Gradient estimates for solutions of the Lame´ ffi system with partially infinite coe cients in dimensions greater than two 6 1 0 2 JiGuang Bao HaiGang Li and YanYan Li ∗ ∗† ‡ n a January 29, 2016 J 8 2 ] Abstract P A We establish upper bounds on the blow-up rate of the gradients of solutions . oftheLame´ system withpartially infinitecoefficients indimensions greater than h twoasthedistancebetweenthesurfacesofdiscontinuity ofthecoefficientsofthe t a systemtendstozero. m [ 1 1 Introduction and main results v 9 7 In this paper, we establish upper bounds on the blow-up rate of the gradients of solu- 8 tionsoftheLame´ systemwithpartiallyinfinitecoefficients indimensionsgreater than 7 0 twoasthedistancebetweenthesurfaces ofdiscontinuityofthecoefficientsofthesys- . 1 temtends tozero. Thiswork isstimulatedbythestudyofBabus˘ka, Andersson,Smith 0 and Levin in [10] concerning initiation and growth of damage in composite materi- 6 1 als. The Lame´ system is assumed and they computationally analyzed the damage and : fracture in composite materials. They observed numerically that the size of the strain v i tensor remains bounded when the distance ǫ, between two inclusions, tends to zero. X This was proved by Li and Nirenberg in [32]. Indeed such ǫ-independent gradient r a estimates was established there for solutions of divergence form second order elliptic systems, including linear systems of elasticity, with piecewise Ho¨lder continuous coefficients inalldimensions. See Bonnetierand Vogelius[16]and Liand Vogelius[33] forcorrespondingresultson divergenceformellipticequations. Theestimatesin[32]and[33]dependontheellipticityofthecoefficients. Ifellip- ticity constants are allowed to deteriorate, the situation is very different. Consider the scalarequation a (x) u = 0 in Ω, k k ∇· ∇ (1.1) u =(cid:16) ϕ (cid:17) on ∂Ω,  k School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and ∗ ComplexSystems,MinistryofEducation,Beijing100875,China. Email:jgbao@bnu.edu.cn. Correspondingauthor.Email:hgli@bnu.edu.cn. † Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854, ‡ USA.Email:yyli@math.rutgers.edu. 1 where Ωis a bounded open set of Rd, d 2, containingtwo ǫ-apart convexinclusions ≥ D and D , ϕ C2(∂Ω)isgiven,and 1 2 ∈ k (0, ) in D D , a (x) = ∈ ∞ 1 ∪ 2 k 1 inΩ D D .  \ 1 ∪ 2 When k = , the L -norm of u for the solutions u of (1.1) generally becomes ∞ unbounded∞as ǫ tends to 0. The|∇bl∞ow| up rate of u is∞respectively ǫ 1/2 in dimen- − sion d = 2, (ǫ lnǫ ) 1 in dimension d = 3, and|∇ǫ 1∞i|n dimension d 4. See Bao, − − | | ≥ Li and Yin [11], as well as Budiansky and Carrier [18], Markenscoff [36], Ammari, Kang and Lim [7], Ammari, Kang, Lee, Lee and Lim [8] and Yun [41, 42]. Further, moredetailed,characterizationsofthesingularbehaviorof u havebeenobtainedby ∇ ∞ Ammari,Ciraolo,Kang,LeeandYun[3],Ammari,Kang,Lee,LimandZribi[9],Bon- netierandTriki[14,15],GorbandNovikov[24]andKang,LimandYun[25,26]. For relatedworks,see[2,4,5,12,15,17,19,20,21,22,28,29,30,31,34,35,37,39,40] and thereferences therein. In this paper, we mainly investigate the gradient estimates for the Lame´ system withpartiallyinfinitecoefficientsindimensiond = 3,aphysicallyrelevantdimension. This paper is a continuation of [13], where the estimate for dimension d = 2, another physically relevant dimension, is established. We prove that (ǫ lnǫ ) 1 is an upper − | | boundoftheblowuprateofthestraintensorindimensionthree,thesameasthescalar equation case mentioned above. New difficulties need to be overcome, and a number of refined estimates, via appropriate iterations, are used in our proof. We also prove that ǫ 1 is an upper bound of the blow up rate of the strain tensor in dimensiond 4, − ≥ whichisalsothesameasthescalarequationcase. Notethatithasbeenprovedin[11] thattheseupperboundsindimensiond 3 are optimalinthescalarequationcase. ≥ We consider the Lame´ system in linear elasticity with piecewise constant coefficients,which isstimulatedbythestudyofcompositemediawithcloselyspaced inter- facial boundaries. Let Ω R3 be a bounded open set with C2 boundary, and D and 1 ⊂ D are two disjointconvex open sets in Ω withC2,γ boundaries, 0 < γ < 1, which are 2 ǫ apart and faraway from∂Ω, that is, D ,D Ω, theprinciplecurvaturesof∂D ,∂D κ > 0, 1 2 ⊂ 1 2 ≥ 0 (1.2) ǫ := dist(D ,D ) > 0, dist(D D ,∂Ω) > κ > 0, 1 2 1 2 1 ∪ whereκ ,κ areconstantsindependentofǫ. WealsoassumethattheC2,γ normsof∂D 0 1 i are bounded by some constant independent of ǫ. This implies that each D contains a i ballofradius r forsomeconstantr > 0independentofǫ. Denote 0∗ 0∗ Ω := Ω D D . 