Gradient estimates for solutions of the Lamé system with partially ...
arXiv:1601.07879v1 [math.AP] 28 Jan 2016
Gradient estimates for solutions of the Lam´e system with partially infinite coefficients in dimensions greater than two JiGuang Bao∗ HaiGang Li∗† and YanYan Li‡ January 29, 2016
Abstract We establish upper bounds on the blow-up rate of the gradients of solutions of the Lam´e system with partially infinite coefficients in dimensions greater than two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero.
1 Introduction and main results In this paper, we establish upper bounds on the blow-up rate of the gradients of solutions of the Lam´e system with partially infinite coefficients in dimensions greater than two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero. This work is stimulated by the study of Babu˘ska, Andersson, Smith and Levin in [10] concerning initiation and growth of damage in composite materials. The Lam´e system is assumed and they computationally analyzed the damage and fracture in composite materials. They observed numerically that the size of the strain tensor remains bounded when the distance ǫ, between two inclusions, tends to zero. This was proved by Li and Nirenberg in [32]. Indeed such ǫ-independent gradient estimates was established there for solutions of divergence form second order elliptic systems, including linear systems of elasticity, with piecewise H¨older continuous coefficients in all dimensions. See Bonnetier and Vogelius [16] and Li and Vogelius [33] for corresponding results on divergence form elliptic equations. The estimates in [32] and [33] depend on the ellipticity of the coefficients. If ellipticity constants are allowed to deteriorate, the situation is very different. Consider the scalar equation ∇ · a (x)∇u k k = 0 in Ω, (1.1) u k = ϕ on ∂Ω,
∗ School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China. Email: [email protected]. † Corresponding author. Email: [email protected]. ‡ Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA. Email: [email protected].
1
where Ω is a bounded open set of Rd , d ≥ 2, containing two ǫ-apart convex inclusions D1 and D2 , ϕ ∈ C 2 (∂Ω) is given, and k ∈ (0, ∞) in D1 ∪ D2 , ak (x) = 1 in Ω \ D1 ∪ D2 .
When k = ∞, the L∞ -norm of |∇u∞ | for the solutions u∞ of (1.1) generally becomes unbounded as ǫ tends to 0. The blow up rate of |∇u∞ | is respectively ǫ −1/2 in dimension d = 2, (ǫ| ln ǫ|)−1 in dimension d = 3, and ǫ −1 in 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, more detailed, characterizations of the singular behavior of ∇u∞ have been obtained by Ammari, Ciraolo, Kang, Lee and Yun [3], Ammari, Kang, Lee, Lim and Zribi [9], Bonnetier and Triki [14, 15], Gorb and Novikov [24] and Kang, Lim and Yun [25, 26]. For related works, see [2, 4, 5, 12, 15, 17, 19, 20, 21, 22, 28, 29, 30, 31, 34, 35, 37, 39, 40] and the references therein. In this paper, we mainly investigate the gradient estimates for the Lam´e system with partially infinite coefficients in dimension d = 3, a physically relevant dimension. This paper is a continuation of [13], where the estimate for dimension d = 2, another physically relevant dimension, is established. We prove that (ǫ| ln ǫ|)−1 is an upper bound of the blow up rate of the strain tensor in dimension three, the same as the scalar 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 dimension d ≥ 4, which is also the same as the scalar equation case. Note that it has been proved in [11] that these upper bounds in dimension d ≥ 3 are optimal in the scalar equation case. We consider the Lam´e system in linear elasticity with piecewise constant coefficients, which is stimulated by the study of composite media with closely spaced interfacial boundaries. Let Ω ⊂ R3 be a bounded open set with C 2 boundary, and D1 and D2 are two disjoint convex open sets in Ω with C 2,γ boundaries, 0 < γ < 1, which are ǫ apart and far away from ∂Ω, that is, D1 , D2 ⊂ Ω, the principle curvatures of ∂D1 , ∂D2 ≥ κ0 > 0, ǫ := dist(D1 , D2 ) > 0, dist(D1 ∪ D2 , ∂Ω) > κ1 > 0,
(1.2)
where κ0 , κ1 are constants independent of ǫ. We also assume that the C 2,γ norms of ∂Di are bounded by some constant independent of ǫ. This implies that each Di contains a ball of radius r0∗ for some constant r0∗ > 0 independent of ǫ. Denote e := Ω \ D1 ∪ D2 . Ω
e and D1 ∪ D2 are occupied, respectively, by two different isotropic and Assume that Ω homogeneous materials with different Lam´e constants (λ, µ) and (λ1 , µ1 ). Then the elasticity tensors for the inclusions and the background can be written, respectively, as C1 and C0 , with Ci1j kl = λ1 δi j δkl + µ1 (δik δ jl + δil δ jk ), and Ci0j kl = λδi j δkl + µ(δik δ jl + δil δ jk ), 2
(1.3)
where i, j, k, l = 1, 2, 3 and δi j is the Kronecker symbol: δi j = 0 for i , j, δi j = 1 for i = j. Let u = (u1 , u2 , u3 )T : Ω → R3 denote the displacement field. For a given vector valued function ϕ, we consider the following Dirichlet problem for the Lam´e system 0 1 ∇ · χ C + χ C e(u) = 0, in Ω, e D1 ∪D2 Ω (1.4) u = ϕ, on ∂Ω,
where χD is the characteristic function of D ⊂ R3 , 1 e(u) = ∇u + (∇u)T 2 is the strain tensor. Assume that the standard ellipticity condition holds for (1.4), that is, µ > 0,
3λ + 2µ > 0;
µ1 > 0,
3λ1 + 2µ1 > 0.
