g-gx0,ρ|Csρ1-(n/s)gH1,s(Bρ+(x0),RN). Obviously, the inequality remains true if we replace gH1,s(Bρ+(x0),RN) by gH1,s(B+,RN), which we will henceforth abbreviate simply as gH1,s.

Next we note that the Poincaré inequality in this setting for x0D,ρ1-|x0| yields (20)Bρ+(x0)|g-gx0,ρ|mdxCpρmBρ+(x0)|Dg|mdx, for a constant Cp which depends only on n.

Finally, we fix an exponent σ(0,1) as follows: if g0, σ can be chosen arbitrarily (but henceforth fixed); otherwise we take σ fixed in (0,1-(n/s)).

Then we establish an appropriate inequality for Caccioppoli.

Theorem 8 (Caccioppoli’s inequality).

Let uW1,m(Ω¯,RN)L(Ω¯,RN) with uLM< and 2a(M)M<λ be a weak solution of systems (1) under assumption conditions (H1)–(H5). Then there exists ρ0(L,M,a(M),s,gH1,s)>0 such that, for all Bρ+(x0)B+, with x0D+, 0<ρ<R<ρ0, there holds (21)Bρ/2+(x0)|Du|2dxC1Bρ+(x0)|u(x)-ux0,R|2ρ2dx+C2αnρn+C3(αnρn)1-(2/s)gH1,s2, where C1 depends only on λ, L, and M and C3 depends on these quantities, and in addition to Cp, C2 depends on λ, L, M, a, b, and gL(B,RN).

Proof.

Consider a cutoff function ηC0(Bρ/2+(x0)), satisfying 0η1,η0 on Bρ/2+(x0) and |η|<4/ρ. Then the function (u-g)η2 is in W01,m(Bρ/2+(x0,RN)) and thus can be taken as a test-function.

Using (H1), (H4), (H5), and Young's inequality and noting that 2a(M)M<λ, we can get from (8) with ε positive but arbitrary (to be fixed later) (22)Bρ+(x0)Aijαβ(·,u)(Dβuj,Dαui)η2dxLBρ+(x0)(1+|u|2)(m-2)/2|Dg||Du|η2dx+2LBρ+(x0)(1+|u|2)(m-2)/2|Dη||Du|η|u-g|dx+aBρ+(x0)|Du|m|u-g|η2dx+bBρ+(x0)|u-g|η2dxεBρ+(x0)(1+|u|2)(m-2)/2|Du|2η2dx+asupBρ+(x0)|u-ux0,ρ|Bρ+(x0)|Du|mη2dx+asupBρ+(x0)|g-gx0,ρ|Bρ+(x0)|Du|mη2dx+L22εBρ+(x0)(1+|u|2)(m-2)/2|Dg|2η2dx+4L2εBρ+(x0)(1+|u|2)(m-2)/2|Dη|2|u-ux0,ρ|2dx+4L2εBρ+(x0)(1+|u|2)(m-2)/2|Dη|2|g-gx0,ρ|2dx+ε2b2Bρ+(x0)ρ2η2dx+1ερ2Bρ+(x0)|u-ux0,ρ|2dx+1ερ2Bρ+(x0)|g-gx0,ρ|2dxεBρ+(x0)(1+|u|2)(m-2)/2|Du|2η2dx+a(M+gL(B+,RN))Bρ+(x0)|Du|mη2dx+64L2+1εBρ+(x0)(1+|u|2)(m-2)/21ρ2|u-ux0,ρ|2dx+ε4b2η2αnρn+2+(L22ε+64L2Cp2ε+4Cpε)·Bρ+(x0)(1+|u|2)(m-2)/2|Dg|2η2dxεBρ+(x0)(1+|u|2)(m-2)/2|Dg|2η2dx+a(M+gL(B+,RN))C(uW1,m(Bρ+(x0)))αnρn+64L2+1εBρ+(x0)(1+|u|2)(m-2)/2(u-ux0,ρρ)2dx+ε4b2η2αnρn+2+(1+M2)(m-2)/2(L22ε+64L2Cp2ε+4Cpε)·Bρ+(x0)|Dg|2η2dx. Using (H2), (19), and (20), we thus have (23)(λ-ε)Bρ+(x0)|Du|2η2dx(λ-ε)Bρ+(x0)(1+|u|2)(m-2)/2|Du|2η2dx64L2+1εBρ+(x0)(1+|u|2)(m-2)/21ρ2|u-ux0|2dx+C(a,M,gL(B+,RN),uW1,m(Bρ+(x0)),b)αnρn+(L,Cp,M)Bρ+(x0)|Dg|2dx64L2+1ε(1+M2)(m-2)/2Bρ+(x0)1ρ2|u-ux0|2dx+C(a,M,gL(B+,RN),uW1,m(Bρ+(x0)),b)αnρn+(L,Cp,M)(αnρn)1-(2/s)  gH1,s2. Thus, we fix ε small enough to yield the desired inequality.

