# Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition

Authors: Shuhong Chen; Zhong Tan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

This paper considers boundary regularity for weak solutions of quasilinear elliptic systems(1)-Dα(Aijαβ(x,u)Dβuj)=Bi(x,u,Du),x∈Ω,
where Ω is a bounded domain in Rn with boundary of class C1,n≥2 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.(H1)
There existsL>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.(H2)
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,ν∈RnN.(H3)
Assume thatAijαβ∈C0(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<∞.(H4)
(Natural growth condition). There exist constantsa and b, with a possibly depending on M>0, such that
(4)|Bi(x,ξ,ν)|≤a(M)|ν|m+bfor allx∈Ω¯,ξ∈RN with |ξ|≤M and ν∈RnN.Further hypothesis (H3) deduces, writingω(·) 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,y∈Ω¯,u,v∈RN with |u|,|v|≤M [1].(H5)
There exists with s>n and a function g∈H1,s(Ω,RN), such that
(6)u|∂Ω=g|∂Ω.Note that we trivially haveg∈H1,2(Ω,RN). Further, by the Sobolev embedding theorem we have g∈C0,κ(Ω,RN) for any κ∈[0,1-(n/s)]. If g|∂Ω≡0, we will take g≡0 on Ω.If the domain we consider is an upper half unit ballB+, the boundary condition becomes as follows. (H5)′
There exists with s>n and a function g∈H1,s(B+,RN), such that
(7)u|D=g|D.Here we writeBρ(x0)={x∈Rn:|x-x0|<ρ}, and further Bρ=Bρ(0), B=B1. Similarly we denote upper half balls as follows: for x0∈Rn-1×{0}, we write Bρ+(x0) for {x∈Rn:xn>0,|x-x0|<ρ} and set Bρ+=Bρ+(0), B+=B1+. For x0∈Rn-1×{0} we further write Dρ(x0) for {x∈Rn: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 u∈W1,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 φ∈C0∞(Ω¯,RN) and, by approximation, for all φ∈W01,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 [2]. 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 [3], there are some previous partial regularity results for quasilinear systems. For example, regularity up to boundary for nonlinear and quasilinear systems [4–6] has been studied by Arkhipova. Wiegner [7] established boundary regularity for systems in diagonal form first, and the proof was generalized and extended by Hildebrandt and Widman [8]. Jost and Meier [9] 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 [14], 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 [15–17] is a natural extension of the harmonic approximation technique, which originated from Simon's proof of Allard's [18] ε-regularity theorem. In this context, using the A-harmonic approximation technique, we obtain the following regularity results.Theorem 2.
Consider a bounded domainΩ in RN, with boundary of class C1. Let u be a bounded weak solution of (1) satisfying the boundary condition (H5), and ∥u∥L∞≤M<∞ 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+∥g∥H1,s2R2(1-(n/s))+R2≤ε02
for some R∈(0,R1] for a given x0∈∂Ω implies u∈C0,γ(B¯R/2(x0)∩Ω¯,RN).Note in particular that the boundary condition (H5) means thatux0,R′ makes sense: in fact, we have ux0,R′=gx0,R′. For ν∈L1(∂Ω), x0∈∂Ω, we set νx0,R′=⨍∂Ω∩B¯R(x0)νdHn-1. In particular, for ν∈L1(Dρ(x0)), x0∈D, we write νx0,ρ′=⨍Dρ(x0)νdHn-1.Combining this result with the analogous interior [19] 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 Theorem2 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 Theorem2 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 ∥u∥L∞≤M<∞ 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+∥g∥H1,s2R2(1-(n/s))+R2≤ε02,
for some R∈(0,R0] for a given x0∈D, implies that there holds: u∈C0,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 [14] and some standard results due to Companato [20].Lemma 5 (A-harmonic approximation lemma).