(1.5)
For ϕ ∈ H 1 (Ω; R3 ), it is well known that there exists a unique solution u ∈ H 1 (Ω; R3 ) of the Dirichlet problem (1.4), which is also the minimizer of the energy functional Z 1 0 1 J1 [u] = χΩe C + χD1 ∪D2 C e(u), e(u) dx 2 Ω on o n Hϕ1 (Ω; R3 ) := u ∈ H 1 (Ω; R3 ) u − ϕ ∈ H01 (Ω; R3 ) . More details can be found in the Appendix in [13]. Introduce the linear space of rigid displacement in R3 , 1 3 3 T Ψ := ψ ∈ C (R ; R ) ∇ψ + (∇ψ) = 0 ,
equivalently, 1 0 0 x2 x3 0 1 2 3 4 5 Ψ = span ψ = 0 , ψ = 1 , ψ = 0 , ψ = −x1 , ψ = 0 , ψ6 = x3 0 0 1 0 −x1 −x2
If ξ ∈ H 1 (D; R3 ), e(ξ) = 0 in D, and D ⊂ R3 is a connected open set, then ξ is a linear combination of {ψα } in D. If an element ξ in Ψ vanishes at three non-collinear points, then ξ ≡ 0, see Lemma 6.1. For fixed λ and µ satisfying µ > 0 and 3λ + 2µ > 0, denoting uλ1 ,µ1 the solution of (1.4). Then, as proved in the Appendix in [13], uλ1 ,µ1 → u in H 1 (Ω; R3 )
as min{µ1 , 3λ1 + 2µ1 } → ∞,
where u is a H 1 (Ω; R3 ) solution of 0 L u := ∇ · C e(u) = 0, λ,µ u = u − , + e(u) = 0, R ∂u · ψα = 0, ∂Di ∂ν0 + u = ϕ,
3
e in Ω, on ∂D1 ∪ ∂D2 , in D1 ∪ D2 , i = 1, 2, α = 1, 2, · · · , 6, on ∂Ω,
(1.6)
.
where
∂u := C0 e(u) ~n = λ (∇ · u) ~n + µ ∇u + (∇u)T ~n. ∂ν0 + and ~n is the unit outer normal of Di , i = 1, 2. Here and throughout this paper the subscript ± indicates the limit from outside and inside the domain, respectively. In this paper we study solutions of (1.6), a Lam´e system with infinite coefficients in D1 ∪ D2 . 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 e R3 ) ∩ C 1 (D1 ∪ D2 ; R3 ). The solution is also the unique H 1 weak solution is in C 1 (Ω; function which has the least energy in appropriate functional spaces, characterized by I∞ [u] = min I∞ [v], v∈A
where
and
1 I∞ [v] := 2
Z C(0) e(v), e(v) dx, e Ω
o n A := u ∈ Hϕ1 (Ω; R3 ) e(u) = 0 in D1 ∪ D2 .
It is well known, see [38], that for any open set O and u, v ∈ C 2 (O), Z Z Z ∂u 0 · v. C e(u), e(v) dx = − Lλ,µ u · v + O O ∂O ∂ν0 +
A calculation gives Lλ,µ u = µ∆uk + (λ + µ)∂xk (∇ · u) , k
k = 1, 2, 3.
(1.7)
(1.8)
We assume that for some δ0 > 0,
δ0 ≤ µ, 3λ + 2µ ≤
1 . δ0
(1.9)
Since D1 and D2 are two strictly convex subdomains of Ω, there exist two points P1 ∈ ∂D1 and P2 ∈ ∂D2 such that dist(P1 , P2 ) = dist(∂D1 , ∂D2 ) = ǫ.
(1.10)
Use P1 P2 to denote the line segment connecting P1 and P2 . Throughout the paper, unless otherwise stated, C denotes a constant, whose values may vary from line to line, depending only on d, κ0, κ1 , γ, δ0, and an upper bound of the C 2 norm of ∂Ω and the C 2,γ norms of ∂D1 and ∂D2 , but not on ǫ. Also, we call a constant having such dependence a universal constant. The main result of this paper is for dimension three. Theorem 1.1. Assume that Ω, D1 , D2 , ǫ are defined in (1.2), λ and µ satisfy (1.9) for e R3 ) be the solution of some δ0 > 0, and ϕ ∈ C 2 (∂Ω; R3 ). Let u ∈ H 1 (Ω; R3 ) ∩ C 1 (Ω; (1.6). Then for 0 < ǫ < 1/2, we have k∇ukL∞ (Ω;R3 ) ≤
C kϕkC2 (∂Ω;R3 ) , ǫ| ln ǫ|
where C is a universal constant. 4
(1.11)
Remark 1.1. The proof of Theorem 1.1 actually gives the following stronger estimates: |∇u(x)| ≤ and
Cdist(x, P1 P2 )
C
e (1.12) kϕkC2 (∂Ω;R3 ) , x ∈ Ω, + | ln ǫ| ǫ + dist2 (x, P1 P2 ) ǫ + dist2 (x, P1 P2 ) |∇u(x)| ≤ CkϕkC2 (∂Ω;R3 ) ,
x ∈ D1 ∪ D2 .