4. The Proof of the Main Theorem

In this section we proceed to the proof of the partial regularity result.

Lemma 9.

Consider uW1,m(Ω-,RN)L(Ω-,RN) to be a weak solution of (1), x0D and yDR(x0), Dρ(y)DR(x0), for R<1-|x0|, and φC0(Bρ/2+(y),RN) with supBρ+(y)|Dφ|1. We have (24)(ρ2)2-nBρ/2+(y)Aijαβ(y,uy,ρ+)(Dβuj,Dαφi)dxC4I(I+ω(I))ρsupBρ/2+(x0)|Dφ|. Here and hereafter, we define (25)I(z,r)=Br+(z)|u-uz,r|2dx+gH1,s2r2(1-(n/s))+r2, for zD,r(0,1-|z|).

Proof.

Using (8) we have (26)Bρ/2+(y)Aijαβ(y,uy,ρ)(Dβuj,Dαφi)dx[aBρ/2+(y)|Du|mdx+2-n-1αnbρn]·ρsupBρ/2+(y)|Dφ|+Bρ/2+(y)|Aijαβ(y,uy,ρ)-Aijαβ(x,u)|·|Du|dxsupBρ/2+(y)|Dφ|.

Applying in turn Young's inequality, (H3), the Caccioppoli inequality (Theorem 8), and Jensen's inequality, we calculate from (26) (27)Bρ/2+(y)Aijαβ(y,uy,ρ)(Dβuj,Dαφi)dx[aBρ/2+(y)|Du|mdx+2-n-1αnbρn]·ρ+[Bρ/2+(y)|Aijαβ(y,uy,ρ)-Aijαβ(x,u)|1/2dx]1/2·[Bρ/2+(y)|Du|2dx]1/2αnρn-12{(aBρ+(y)|Du|mx+2-nb)ρ2}+αnρn-1ω(ρm+Mm-2Bρ+(y)|u-uy,ρ|2dx)·{C1Bρ+(y)|u-uy,ρ|2dx+C3gH1,s2ρ2(1-(n/s))+C2ρ2Bρ+(y)|u-uy,ρ|2dxC3gH1,s2ρ2(1-(n/s))}1/2αnρn-12C5I+αnρn-12C6ω(I)IC7αnρn-1(I+ω(I)I), where C5=auW1,m+b,C6=max{C1,C2,C3}, and C7=(1/2)(C5+C6), for zD, r(0,1-|z|). We introduce the notation (28)I(z,r)=Br+(z)|u-uz,r|2dz+gH1,s2r2(1-(n/s))+r2 and further write I for I(y,ρ). For arbitrary φC0(Ω,RN) we thus have, by rescalling, (29)Bρ/2+(y)Aijαβ(y,uy,ρ)(Dβuj,Dαφi)dxC7αnρn-1I(I+ω(I)).

