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

0.28 MB·

by  JiGuang Bao

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 co-
efficients 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:[email protected].
Correspondingauthor.Email:[email protected].
†
Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854,
‡
USA.Email:[email protected].
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 coeffi-
cients,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