1 2 \ ∪ Assume that Ω and D D are oeccupied, respectively, by two different isotropicand 1 2 ∪ homogeneous materials with different Lame´ constants (λ,µ) and (λ ,µ ). Then the 1 1 e elasticitytensorsfortheinclusionsandthebackgroundcanbewritten,respectively,as C1 and C0, with C1 = λ δ δ +µ (δ δ +δ δ ), ijkl 1 ij kl 1 ik jl il jk and C0 = λδ δ +µ(δ δ +δ δ ), (1.3) ijkl ij kl ik jl il jk 2 where i, j,k,l = 1,2,3 and δ is the Kronecker symbol: δ = 0 for i , j, δ = 1 for ij ij ij i = j. Letu = (u ,u ,u )T : Ω R3 denotethedisplacementfield. Foragivenvector 1 2 3 → valuedfunctionϕ, weconsiderthefollowingDirichletproblemfortheLame´ system χ C0 +χ C1 e(u) = 0, inΩ, ∇·(cid:18)(cid:16) Ω D1∪D2 (cid:17) (cid:19) (1.4) u = ϕ, e on∂Ω, whereχ is thecharacteristicfunctionof D R3, D ⊂ 1 e(u) = u+( u)T 2 ∇ ∇ (cid:16) (cid:17) isthestraintensor. Assumethat thestandard ellipticityconditionholdsfor(1.4), thatis, µ > 0, 3λ+2µ > 0; µ > 0, 3λ +2µ > 0. (1.5) 1 1 1 For ϕ H1(Ω;R3), it is well known that there exists a unique solution u H1(Ω;R3) ∈ ∈ oftheDirichletproblem(1.4),which isalso theminimizeroftheenergy functional 1 J [u] = χ C0 +χ C1 e(u),e(u) dx 1 2 ZΩ(cid:18)(cid:16) Ω D1∪D2 (cid:17) (cid:19) e on H1(Ω;R3) := u H1(Ω;R3) u ϕ H1(Ω;R3) . ϕ ∈ − ∈ 0 n (cid:12) o Moredetailscan befoundin theAppendixin [1(cid:12)3]. (cid:12) Introducethelinearspaceofrigiddisplacementin R3, Ψ := ψ C1(R3;R3) ψ+( ψ)T = 0 , ∈ ∇ ∇ (cid:26) (cid:27) (cid:12) (cid:12) equivalently, (cid:12) 1 0 0 x x 0 2 3 Ψ = span ψ1 = 0 , ψ2 = 1 , ψ3 = 0 , ψ4 = x , ψ5 = 0 , ψ6 = x .         1    3   If ξ H1(D;R3),e(0ξ) = 0 in D0, and D R13 is a conne−c0ted open set−, txh1en ξ is ali−nxe2ar  ∈ ⊂ combination of ψα in D. If an element ξ in Ψ vanishes at three non-collinear points, { } then ξ 0,seeLemma6.1. ≡ For fixed λ and µ satisfying µ > 0 and 3λ+2µ > 0, denoting u the solutionof λ1,µ1 (1.4). Then,as provedintheAppendixin[13], u u in H1(Ω;R3) as min µ ,3λ +2µ , λ1,µ1 → { 1 1 1} → ∞ whereu isa H1(Ω;R3)solutionof u := C0e(u) = 0, in Ω, λ,µ L ∇· u = u , (cid:16) (cid:17) on ∂D ∂D , e((cid:12)(cid:12)(cid:12)+u)∂=u 0(cid:12)(cid:12)(cid:12)−, ψα = 0, iin=De11,∪21,∪Dα2,=21,2, ,6, (1.6) Ru∂D=i ϕ∂ν,0(cid:12)(cid:12)(cid:12)+ · on ∂Ω, ··· 3 where ∂u := C0e(u) ~n = λ( u)~n+µ u+( u)T ~n. ∂ν0(cid:12)(cid:12)+ (cid:16) (cid:17) ∇· (cid:16)∇ ∇ (cid:17) and ~n is the unit out(cid:12)(cid:12)er normal of D, i = 1,2. Here and throughout this paper the (cid:12) i subscript indicatesthelimitfromoutsideandinsidethedomain,respectively. Inthis ± paperwestudysolutionsof (1.6),aLame´ systemwithinfinitecoefficients in D D . 