Multiplying (29) through by (ρ/2)2-n yields(30)|(ρ2)2-nBρ/2+(y)Aijαβ(y,uy,ρ)(Dβuj,Dαφi)dx|C4I(I+ω(I))ρsupBρ/2+(x0)|Dφ|, for C4 defined by C4=2n-3αnC7.

Lemma 10.

Consider u satisfying the conditions of Theorem 2 and σ fixed; then we can find δ and s0 together, with positive constants C8 such that the smallness conditions: 0<ω(s0)δ/2 and I(x0,R)C8-1min{δ2/4,s0} together, imply the growth condition (31)I(y,θρ)θ2σI(y,ρ).

Proof.

We now set w=u-g, using in turn (H1), Young's inequality, and Hölder's inequality. We have from (30) (32)|(ρ2)2-nBρ/2+(y)Aijαβ(y,uy,ρ)(Dβwj,Dαφi)dx||(ρ2)2-nBρ/2+(y)Aijαβ(y,uy,ρ)(Dβuj,Dαφi)dx|+|(ρ2)2-nBρ/2+(y)Aijαβ(y,uy,ρ)(Dβgj,Dαφi)dx|C9I[I+ω(I)]ρsupBρ/2+(x0)|Dφ|, for C9=max{C4,(αn/2)1-(n/s)}.

We now set v=w/γ, for γ=C9I. From (32) we then have (33)|(ρ2)2-nBρ/2+(y)Aijαβ(y,uy,ρ)(Dβvj,Dαφi)dx|(I+ω(I))ρsupBρ/2+(x0)|Dφ|, and from (32) we observe from the definition of C9 (recalling also the definition of γ) (34)(ρ2)2-nBρ/2+(y)|Dv|2dx<1. Further we note (35)v|Dρ(y)=1γw|Dρ(y)=1γ(u-g)|Dρ(y)0.

For ε>0 we take δ=δ(n,N,λ,L,ε) to be the corresponding δ from the A-harmonic approximation lemma. Suppose that we could ensure that the smallness condition (36)I+ω(I)δ holds. Then in view of (33), (34), and (35) we would be able to apply Lemma 5 to conclude the existence of a function hH1,2(Bρ/2+(y),RN) which is Aijαβ(y,uy,ρ)-harmonic, with h|Dρ/2(y)0 such that (37)(ρ2)2-nBρ/2+(y)|Dh|2dx1,(38)(ρ2)-nBρ/2+(y)|v-h|2dxε.

For θ(0,1/4] arbitrary (to be fixed later), we have from the Campanato theorem, noting (37) and recalling also that h(y)=0, (39)supBθρ+(y)|h|2θ2ρ2supBρ/4+(y)|Dh|24C0θ2.

Using (38) and (39) we observe (40)(θρ)-nBθρ+(y)|v|2dx2(θρ)-n[Bθρ+(y)|v-h|2dx+Bθρ+(y)|h|2dx]2(θρ)-n[(ρ2)nε+12αn(θρ)nsupBθρ+(y)|h|2]21-nθ-nε+4αnC0θ2, and, hence, on multiplying this through by γ2, we obtain the estimate (41)(θρ)-nBθρ+(y)|w|2dxC92(21-nθ-nε+4αnC0θ2)I.

For the time being, we restrict to the case that g does not vanish identically. Recalling that w=u-g, using in turn Poincaré’s, Sobolev’s, and then Hölder’s inequalities, and noting also that uy,θρ=gy,θρ, thus from (41) we get (42)(θρ)-nBθρ+(y)|u-uy,θρ|2dx2(θρ)-n[Bθρ+(y)|u-g|2dx+Bθρ+(y)|g-gy,θρ|2dx]2C92(21-nθ-nε+4αnC0θ2)I+2Cp(θρ)2-n[12αn(θρ)n]1-(2/s)gH1,s2C10(θ-nε+θ2)I+C10θ2(1-(n/s))I, for C10=max{8αnC0C92,22/sCpαn1-(2/s)}, and provided ε=θn+2, we have (43)(θρ)-nBθρ+(y)|u-uy,θρ|2dx3C10θ2(1-(n/s))I.