(1.13)
Remark 1.2. The strict convexity assumption on ∂D1 and ∂D2 can be replaced by a weaker relative strict convexity assumption, see (3.5) in Section 3. Remark 1.3. Here ϕ ∈ C 2 (∂Ω; R3 ) can be replaced by ϕ ∈ H 1/2 (∂Ω; R3 ). Indeed, the H 1 norm of the solution u in Ω is bounded by a universal n constant. o κ Then standard κelliptic 2 estimates give a universal bound of u in C norm in x ∈ Ω 41 < dist(x, ∂Ω) < 21 . We o n apply the theorem in Ω′ := x ∈ Ω dist(x, ∂Ω) > κ31 with ϕ′ := u ∂Ω′ . Remark 1.4. Since the blow up rate of |∇u∞ | for solutions of the scalar equation (1.1) when k = ∞ is known to reach the magnitude (ǫ| ln ǫ|)−1 in dimension three, see [11], estimate (1.11) is expected to be optimal.
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 C 2 boundary, and D1 and D2 are two disjoint convex open sets in Ω with C 2,γ boundaries, satisfying (1.2). Let C0 be given by (1.3) with i, j, k, l = 1, 2, · · · , d, where λ and µ satisfy µ > 0, dλ + 2µ > 0, and
Ψ := ψ ∈ C 1 (Rd ; Rd ) ∇ψ + (∇ψ)T = 0
(1.14)
be the linear space of rigid displacement in Rd . With e1 , · · · , ed denoting the standard basis of Rd , o n ei , x j ek − xk e j 1 ≤ i ≤ d, 1 ≤ j < k ≤ d
is a basis of Ψ. Denote the basis of Ψ as {ψα }, α = 1, 2, · · · , d(d+1) . Consider 2 0 e L u := ∇ · C e(u) = 0, in Ω, λ,µ on ∂D1 ∪ ∂D2 , u + = u − , e(u) = 0, in D1 ∪ D2 , R ∂u α · ψ = 0, i = 1, 2, α = 1, 2, · · · , d(d+1) , ∂ν0 + 2 ∂D i u = ϕ, on ∂Ω.
(1.15)
Then we have
Theorem 1.2. Assume as above, and ϕ ∈ C 2 (∂Ω; Rd ), d ≥ 4. Let u ∈ H 1 (Ω; Rd ) ∩ e Rd ) be the solution of (1.15). Then for 0 < ǫ < 1/2, we have C 1 (Ω; k∇ukL∞ (Ω;Rd ) ≤
where C is a universal constant. 5
C kϕkC2 (∂Ω;Rd ) , ǫ
(1.16)
Remark 1.5. The proof of Theorem 1.2 actually gives the following stronger estimate in dimension d ≥ 4: C e kϕkC2 (∂Ω;Rd ) , x ∈ Ω, 2 ǫ + dist (x, P1 P2 ) |∇u(x)| ≤ CkϕkC2 (∂Ω;Rd ) , x ∈ D1 ∪ D2 .
We also have Remarks 1.2–1.4 accordingly.
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 The proof of Theorem 1.1 makes use of the following decomposition. By the third line of (1.6), u is a linear combination of {ψα } in D1 and D2 , respectively. Since it is clear e and ξ = 0 on ∂Ω e imply that ξ = 0 in Ω, e we decompose the solution that Lλ,µ ξ = 0 in Ω of (1.6), as in [13], as follows: 6 P C1α ψα , in D1 , α=1 6 P (2.1) u= C2α ψα , in D2 , α=1 6 6 P P α α e C2α vα2 + v0 , in Ω, C v + 1 1 α=1
α=1
e R3 ), i = 1, 2, α = 1, 2, · · · , 6, and v0 ∈ C 1 (Ω; e R3 ) are respectively the where vαi ∈ C 1 (Ω; solution of e Lλ,µ vαi = 0, in Ω, α (2.2) vi = ψα , on ∂Di , α v = 0, on ∂D j ∪ ∂Ω, j , i, i
and
Lλ,µ v0 = 0, v0 = 0, v0 = ϕ,
e in Ω, on ∂D1 ∪ ∂D2 , on ∂Ω.
(2.3)
The constants Ciα := Ciα (ǫ), i = 1, 2, α = 1, 2, · · · , 6, are uniquely determined by u. By the decomposition (2.1), we write ∇u =
3 X α=1
X XX C1α − C2α ∇vα1 + C2α ∇(vα1 + vα2 ) + Ciα ∇vαi + ∇v0 , 3
2
α=1
6
i=1 α=4
6
e in Ω,
(2.4)
then |∇u| ≤
6 2 X 3 3 X X X C α ∇vα + ∇v , C α ∇(vα + vα ) + C α − C α ∇vα + 0 i i 1 2 2 1 1 2 i=1 α=4
α=1
α=1
e in Ω.