1 2 ∪ The existence, uniqueness and regularity of weak solutions of (1.6), as well as a variational formulation, can be found in the Appendix in [13]. In particular, the H1 weak solution is in C1(Ω;R3) C1(D D ;R3). The solution is also the unique 1 2 ∩ ∪ functionwhichhas theleastenergy inappropriatefunctionalspaces, characterized by e I [u] = minI [v], ∞ v ∞ ∈A where 1 I [v] := C(0)e(v),e(v) dx, ∞ 2 ZΩ (cid:16) (cid:17) and e := u H1(Ω;R3) e(u) = 0 in D D . A ∈ ϕ 1 ∪ 2 n (cid:12) o It is wellknown,see[38], that forany(cid:12)open setO and u,v C2(O), (cid:12) ∈ ∂u C0e(u),e(v) dx = u v+ v. (1.7) λ,µ ZO(cid:16) (cid:17) −ZO(cid:16)L (cid:17)· Z∂O ∂ν0(cid:12)(cid:12)+ · (cid:12) (cid:12) A calculationgives (cid:12) u = µ∆u +(λ+µ)∂ ( u), k = 1,2,3. (1.8) Lλ,µ k k xk ∇· (cid:16) (cid:17) Weassumethatforsomeδ > 0, 0 1 δ µ,3λ+2µ . (1.9) 0 ≤ ≤ δ 0 Since D and D are two strictlyconvex subdomainsof Ω, there exist two points P 1 2 1 ∈ ∂D and P ∂D suchthat 1 2 2 ∈ dist(P ,P ) = dist(∂D ,∂D ) = ǫ. (1.10) 1 2 1 2 Use P P to denote the line segment connecting P and P . Throughout the paper, 1 2 1 2 unlessotherwisestated,C denotesaconstant,whosevaluesmayvaryfromlinetoline, dependingonlyond,κ ,κ ,γ,δ ,andanupperboundoftheC2normof∂ΩandtheC2,γ 0 1 0 normsof∂D and ∂D , but notonǫ. Also,wecall aconstanthaving suchdependence 1 2 auniversalconstant. Themainresult ofthispaper isfordimensionthree. Theorem 1.1. Assume that Ω, D ,D , ǫ are defined in (1.2), λ and µ satisfy (1.9) for 1 2 some δ > 0, and ϕ C2(∂Ω;R3). Let u H1(Ω;R3) C1(Ω;R3) be the solution of 0 ∈ ∈ ∩ (1.6). Then for0 < ǫ < 1/2, wehave e C u ϕ , (1.11) k∇ kL∞(Ω;R3) ≤ ǫ lnǫ k kC2(∂Ω;R3) | | whereC is auniversalconstant. 4 Remark1.1. TheproofofTheorem1.1actuallygivesthefollowingstrongerestimates: C Cdist(x,P P ) u(x) + 1 2 ϕ , x Ω, (1.12) |∇ | ≤ (cid:18) lnǫ ǫ +dist2(x,P P ) ǫ +dist2(x,P P )(cid:19)k kC2(∂Ω;R3) ∈ 1 2 1 2 | | e (cid:0) (cid:1) and u(x) C ϕ , x D D . (1.13) C2(∂Ω;R3) 1 2 |∇ | ≤ k k ∈ ∪ Remark 1.2. The strict convexity assumption on ∂D and ∂D can be replaced by a 1 2 weaker relativestrict convexityassumption,see(3.5)inSection 3. Remark1.3. Hereϕ C2(∂Ω;R3)canbereplacedbyϕ H1/2(∂Ω;R3). Indeed,theH1 ∈ ∈ norm of the solutionu in Ω is bounded by a universalconstant. Then standard elliptic estimatesgiveauniversalboundofuinC2 normin x Ω κ1 < dist(x,∂Ω) < κ1 . We ∈ 4 2 applythetheoremin Ω′ := x ∈ Ω dist(x,∂Ω) > κ31n withϕ(cid:12)(cid:12)(cid:12)′ := u∂Ω′. o n (cid:12) o (cid:12) Remark 1.4. Since the blow up rat(cid:12)e of u for solutions of the(cid:12)scalar equation (1.1) when k = is known to reach the(cid:12)mag|∇nitu∞d|e (ǫ lnǫ ) 1 in dimen(cid:12)sion three, see [11], − ∞ | | estimate(1.11)isexpected tobeoptimal. Following arguments in the proof of Theorem 1.1, we establish the corresponding estimates for higher dimensions d 4. Let Ω Rd, d 4 be a bounded open set ≥ ⊂ ≥ with C2 boundary, and D and D are two disjoint convex open sets in Ω with C2,γ 1 2 boundaries,satisfying(1.2). Let C0 begivenby(1.3)withi, j,k,l = 1,2, ,d,where ··· λ and µsatisfy µ > 0, dλ+2µ > 0, and Ψ := ψ C1(Rd;Rd) ψ+( ψ)T = 0 (1.14) ∈ ∇ ∇ (cid:26) (cid:27) (cid:12) be the linear space of rigid displacement in R(cid:12)(cid:12)d. With e , ,e denoting the standard 1 d ··· basisofRd, e, x e x e 1 i d, 1 j < k d i j k k j − ≤ ≤ ≤ ≤ n (cid:12) o isa basisofΨ. DenotethebasisofΨ(cid:12)(cid:12) as ψα , α = 1,2, , d(d+1). Consider { } ··· 2 u := C0e(u) = 0, in Ω, λ,µ L ∇· u = u , (cid:16) (cid:17) on ∂D ∂D , e((cid:12)(cid:12)(cid:12)+u)∂=u 0(cid:12)(cid:12)(cid:12)−, ψα = 0, iin=De11,∪21,∪Dα2,=21,2, , d(d+1), (1.15) Then wehaveRu∂D=i ϕ∂ν,0(cid:12)(cid:12)(cid:12)+ · on ∂Ω. ··· 2 Theorem 1.2. Assume as above, and ϕ C2(∂Ω;Rd), d 4. Let u H1(Ω;Rd) ∈ ≥ ∈ ∩ C1(Ω;Rd)bethesolutionof (1.15). Then for0 < ǫ < 1/2,we have e C u ϕ , (1.16) k∇ kL∞(Ω;Rd) ≤ ǫ k kC2(∂Ω;Rd) whereC is auniversalconstant. 5 Remark 1.5. The proof of Theorem 1.2 actually gives the following stronger estimate in dimensiond 4: ≥ C ϕ , x Ω, ǫ +dist2(x,P P )k kC2(∂Ω;Rd) ∈ 1 2 |∇u(x)| ≤  e C ϕ , x D D . Wealso haveRemarks 1.2–k1.4kCa2c(∂cΩo;Rrdd)ingly. ∈ 1 ∪ 2 The rest of this paper is organized as follows. In Section 2, we first introduce a setup for the proof of Theorem 1.1. Then we state a proposition, Proposition 2.1, containing key estimates, and deduce Theorem 1.1 from the proposition. In Sections 3 and 4, we prove Proposition 2.1. The proof of Theorem 1.2 is given in Section 5. A linear algebra lemma, Lemma 6.2, used in the proof of Theorem 1.1, is given in Section 6. 2 Outline of the Proof of Theorem 1.1 TheproofofTheorem1.1makesuseofthefollowingdecomposition. Bythethirdline of (1.6), u is a linear combination of ψα in D and D , respectively. Since it is clear 1 2 { } that ξ = 0inΩandξ = 0on∂Ωimplythatξ = 0inΩ,wedecomposethesolution λ,µ L of (1.6), as in[13], as follows: e e e 6 Cαψα, in D , α=1 1 1 u = αPP=661C2αψα, 6 in D2, (2.1) αP=1C1αvα1 +αP=1C2αvα2 +v0, inΩe, wherevα C1(Ω;R3),i = 1,2,α = 1,2, ,6,andv C1(Ω;R3)arerespectivelythe i ∈ ··· 0 ∈ solutionof e vα = 0, inΩ, e Lλ,µ i vα = ψα, on ∂D, (2.2) vαi = 0, one∂Di ∂Ω, j , i, and  i j ∪ v = 0, inΩ, λ,µ 