Note that fix ε=θn+2, which is also fixed δ. Since ρ1, we see from the definition of I(44)gH1,s2(θρ)2(1-(n/s))θ2(1-(n/s))I, and further (45)(θρ)2θ2I.

Combining these estimates with (43), we can get (46)I(y,θρ)3(C10+1)θ2(1-(n/s))I.

Choose θ(0,1/4] sufficiently small that there holds: 3(C10+1)θ2(1-(n/s))θ2σ.

We can see from (46) (47)I(y,θρ)θ2σI.

We now choose s0>0 such that 0<ω(s0)<(δ/2) and define C8 by (48)C8=max{2n-1,2C92+1,2Cs2+1}. Suppose that we have (49)I(x0,R)C8-1min{δ24,s0}, for some R(0,R0], where R0=min{2s0,1-|x0|}.

For any yDR/2(x0) we use the Sobolev inequality to calculate (50)αnRn2n+1|ux0,R-uy,R/2|2=BR/2+|ux0,R-uy,R/2|2dx=BR/2+|gx0,R-gy,R/2|2dx2BR/2+|g-gx0,R|2dx+2BR/2+|g-gy,R/2|2dx2αnCs2gH1,s2Rn+2(1-(n/s)).

Then we can calculate (51)I(y,12R)2n-1BR/2+(y)|u-ux0,R|2dx+(2Cs2+1)gH1,s2R2(1-(n/s))+14R2C8I(x0,R).

Then we have (52)I(y,12R)+ω(I(y,12R))C8I(x0,R)+ω(C8I(x0,R))12δ+ω(s0)δ, which means that the condition (49) is sufficient to guarantee the smallness condition (37) for ρ=R/2, for all yDR/2(x0). We can thus conclude that (46) holds in this situation. From (46) we thus have (53)I(y,θρ2)+ω(I(y,θρ2))I(y,12R)+ω(I(y,12R))δ, meaning that we can apply (46) on Bθρ/2+(y) as well, yielding (54)I(y,θ2R2)θ4σI(y,R2), and inductively (55)I(y,θkR2)θ2kσI(y,R2).

The next step is to go from a discrete to a continuous version of the decay estimate. Given ρ(0,R/2], we can find kN0 such that θk+1R/2<ρθkR/2. Firstly we use the Sobolev inequality, to see(56)Bρ+(y)|uy,ρ-uy,θkR/2|2dx2αn(12θkR)nCs2gH1,s2(12θkR)2(1-(n/s)), which allows us to deduce (57)Bρ+(y)|u-uy,ρ|2dx2Bρ+(y)|u-uy,θkR/2|2dx+4αn(12θkR)nCs2gH1,s2(12θkR)2(1-(n/s)), and, hence, (58)I(y,ρ)C11I(y,θkR2), for C11=8θ-nCs2+1. Combining this with (55) and (51), we have (59)I(y,ρ)C11θ2kσI(y,R2)C8C11θ-2σ(2ρR)2σI(x0,R)C8C11(2θ)I(x0,R)(ρR)2σ, and more particularly (60)infμRNBρ+(y)|u-μ|2dxC12I(x0,R)(ρR)2σ, for C12=C8C11(2/θ)2σ. Recall that this estimate is valid for all yD and ρ with Dρ(y)DR/2(x0); assume only the condition (49) on I(x0,R). This yields after replacing R with 6R the boundary estimate (15) which requires to apply Lemma 6.

Combining the boundary and interior estimates  we can derive the desired result. As the argument for combining the boundary and interior regularity results is relatively standard, we omit it. Hence we can apply Lemma 6 and conclude the desired Hölder continuity.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11201415, 11271305), Natural Science Foundation of Fujian Province (2012J01027), and Training Programme Foundation for Excellent Youth Researching Talents of Fujian's Universities (JA12205).

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