Consider fixed positiveλ and L, and n,N∈N with n≥2. Then for any given ε>0 there exists δ=δ(n,N,λ,L,ε)∈(0,1] with the following property: for any A∈Bil(RnN) satisfying
(11)A(ν,ν)≥λ|ν|2forallν∈RnN,|A(ν,ν-)|≤L|ν||ν-|forallν,ν-∈RnN
for any w∈H1,2(Bρ+(x0),RN)   (for some ρ>0,x0∈Rn) satisfying
(12)ρ2-n∫Bρ+(x0)|Dw|2dx≤1,|ρ2-n∫Bρ+(x0)‍A(Dw,Dφ)dx|≤δρsupBρ+(x0)|Dφ|,w|Dρ(x0)=0
for all φ∈C01(Bρ+(x0),RN), there exists an A-harmonic function
(13)v∈H~={·|ρ2-n∫Bρ+(x0)‍|Dw~|2dx≤1,w~|Dρ(x0)w~∈H1,2(Bρ+(x0),RN)·|ρ2-n∫Bρ+(x0)‍|Dw~|2dx≤1,w~|Dρ(x0)≡0}
with
(14)ρ-n∫Bρ+(x0)‍|v-w|2dx≤ε.Next we recall a slight modification of a characterization of Hölder continuous functions originally due to Campanato [21].Lemma 6.
Considern∈N,n≥2, and x0∈Rn-1×{0}. Suppose that there are positive constants κ and α, with α∈(0,1] such that, for some ν∈L2(B6R+(x0)), there holds the following:
(15)infμ∈R{ρ-n∫Bρ+(y)‍|ν-μ|2dx}≤κ2(ρR)2α,
for all y∈D2R(x0) and ρ≤4R; and
(16)infμ∈Rρ-n{∫Bρ(y)‍|ν-μ|2dx}≤κ2(ρR)2α,
for all y∈B2R+(x0) and Bρ(y)⊂B2R+(x0).
Then there exists a Hölder continuous representative of theL2-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,z∈B¯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 [20].Lemma 7.
Consider fixed positiveλ and L, and n,N∈N with n≥2. Then there exists C0 depending only on n, N, λ, and L   (without loss of generality we take C0≥1) such that, for A∈Bil(RnN) satisfying (11), any A-harmonic function h on Bρ+(x0) with h|Dρ(x0)≡0 satisfies
(18)ρ2supBρ/2+(x0)|Dh|2≤C0ρ2-n∫Bρ+(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 constantCs depending only on s, n, and N such that for x0∈D,ρ≤1-|x0| there holds
(19)supBρ+(x0)|g-gx0,ρ′|≤Csρ1-(n/s)∥g∥H1,s(Bρ+(x0),RN).
Obviously, the inequality remains true if we replace ∥g∥H1,s(Bρ+(x0),RN) by ∥g∥H1,s(B+,RN), which we will henceforth abbreviate simply as ∥g∥H1,s.Next we note that the Poincaré inequality in this setting forx0∈D,ρ≤1-|x0| yields
(20)∫Bρ+(x0)‍|g-gx0,ρ|mdx≤Cpρm∫Bρ+(x0)|Dg|m‍dx,
for a constant Cp which depends only on n.Finally, we fix an exponentσ∈(0,1) as follows: if g≡0, 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).
Letu∈W1,m(Ω¯,RN)∩L∞(Ω¯,RN) with ∥u∥L∞≤M<∞ 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,∥g∥H1,s)>0 such that, for all Bρ+(x0)⊂B+, with x0∈D+, 0<ρ<R<ρ0, there holds
(21)∫Bρ/2+(x0)|Du|2dx≤C1∫Bρ+(x0)|u(x)-ux0,R′|2ρ2dx+C2αnρn+C3(αnρn)1-(2/s)∥g∥H1,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 ∥g∥L∞(B,RN).Proof.