(2.5) The proof of Theorem 1.1 can be reduced to the following proposition. Without loss of generality, we only need to prove Theorem 1.1 for kϕkC2 (∂Ω) = 1, and for the general case by considering u/kϕkC2 (∂Ω) if kϕkC2 (∂Ω) > 0. If ϕ ∂Ω = 0, then u = 0.
Proposition 2.1. Under the hypotheses of Theorem 1.1, and the normalization kϕkC2 (∂Ω) = 1, let vαi and v0 be the solution to (2.2) and (2.3), respectively. Then for 0 < ǫ < 1/2, we have
∇v0 ∞ e ≤ C; (2.6)
α L (Ω)α
∇(v + v ) ≤ C, α = 1, 2, 3; (2.7) 1
2
α ∇vi (x) ≤
and
α ∇vi (x) ≤
e L∞ (Ω)
C
ǫ + dist2 (x, P1 P2 ) Cdist(x, P1 P2 )
e i = 1, 2, α = 1, 2, 3, x ∈ Ω;
,
ǫ + dist2 (x, P1 P2 )
+ C,
e i = 1, 2, α = 4, 5, 6, x ∈ Ω;
α Ci ≤ C, i = 1, 2, α = 1, 2, · · · , 6; α C C1 − C2α ≤ , α = 1, 2, 3. | ln ǫ|
(2.8)
(2.9)
(2.10) (2.11)
Proof of Theorem 1.1 by using Proposition 2.1. Clearly, we only need to prove the theorem under the normalization kϕkC2 (∂Ω) = 1. Since Ci4 Ci5 0 4 6 0 Ci , in Di , i = 1, 2, ∇u = −Ci −Ci5 −Ci6 0
estimate (1.13) follows from (2.10). By (2.5) and Proposition 2.1, we have
3 2 X 6 X α X C α ∇vα + C α α ∇u(x) ≤ C1 − C2 ∇v1 (x) + i i α=1
≤
i=1 α=4
Cdist(x, P1 P2 ) C + . ǫ + dist2 (x, P1 P2 ) | ln ǫ| ǫ + dist2 (x, P1 P2 )
Theorem 1.1 follows immediately.
(2.12)
To complete this section, we recall some properties of the tensor C. For the isotropic elastic material, let C := (Ci j kl ) = λδi j δkl + µ δik δ jl + δil δ jk , µ > 0, dλ + 2µ > 0. (2.13) 7
The components Ci j kl satisfy the following symmetric condition: Ci j kl = Ckl i j = Ckl j i ,
i, j, k, l = 1, 2, · · · , d.
(2.14)
We will use the following notations: (CA)i j =
d X
Ci j kl Akl ,
and
(A, B) := A : B =
d X
Ai j Bi j ,
i, j=1
k,l=1
for every pair of d ×d matrices A = (Ai j ), B = (Bi j ). By the symmetric condition (2.14), we have (CA, B) = (A, CB), (2.15) (CA, B) = (CAT , B) = (CA, C) = (CAT , C). For an arbitrary d × d real symmetric matrix η = (ηi j ), we have Ci j kl ηkl ηi j = λ ηii ηkk + 2µ ηk j ηk j . It follows from (2.13) that C satisfies the ellipticity condition n o n o min 2µ, dλ + 2µ |η|2 ≤ Ci j kl ηkl ηi j ≤ max 2µ, dλ + 2µ |η|2 , where |η|2 =
d P
i, j=1
(2.16)
η2i j . In particular,
n o 2 min 2µ, dλ + 2µ A + AT ≤ C A + AT , A + AT .
(2.17)
3 Estimates of |∇v0|, |∇vαi |, and |∇(vα1 + vα2 )|
We first fix notations. Use (x1 , x2 , x3 ) to denote a point in R3 and x′ = (x1 , x2 ). By a translation and rotation if necessary, we may assume without loss of generality that the points P1 and P2 in (1.10) satisfy ǫ ǫ P1 = 0 ′ , ∈ ∂D1 , and P2 = 0′ , − ∈ ∂D2 . 2 2 Fix a small universal constant R, such that the portion of ∂D1 and ∂D2 near P1 and P2 , respectively, can be represented by x3 =
ǫ + h1 (x′ ), 2
and
ǫ x3 = − + h2 (x′ ), 2
for |x′ | < 2R.
(3.1)
Then by the smoothness assumptions on ∂D1 and ∂D2 , the functions h1 (x′ ) and h2 (x′ ) are of class C 2,γ (BR (0′ )), satisfying ǫ ǫ + h1 (x′ ) > − + h2 (x′ ), 2 2 h1 (0′ ) = h2 (0′ ) = 0,
for |x′ | < 2R,
∇h1 (0′ ) = ∇h2 (0′ ) = 0, 8
(3.2)
∇2 h1 (0′ ) ≥ κ0 I,
∇2 h2 (0′ ) ≤ −κ0 I,
(3.3)
and kh1 kC2,γ (B′2R ) + kh2 kC2,γ (B′2R ) ≤ C.