0 L v = 0, on∂D ∂D , (2.3) v0 = ϕ, one∂Ω.1 ∪ 2 0 TheconstantsCα :=Cα(ǫ), i= 1,2,α = 1,2, ,6,are uniquelydeterminedbyu. i i ··· By thedecomposition(2.1), wewrite 3 3 2 6 u = Cα Cα vα + Cα (vα +vα)+ Cα vα + v , inΩ, (2.4) ∇ 1 − 2 ∇ 1 2∇ 1 2 i ∇ i ∇ 0 Xα=1 (cid:0) (cid:1) Xα=1 Xi=1 Xα=4 e 6 then 3 3 2 6 u Cα Cα vα + Cα (vα +vα) + Cα vα + v , inΩ. |∇ | ≤ 1 − 2 ∇ 1 2 ∇ 1 2 i ∇ i ∇ 0 Xα=1 (cid:12) (cid:12)(cid:12) (cid:12) Xα=1 (cid:12) (cid:12)(cid:12) (cid:12) Xi=1 Xα=4 (cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (2e.5) The proof of Theorem 1.1 can be reduced to the following proposition. Without loss ofgenerality,weonlyneedtoproveTheorem1.1for ϕ = 1,andforthegeneral C2(∂Ω) k k caseby consideringu/ ϕ if ϕ > 0. If ϕ = 0,then u = 0. k kC2(∂Ω) k kC2(∂Ω) ∂Ω (cid:12) Proposition2.1. UnderthehypothesesofTheorem1(cid:12)(cid:12).1,andthenormalization ϕ C2(∂Ω) = k k 1, let vα and v be the solution to (2.2) and (2.3), respectively. Then for 0 < ǫ < 1/2, i 0 we have v C; (2.6) ∇ 0 L∞(Ω) ≤ (cid:13)(cid:13)(cid:13)∇(vα1(cid:13)(cid:13)(cid:13)+veα2) L∞(Ω) ≤ C, α = 1,2,3; (2.7) (cid:13)(cid:13) (cid:13)(cid:13) C (cid:13) vα(x) (cid:13) e , i = 1,2, α = 1,2,3, x Ω; (2.8) ∇ i ≤ ǫ +dist2(x,P P ) ∈ (cid:12) (cid:12) 1 2 (cid:12)(cid:12) (cid:12)(cid:12) Cdist(x,P P ) e vα(x) 1 2 +C, i = 1,2, α = 4,5,6, x Ω; (2.9) ∇ i ≤ ǫ +dist2(x,P P ) ∈ (cid:12) (cid:12) 1 2 (cid:12) (cid:12) (cid:12) (cid:12) e and Cα C, i = 1,2, α = 1,2, ,6; (2.10) i ≤ ··· (cid:12) (cid:12) (cid:12) (cid:12) C (cid:12) C(cid:12)α Cα , α = 1,2,3. (2.11) 1 − 2 ≤ lnǫ (cid:12) (cid:12) | | (cid:12) (cid:12) Proofof Theorem 1.1 byus(cid:12)ingProp(cid:12)osition2.1. Clearly,weonlyneedtoprovethethe- orem underthenormalization ϕ = 1. C2(∂Ω) k k Since 0 C4 C5 i i   estimate(1.13)follow∇suf=rom−−CC(2ii45.10−).0Ci6 C0i6, in Di, i = 1,2, By (2.5)and Proposition2.1, wehave 3 2 6 u(x) Cα Cα vα(x) + Cα vα +C ∇ ≤ 1 − 2 ∇ 1 i ∇ i (cid:12) (cid:12) Xα=1 (cid:12) (cid:12)(cid:12) (cid:12) Xi=1 Xα=4(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) C Cdist(x,P P ) + 1 2 . (2.12) ≤ lnǫ ǫ +dist2(x,P P ) ǫ +dist2(x,P P ) 1 2 1 2 | | (cid:16) (cid:17) Theorem 1.1followsimmediately. (cid:3) To complete this section, we recall some properties of the tensor C. For the isotropicelasticmaterial,let C := (C ) = λδ δ +µ δ δ +δ δ , µ > 0, dλ+2µ > 0. (2.13) ijkl ij kl ik jl il jk (cid:16) (cid:16) (cid:17)(cid:17) 7 ThecomponentsC satisfythefollowingsymmetriccondition: ijkl C = C = C , i, j,k,l= 1,2, ,d. (2.14) ijkl klij klji ··· Wewillusethefollowingnotations: d d (CA) = C A , and (A,B) := A : B = A B , ij ijkl kl ij ij kX,l=1 iX,j=1 foreverypairofd dmatricesA = (A ),B = (B ). Bythesymmetriccondition(2.14), ij ij × wehave (CA,B) = (A,CB), (2.15) (CA,B) = (CAT,B) = (CA,C) = (CAT,C). Foran arbitrary d d real symmetricmatrixη = (η ), wehave ij × C η η = λη η +2µη η . ijkl kl ij ii kk kj kj It followsfrom (2.13)that Csatisfiestheellipticitycondition min 2µ,dλ+2µ η2 C η η max 2µ,dλ+2µ η2, (2.16) ijkl kl ij | | ≤ ≤ | | n o n o d where η2 = η2. In particular, | | i,j=1 ij P min 2µ,dλ+2µ A+AT 2 C A+AT , A+AT . (2.17) ≤ n o(cid:12) (cid:12) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) (cid:12) (cid:12) (cid:12) (cid:12) 3 Estimates of v , vα , and (vα + vα) 0 |∇ | |∇ i | |∇ 1 2 | We first fix notations. Use (x ,x ,x ) to denote a point in R3 and x = (x ,x ). By a 1 2 3 ′ 1 2 translationandrotationifnecessary,wemayassumewithoutlossofgeneralitythatthe points P and P in(1.10)satisfy 1 2 ǫ ǫ P = 0 , ∂D , and P = 0 , ∂D . 1 ′ 1 2 ′ 2 2 ∈ −2 ∈ (cid:18) (cid:19) (cid:18) (cid:19) Fix asmalluniversalconstantR, suchthattheportionof∂D and ∂D near P and P , 1 2 1 2 respectively,can berepresented by ǫ ǫ x = +h (x ), and x = +h (x ), for x < 2R. (3.1) 3 1 ′ 3 2 ′ ′ 2 −2 | | Then by the smoothness assumptionson ∂D and ∂D , the functions h (x ) and h (x ) 1 2 1 ′ 2 ′ are ofclassC2,γ(B (0 )), satisfying R ′ ǫ ǫ +h (x ) > +h (x ), for x < 2R, 1 ′ 2 ′ ′ 2 −2 | | h (0 ) = h (0 ) = 0, h (0 ) = h (0 ) = 0, (3.2) 1 ′ 2 ′ 1 ′ 2 ′ ∇ ∇ 8 2h (0 ) κ I, 2h (0 ) κ I, (3.3) 1 ′ 0 2 ′ 0 ∇ ≥ ∇ ≤ − and h + h C. (3.4) 1 C2,γ(B ) 2 C2,γ(B ) k k ′2R k k ′2R ≤ In particular, we only use a weaker relative strict convexity assumption of ∂D and 1 ∂D , thatis 2 h (x ) h (x ) κ x 2, if x < 2R. (3.5) 1 ′ 2 ′ 0 ′ ′ − ≥ | | | | For0 r 2R, denote ≤ ≤ ǫ ǫ Ω := (x ,x ) R3 +h (x ) < x < +h (x ), x < r . r ′ 3 2 ′ 3 1 ′ ′ ∈ − 2 2 | | (cid:26) (cid:27) (cid:12) (cid:12) For0 z < R, let (cid:12) ′ ≤ | | ǫ ǫ Ω (z ) := (x ,x ) R3 +h (x ) < x < +h (x ), x z < s . (3.6) s ′ ′ 3 2 ′ 3 1 ′ ′ ′ ∈ − 2 2 | − | (cid:26) (cid:27) (cid:12) b (cid:12)(cid:12) 3.1 Estimates of v , vα for α = 1,2,3, and (vα + vα) |∇ 0| |∇ i | |∇ 1 2 | Lemma 3.1. v + v C. (3.7) k 0kL∞(Ω) k∇ 0kL∞(Ω) ≤ vα +vα + (vαe+vα) e C, α = 1,2, ,6. (3.8) k 1 2kL∞(Ω) k∇ 1 2 kL∞(Ω) ≤ ··· The proof of Lemma 3e.1 is essentially theesame as in [13] for dimension two. We omitithere. By