Consider a cutoff functionη∈C0∞(Bρ/2+(x0)), satisfying 0≤η≤1,η≡0 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 that2a(M)M<λ, we can get from (8) with ε positive but arbitrary (to be fixed later)
(22)∫Bρ+(x0)‍Aijαβ(·,u)(Dβuj,Dαui)η2dx≤L∫Bρ+(x0)‍(1+|u|2)(m-2)/2|Dg||Du|η2dx+2L∫Bρ+(x0)‍(1+|u|2)(m-2)/2|Dη||Du|η|u-g|dx+a∫Bρ+(x0)‍|Du|m|u-g|η2dx+b∫Bρ+(x0)‍|u-g|η2dx≤ε∫Bρ+(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+L22ε∫Bρ+(x0)‍(1+|u|2)(m-2)/2|Dg|2η2dx+4L2ε∫Bρ+(x0)‍(1+|u|2)(m-2)/2|Dη|2|u-ux0,ρ′|2dx+4L2ε∫Bρ+(x0)‍(1+|u|2)(m-2)/2|Dη|2|g-gx0,ρ′|2dx+ε2b2∫Bρ+(x0)‍ρ2η2dx+1ερ2∫Bρ+(x0)‍|u-ux0,ρ′|2dx+1ερ2∫Bρ+(x0)‍|g-gx0,ρ′|2dx≤ε∫Bρ+(x0)‍(1+|u|2)(m-2)/2|Du|2η2dx+a(M+∥g∥L∞(B+,RN))∫Bρ+(x0)‍|Du|mη2dx+64L2+1ε∫Bρ+(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|2η2dx≤ε∫Bρ+(x0)‍(1+|u|2)(m-2)/2|Dg|2η2dx+a(M+∥g∥L∞(B+,RN))C(∥u∥W1,m(Bρ+(x0)))αnρn+64L2+1ε∫Bρ+(x0)‍(1+|u|2)(m-2)/2(u-ux0,ρ′ρ)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|2η2dx≤64L2+1ε∫Bρ+(x0)(1+|u|2)(m-2)/21ρ2|u-ux0′|2dx+C(a,M,∥g∥L∞(B+,RN),∥u∥W1,m(Bρ+(x0)),b)αnρn+(L,Cp,M)∫Bρ+(x0)|Dg|2dx≤64L2+1ε(1+M2)(m-2)/2∫Bρ+(x0)1ρ2|u-ux0′|2dx+C(a,M,∥g∥L∞(B+,RN),∥u∥W1,m(Bρ+(x0)),b)αnρn+(L,Cp,M)(αnρn)1-(2/s)∥g∥H1,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.
Consideru∈W1,m(Ω-,RN)∩L∞(Ω-,RN) to be a weak solution of (1), x0∈D and y∈DR(x0), Dρ(y)⊂⊂DR(x0), for R<1-|x0|, and φ∈C0∞(Bρ/2+(y),RN) with supBρ+(y)|Dφ|≤1. We have
(24)(ρ2)2-n∫Bρ/2+(y)‍Aijαβ(y,uy,ρ+)(Dβuj,Dαφi)dx≤C4I(I+ω(I))ρsupBρ/2+(x0)|Dφ|.
Here and hereafter, we define
(25)I(z,r)=⨍Br+(z)|u-uz,r′|2dx+∥g∥H1,s2r2(1-(n/s))+r2,
for z∈D,r∈(0,1-|z|).Proof.
Using (8) we have
(26)∫Bρ/2+(y)Aijαβ(y,uy,ρ′)(Dβuj,Dαφi)dx≤[a∫Bρ/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 (Theorem8), and Jensen's inequality, we calculate from (26)
(27)∫Bρ/2+(y)Aijαβ(y,uy,ρ′)(Dβuj,Dαφi)dx≤[a∫Bρ/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]1/2≤αnρn-12{(a⨍Bρ+(y)|Du|mx+2-nb)ρ2}+αnρn-1ω(ρm+Mm-2⨍Bρ+(y)|u-uy,ρ′|2dx)·{C1⨍Bρ+(y)|u-uy,ρ′|2dx+C3∥g∥H1,s2ρ2(1-(n/s))+C2ρ2∫Bρ+(y)|u-uy,ρ′|2dxC3∥g∥H1,s2ρ2(1-(n/s))}1/2≤αnρn-12C5I+αnρn-12C6ω(I)I≤C7αnρn-1(I+ω(I)I),
where C5=a∥u∥W1,m+b,C6=max{C1,C2,C3}, and C7=(1/2)(C5+C6), for z∈D, r∈(0,1-|z|). We introduce the notation
(28)I(z,r)=⨍Br+(z)|u-uz,r′|2dz+∥g∥H1,s2r2(1-(n/s))+r2
and further write I for I(y,ρ). For arbitrary φ∈C0∞(Ω,RN) we thus have, by rescalling,
(29)∫Bρ/2+(y)‍Aijαβ(y,uy,ρ′)(Dβuj,Dαφi)dx≤C7αnρn-1I(I+ω(I)).