(3.4)
In particular, we only use a weaker relative strict convexity assumption of ∂D1 and ∂D2 , that is h1 (x′ ) − h2 (x′ ) ≥ κ0 |x′ |2 , if |x′ | < 2R. (3.5) For 0 ≤ r ≤ 2R, denote ǫ ǫ Ωr := (x′ , x3 ) ∈ R3 − + h2 (x′ ) < x3 < + h1 (x′ ), |x′ | < r . 2 2
For 0 ≤ |z′ | < R, let b s (z′ ) := (x′ , x3 ) ∈ R3 − ǫ + h2 (x′ ) < x3 < ǫ + h1 (x′ ), |x′ − z′ | < s . Ω 2 2
(3.6)
3.1 Estimates of |∇v0 |, |∇vαi | for α = 1, 2, 3, and |∇(vα1 + vα2 )|
Lemma 3.1. kvα1
+
vα2 kL∞ (Ω) e
kv0 kL∞ (Ω) e + k∇v0 k L∞ (Ω) e ≤ C.
+
k∇(vα1
+
vα2 )kL∞ (Ω) e
≤ C,
α = 1, 2, · · · , 6.
(3.7) (3.8)
The proof of Lemma 3.1 is essentially the same as in [13] for dimension two. We omit it here. By Lemma 3.1, (2.6) and (2.7) is proved. To estimate (2.8), we introduce a scalar function u¯ ∈ C 2 (R3 ), such that u¯ = 1 on ∂D1 , u¯ = 0 on ∂D2 ∪ ∂Ω, x3 − h2 (x′ ) + 2ǫ , u¯ (x) = ǫ + h1 (x′ ) − h2 (x′ )
in Ω2R ,
(3.9)
and k¯ukC2 (R3 \ΩR ) ≤ C.
(3.10)
Define u¯ α1 = u¯ ψα ,
α = 1, 2, 3,
e in Ω,
(3.11)
u¯ α2 = uψα ,
α = 1, 2, 3,
e in Ω,
(3.12)
in Ω2R ,
(3.13)
e then u¯ α1 = vα1 on ∂Ω. Similarly, we define
e where u is a scalar function in C 2 (R3 ) satisfying u = 1 on such that u¯ α2 = vα2 on ∂Ω, ∂D2 , u = 0 on ∂D1 ∪ ∂Ω, u(x) =
−x3 + h1 (x′ ) + 2ǫ , ǫ + h1 (x′ ) − h2 (x′ )
and kukC2 (R3 \ΩR ) ≤ C. In order to prove (2.8), it suffices to prove the following proposition. 9
(3.14)
e R3 ) be the weak solution of (2.2) Proposition 3.2. Assume the above, let vαi ∈ H 1 (Ω; with α = 1, 2, 3. Then for i = 1, 2, α = 1, 2, 3, Z ∇(vα − u¯ α ) 2 dx ≤ C; (3.15) i i e Ω
and
Consequently,
α
∇vi L∞ (Ω\Ω e R ) ≤ C, C √ √ , |x′ | ≤ ǫ, α ǫ ∇(vi − u¯ αi )(x) ≤ √ C ǫ < |x′ | ≤ R, ′, |x | α ∇vi (x) ≤
and
C , ǫ + |x′ |2
(3.16)
∀ x ∈ ΩR .
(3.17)
∀ x ∈ ΩR ,
(3.18)
C √ ′ ǫ, , |x | ≤ √ ǫ ∇x′ vαi (x) ≤ √ C ǫ < |x′ | ≤ R. ′, |x |
(3.19)
A direct calculation gives, in view of (3.2)-(3.5), that |∂ xk u¯ (x)| ≤
C|xk | , k = 1, 2, ǫ + |x′ |2
Thus |∇¯uαi (x)| ≤ For k, l = 1, 2, |∂ xk xl u¯ (x)| ≤
C , ǫ + |x′ |2
C , ǫ + |x′ |2
|∂ xk x3 u(x)| ¯ ≤
|∂ x3 u(x)| ¯ ≤
C , ǫ + |x′ |2
i = 1, 2, α = 1, 2, 3,
C|x′ | , (ǫ + |x′ |2 )2
x ∈ ΩR . x ∈ ΩR .
∂ x3 x3 u¯ (x) = 0,
(3.20)
(3.21)
x ∈ ΩR . (3.22)
For u¯ αi , defined by (3.11) and (3.12), making use of (1.8) and (3.22), we have, for i = 1, 2, α = 1, 2, 3, X ∂ u(x) Lλ,µ u¯ αi (x) ≤ C ≤ xk xl ¯ k+l
Gradient estimates for solutions of the Lam´e system with partially infinite coefficients in dimensions greater than two JiGuang Bao∗ HaiGang Li∗† and YanYan Li‡ January 29, 2016
Abstract We establish upper bounds on the blow-up rate of the gradients of solutions of the Lam´e system with partially infinite coefficients in dimensions greater than two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero.