Lemma3.1, (2.6)and (2.7)is proved. To estimate (2.8), we introduce a scalar function u¯ C2(R3), such that u¯ = 1 on ∈ ∂D , u¯ = 0 on∂D ∂Ω, 1 2 ∪ x h (x )+ ǫ u¯(x) = 3 − 2 ′ 2 , in Ω , (3.9) ǫ +h (x ) h (x ) 2R 1 ′ 2 ′ − and u¯ C. (3.10) k kC2(R3\ΩR) ≤ Define u¯α = u¯ψα, α = 1,2,3, in Ω, (3.11) 1 then u¯α = vα on ∂Ω. e 1 1 Similarly,wedefine e u¯α = uψα, α = 1,2,3, in Ω, (3.12) 2 such that u¯α = vα on ∂Ω, where u is a scalar function ineC2(R3) satisfying u = 1 on 2 2 ∂D , u = 0 on∂D ∂Ω, 2 1 ∪ e x +h (x )+ ǫ u(x) = − 3 1 ′ 2 , in Ω , (3.13) ǫ +h (x ) h (x ) 2R 1 ′ 2 ′ − and u C. (3.14) k kC2(R3\ΩR) ≤ In orderto prove(2.8),it suffices toprovethefollowingproposition. 9 Proposition 3.2. Assume the above, let vα H1(Ω;R3) be the weak solution of (2.2) i ∈ with α = 1,2,3. Then fori = 1,2, α = 1,2,3, e (vα u¯α)2dx C; (3.15) ZΩ ∇ i − i ≤ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and e vα C, (3.16) (cid:13)∇ i (cid:13)L∞(Ω\ΩR) ≤ (cid:13)(cid:13) C (cid:13)(cid:13) e , x √ǫ, ′ (vα u¯α)(x)  √ǫ | | ≤ x Ω . (3.17) Consequently, (cid:12)(cid:12)(cid:12)∇ i − i (cid:12)(cid:12)(cid:12) ≤ |Cx′|, √ǫ < |x′| ≤ R, ∀ ∈ R C vα(x) , x Ω , (3.18) ∇ i ≤ ǫ + x 2 ∀ ∈ R (cid:12) (cid:12) | ′| (cid:12) (cid:12) and (cid:12) (cid:12) C , x √ǫ, ′ vα(x)  √ǫ | | ≤ (3.19) A direct calculationgi(cid:12)(cid:12)(cid:12)∇vexs′,iin v(cid:12)(cid:12)(cid:12)ie≤wo|fCx′(|3,.2)-√(3ǫ.5<),|xth′|a≤t R. C x C ∂ u¯(x) | k| , k = 1,2, ∂ u¯(x) , x Ω . (3.20) | xk | ≤ ǫ + x 2 | x3 | ≤ ǫ + x 2 ∈ R ′ ′ | | | | Thus C u¯α(x) , i = 1,2, α = 1,2,3, x Ω . (3.21) |∇ i | ≤ ǫ + x 2 ∈ R ′ | | Fork,l = 1,2, C C x ∂ u¯(x) , ∂ u¯(x) | ′| , ∂ u¯(x) = 0, x Ω . (3.22) | xkxl | ≤ ǫ + x 2 | xkx3 | ≤ (ǫ + x 2)2 x3x3 ∈ R ′ ′ | | | | For u¯α, defined by (3.11) and (3.12), making use of (1.8) and (3.22), we have, for i i = 1,2, α = 1,2,3, C C x u¯α(x) C ∂ u¯(x) + | ′| , x Ω . (3.23) Lλ,µ i ≤ xkxl ≤ ǫ + x 2 (ǫ + x 2)2 ∈ R (cid:12) (cid:12) kX+l<6(cid:12) (cid:12) | ′| | ′| (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) For z 2R, wealwaysuseδto denote ′ | | ≤ ǫ +h (z ) h (z ) δ := δ(z ) = 1 ′ − 2 ′ . (3.24) ′ 2 By (3.2)-(3.5), 1 ǫ + z 2 δ(z ) C ǫ + z 2 . (3.25) ′ ′ ′ C | | ≤ ≤ | | (cid:16) (cid:17) (cid:16) (cid:17) 10

Gradient estimates for solutions of the Lam\'e system with partially infinite coefficients in dimensions greater than two

Similar Gradient estimates for solutions of the Lam\'e system with partially infinite coefficients in dimensions greater than two