Multiplying (29) through by (ρ/2)2-n yields(30)|(ρ2)2-n∫Bρ/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.
Consideru 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 setw=u-g, using in turn (H1), Young's inequality, and Hölder's inequality. We have from (30)
(32)|(ρ2)2-n∫Bρ/2+(y)‍Aijαβ(y,uy,ρ′)(Dβwj,Dαφi)dx|≤|(ρ2)2-n∫Bρ/2+(y)‍Aijαβ(y,uy,ρ′)(Dβuj,Dαφi)dx|+|(ρ2)2-n∫Bρ/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 setv=w/γ, for γ=C9I. From (32) we then have
(33)|(ρ2)2-n∫Bρ/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-n∫Bρ/2+(y)‍|Dv|2dx<1.
Further we note
(35)v|Dρ(y)=1γw|Dρ(y)=1γ(u-g)|Dρ(y)≡0.
Forε>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 h∈H1,2(Bρ/2+(y),RN) which is Aijαβ(y,uy,ρ′)-harmonic, with h|Dρ/2(y)≡0 such that
(37)(ρ2)2-n∫Bρ/2+(y)‍|Dh|2dx≤1,(38)(ρ2)-n∫Bρ/2+(y)‍|v-h|2dx≤ε.
Forθ∈(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|2≤θ2ρ2supBρ/4+(y)|Dh|2≤4C0θ2.
Using (38) and (39) we observe
(40)(θρ)-n∫Bθρ+(y)|v|2dx≤2(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)-n∫Bθρ+(y)|w|2‍dx≤C92(21-nθ-nε+4αnC0θ2)I.
For the time being, we restrict to the case thatg 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)-n∫Bθρ+(y)|u-uy,θρ′|2dx≤2(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)∥g∥H1,s2≤C10(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)-n∫Bθρ+(y)|u-uy,θρ′|2‍dx≤3C10θ2(1-(n/s))I.
Note that fixε=θn+2, which is also fixed δ. Since ρ≤1, we see from the definition of I(44)∥g∥H1,s2(θρ)2(1-(n/s))≤θ2(1-(n/s))I,
and further
(45)(θρ)2≤θ2I.
Combining these estimates with (43), we can get
(46)I(y,θρ)≤3(C10+1)θ2(1-(n/s))I.
Chooseθ∈(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 chooses0>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 anyy∈DR/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′|2dx≤2∫BR/2+|g-gx0,R′|2dx+2∫BR/2+|g-gy,R/2′|2dx≤2αnCs2∥g∥H1,s2Rn+2(1-(n/s)).
Then we can calculate(51)I(y,12R)≤2n-1⨍BR/2+(y)|u-ux0,R′|2dx+(2Cs2+1)∥g∥H1,s2R2(1-(n/s))+14R2≤C8I(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 y∈DR/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. Givenρ∈(0,R/2], we can find k∈N0 such that θk+1R/2<ρ≤θkR/2. Firstly we use the Sobolev inequality, to see(56)∫Bρ+(y)‍|uy,ρ′-uy,θkR/2′|2dx≤2αn(12θkR)nCs2∥g∥H1,s2(12θkR)2(1-(n/s)),
which allows us to deduce
(57)∫Bρ+(y)‍|u-uy,ρ′|2dx≤2∫Bρ+(y)‍|u-uy,θkR/2′|2dx+4αn(12θkR)nCs2∥g∥H1,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)infμ∈RN∫Bρ+(y)‍|u-μ|2dx≤C12I(x0,R)(ρR)2σ,
for C12=C8C11(2/θ)2σ. Recall that this estimate is valid for all y∈D 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 [19] 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.



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*Source: 676215-2013-04-28.xml*