1 Introduction and main results In this paper, we establish upper bounds on the blow-up rate of the gradients of solutions of the Lam´e system with partially infinite coefficients in dimensions greater than two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero. This work is stimulated by the study of Babu˘ska, Andersson, Smith and Levin in [10] concerning initiation and growth of damage in composite materials. The Lam´e system is assumed and they computationally analyzed the damage and fracture in composite materials. They observed numerically that the size of the strain tensor remains bounded when the distance ǫ, between two inclusions, tends to zero. This was proved by Li and Nirenberg in [32]. Indeed such ǫ-independent gradient estimates was established there for solutions of divergence form second order elliptic systems, including linear systems of elasticity, with piecewise H¨older continuous coefficients in all dimensions. See Bonnetier and Vogelius [16] and Li and Vogelius [33] for corresponding results on divergence form elliptic equations. The estimates in [32] and [33] depend on the ellipticity of the coefficients. If ellipticity constants are allowed to deteriorate, the situation is very different. Consider the scalar equation ∇ · a (x)∇u k k = 0 in Ω, (1.1) u k = ϕ on ∂Ω,
∗ School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China. Email: [email protected]. † Corresponding author. Email: [email protected]. ‡ Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA. Email: [email protected].
1
where Ω is a bounded open set of Rd , d ≥ 2, containing two ǫ-apart convex inclusions D1 and D2 , ϕ ∈ C 2 (∂Ω) is given, and k ∈ (0, ∞) in D1 ∪ D2 , ak (x) = 1 in Ω \ D1 ∪ D2 .
When k = ∞, the L∞ -norm of |∇u∞ | for the solutions u∞ of (1.1) generally becomes unbounded as ǫ tends to 0. The blow up rate of |∇u∞ | is respectively ǫ −1/2 in dimension d = 2, (ǫ| ln ǫ|)−1 in dimension d = 3, and ǫ −1 in 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, more detailed, characterizations of the singular behavior of ∇u∞ have been obtained by Ammari, Ciraolo, Kang, Lee and Yun [3], Ammari, Kang, Lee, Lim and Zribi [9], Bonnetier and Triki [14, 15], Gorb and Novikov [24] and Kang, Lim and Yun [25, 26]. For related works, see [2, 4, 5, 12, 15, 17, 19, 20, 21, 22, 28, 29, 30, 31, 34, 35, 37, 39, 40] and the references therein. In this paper, we mainly investigate the gradient estimates for the Lam´e system with partially infinite coefficients in dimension d = 3, a physically relevant dimension. This paper is a continuation of [13], where the estimate for dimension d = 2, another physically relevant dimension, is established. We prove that (ǫ| ln ǫ|)−1 is an upper bound of the blow up rate of the strain tensor in dimension three, the same as the scalar 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 dimension d ≥ 4, which is also the same as the scalar equation case. Note that it has been proved in [11] that these upper bounds in dimension d ≥ 3 are optimal in the scalar equation case. We consider the Lam´e system in linear elasticity with piecewise constant coefficients, which is stimulated by the study of composite media with closely spaced interfacial boundaries. Let Ω ⊂ R3 be a bounded open set with C 2 boundary, and D1 and D2 are two disjoint convex open sets in Ω with C 2,γ boundaries, 0 < γ < 1, which are ǫ apart and far away from ∂Ω, that is, D1 , D2 ⊂ Ω, the principle curvatures of ∂D1 , ∂D2 ≥ κ0 > 0, ǫ := dist(D1 , D2 ) > 0, dist(D1 ∪ D2 , ∂Ω) > κ1 > 0,
(1.2)
where κ0 , κ1 are constants independent of ǫ. We also assume that the C 2,γ norms of ∂Di are bounded by some constant independent of ǫ. This implies that each Di contains a ball of radius r0∗ for some constant r0∗ > 0 independent of ǫ. Denote e := Ω \ D1 ∪ D2 . Ω
e and D1 ∪ D2 are occupied, respectively, by two different isotropic and Assume that Ω homogeneous materials with different Lam´e constants (λ, µ) and (λ1 , µ1 ). Then the elasticity tensors for the inclusions and the background can be written, respectively, as C1 and C0 , with Ci1j kl = λ1 δi j δkl + µ1 (δik δ jl + δil δ jk ), and Ci0j kl = λδi j δkl + µ(δik δ jl + δil δ jk ), 2
(1.3)
where i, j, k, l = 1, 2, 3 and δi j is the Kronecker symbol: δi j = 0 for i , j, δi j = 1 for i = j. Let u = (u1 , u2 , u3 )T : Ω → R3 denote the displacement field. For a given vector valued function ϕ, we consider the following Dirichlet problem for the Lam´e system 0 1 ∇ · χ C + χ C e(u) = 0, in Ω, e D1 ∪D2 Ω (1.4) u = ϕ, on ∂Ω,
where χD is the characteristic function of D ⊂ R3 , 1 e(u) = ∇u + (∇u)T 2 is the strain tensor. Assume that the standard ellipticity condition holds for (1.4), that is, µ > 0,
3λ + 2µ > 0;
µ1 > 0,
3λ1 + 2µ1 > 0.
(1.5)
For ϕ ∈ H 1 (Ω; R3 ), it is well known that there exists a unique solution u ∈ H 1 (Ω; R3 ) of the Dirichlet problem (1.4), which is also the minimizer of the energy functional Z 1 0 1 J1 [u] = χΩe C + χD1 ∪D2 C e(u), e(u) dx 2 Ω on o n Hϕ1 (Ω; R3 ) := u ∈ H 1 (Ω; R3 ) u − ϕ ∈ H01 (Ω; R3 ) . More details can be found in the Appendix in [13]. Introduce the linear space of rigid displacement in R3 , 1 3 3 T Ψ := ψ ∈ C (R ; R ) ∇ψ + (∇ψ) = 0 ,
equivalently, 1 0 0 x2 x3 0 1 2 3 4 5 Ψ = span ψ = 0 , ψ = 1 , ψ = 0 , ψ = −x1 , ψ = 0 , ψ6 = x3 0 0 1 0 −x1 −x2
If ξ ∈ H 1 (D; R3 ), e(ξ) = 0 in D, and D ⊂ R3 is a connected open set, then ξ is a linear combination of {ψα } in D. If an element ξ in Ψ vanishes at three non-collinear points, then ξ ≡ 0, see Lemma 6.1. For fixed λ and µ satisfying µ > 0 and 3λ + 2µ > 0, denoting uλ1 ,µ1 the solution of (1.4). Then, as proved in the Appendix in [13], uλ1 ,µ1 → u in H 1 (Ω; R3 )
as min{µ1 , 3λ1 + 2µ1 } → ∞,
where u is a H 1 (Ω; R3 ) solution of 0 L u := ∇ · C e(u) = 0, λ,µ u = u − , + e(u) = 0, R ∂u · ψα = 0, ∂Di ∂ν0 + u = ϕ,
3
e in Ω, on ∂D1 ∪ ∂D2 , in D1 ∪ D2 , i = 1, 2, α = 1, 2, · · · , 6, on ∂Ω,
(1.6)
.
where
∂u := C0 e(u) ~n = λ (∇ · u) ~n + µ ∇u + (∇u)T ~n. ∂ν0 + and ~n is the unit outer normal of Di , i = 1, 2. Here and throughout this paper the subscript ± indicates the limit from outside and inside the domain, respectively. In this paper we study solutions of (1.6), a Lam´e system with infinite coefficients in D1 ∪ D2 . 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 e R3 ) ∩ C 1 (D1 ∪ D2 ; R3 ). The solution is also the unique H 1 weak solution is in C 1 (Ω; function which has the least energy in appropriate functional spaces, characterized by I∞ [u] = min I∞ [v], v∈A
where
and
1 I∞ [v] := 2
Z C(0) e(v), e(v) dx, e Ω
o n A := u ∈ Hϕ1 (Ω; R3 ) e(u) = 0 in D1 ∪ D2 .
It is well known, see [38], that for any open set O and u, v ∈ C 2 (O), Z Z Z ∂u 0 · v. C e(u), e(v) dx = − Lλ,µ u · v + O O ∂O ∂ν0 +
A calculation gives Lλ,µ u = µ∆uk + (λ + µ)∂xk (∇ · u) , k
k = 1, 2, 3.
(1.7)
(1.8)
We assume that for some δ0 > 0,
δ0 ≤ µ, 3λ + 2µ ≤
1 . δ0
(1.9)
Since D1 and D2 are two strictly convex subdomains of Ω, there exist two points P1 ∈ ∂D1 and P2 ∈ ∂D2 such that dist(P1 , P2 ) = dist(∂D1 , ∂D2 ) = ǫ.
(1.10)
Use P1 P2 to denote the line segment connecting P1 and P2 . Throughout the paper, unless otherwise stated, C denotes a constant, whose values may vary from line to line, depending only on d, κ0, κ1 , γ, δ0, and an upper bound of the C 2 norm of ∂Ω and the C 2,γ norms of ∂D1 and ∂D2 , but not on ǫ. Also, we call a constant having such dependence a universal constant. The main result of this paper is for dimension three. Theorem 1.1. Assume that Ω, D1 , D2 , ǫ are defined in (1.2), λ and µ satisfy (1.9) for e R3 ) be the solution of some δ0 > 0, and ϕ ∈ C 2 (∂Ω; R3 ). Let u ∈ H 1 (Ω; R3 ) ∩ C 1 (Ω; (1.6). Then for 0 < ǫ < 1/2, we have k∇ukL∞ (Ω;R3 ) ≤
C kϕkC2 (∂Ω;R3 ) , ǫ| ln ǫ|
where C is a universal constant. 4
(1.11)
Remark 1.1. The proof of Theorem 1.1 actually gives the following stronger estimates: |∇u(x)| ≤ and
Cdist(x, P1 P2 )
C
e (1.12) kϕkC2 (∂Ω;R3 ) , x ∈ Ω, + | ln ǫ| ǫ + dist2 (x, P1 P2 ) ǫ + dist2 (x, P1 P2 ) |∇u(x)| ≤ CkϕkC2 (∂Ω;R3 ) ,
x ∈ D1 ∪ D2 .
(1.13)
Remark 1.2. The strict convexity assumption on ∂D1 and ∂D2 can be replaced by a weaker relative strict convexity assumption, see (3.5) in Section 3. Remark 1.3. Here ϕ ∈ C 2 (∂Ω; R3 ) can be replaced by ϕ ∈ H 1/2 (∂Ω; R3 ). Indeed, the H 1 norm of the solution u in Ω is bounded by a universal n constant. o κ Then standard κelliptic 2 estimates give a universal bound of u in C norm in x ∈ Ω 41 < dist(x, ∂Ω) < 21 . We o n apply the theorem in Ω′ := x ∈ Ω dist(x, ∂Ω) > κ31 with ϕ′ := u ∂Ω′ . Remark 1.4. Since the blow up rate of |∇u∞ | for solutions of the scalar equation (1.1) when k = ∞ is known to reach the magnitude (ǫ| ln ǫ|)−1 in dimension three, see [11], estimate (1.11) is expected to be optimal.
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 C 2 boundary, and D1 and D2 are two disjoint convex open sets in Ω with C 2,γ boundaries, satisfying (1.2). Let C0 be given by (1.3) with i, j, k, l = 1, 2, · · · , d, where λ and µ satisfy µ > 0, dλ + 2µ > 0, and
Ψ := ψ ∈ C 1 (Rd ; Rd ) ∇ψ + (∇ψ)T = 0
(1.14)
be the linear space of rigid displacement in Rd . With e1 , · · · , ed denoting the standard basis of Rd , o n ei , x j ek − xk e j 1 ≤ i ≤ d, 1 ≤ j < k ≤ d
is a basis of Ψ. Denote the basis of Ψ as {ψα }, α = 1, 2, · · · , d(d+1) . Consider 2 0 e L u := ∇ · C e(u) = 0, in Ω, λ,µ on ∂D1 ∪ ∂D2 , u + = u − , e(u) = 0, in D1 ∪ D2 , R ∂u α · ψ = 0, i = 1, 2, α = 1, 2, · · · , d(d+1) , ∂ν0 + 2 ∂D i u = ϕ, on ∂Ω.
(1.15)
Then we have
Theorem 1.2. Assume as above, and ϕ ∈ C 2 (∂Ω; Rd ), d ≥ 4. Let u ∈ H 1 (Ω; Rd ) ∩ e Rd ) be the solution of (1.15). Then for 0 < ǫ < 1/2, we have C 1 (Ω; k∇ukL∞ (Ω;Rd ) ≤
where C is a universal constant. 5
C kϕkC2 (∂Ω;Rd ) , ǫ
(1.16)
Remark 1.5. The proof of Theorem 1.2 actually gives the following stronger estimate in dimension d ≥ 4: C e kϕkC2 (∂Ω;Rd ) , x ∈ Ω, 2 ǫ + dist (x, P1 P2 ) |∇u(x)| ≤ CkϕkC2 (∂Ω;Rd ) , x ∈ D1 ∪ D2 .
We also have Remarks 1.2–1.4 accordingly.
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 The proof of Theorem 1.1 makes use of the following decomposition. By the third line of (1.6), u is a linear combination of {ψα } in D1 and D2 , respectively. Since it is clear e and ξ = 0 on ∂Ω e imply that ξ = 0 in Ω, e we decompose the solution that Lλ,µ ξ = 0 in Ω of (1.6), as in [13], as follows: 6 P C1α ψα , in D1 , α=1 6 P (2.1) u= C2α ψα , in D2 , α=1 6 6 P P α α e C2α vα2 + v0 , in Ω, C v + 1 1 α=1
α=1
e R3 ), i = 1, 2, α = 1, 2, · · · , 6, and v0 ∈ C 1 (Ω; e R3 ) are respectively the where vαi ∈ C 1 (Ω; solution of e Lλ,µ vαi = 0, in Ω, α (2.2) vi = ψα , on ∂Di , α v = 0, on ∂D j ∪ ∂Ω, j , i, i
and
Lλ,µ v0 = 0, v0 = 0, v0 = ϕ,
e in Ω, on ∂D1 ∪ ∂D2 , on ∂Ω.
(2.3)
The constants Ciα := Ciα (ǫ), i = 1, 2, α = 1, 2, · · · , 6, are uniquely determined by u. By the decomposition (2.1), we write ∇u =
3 X α=1
X XX C1α − C2α ∇vα1 + C2α ∇(vα1 + vα2 ) + Ciα ∇vαi + ∇v0 , 3
2
α=1
6
i=1 α=4
6
e in Ω,
(2.4)
then |∇u| ≤
6 2 X 3 3 X X X C α ∇vα + ∇v , C α ∇(vα + vα ) + C α − C α ∇vα + 0 i i 1 2 2 1 1 2 i=1 α=4
α=1
α=1
e in Ω.
(2.5) The proof of Theorem 1.1 can be reduced to the following proposition. Without loss of generality, we only need to prove Theorem 1.1 for kϕkC2 (∂Ω) = 1, and for the general case by considering u/kϕkC2 (∂Ω) if kϕkC2 (∂Ω) > 0. If ϕ ∂Ω = 0, then u = 0.
Proposition 2.1. Under the hypotheses of Theorem 1.1, and the normalization kϕkC2 (∂Ω) = 1, let vαi and v0 be the solution to (2.2) and (2.3), respectively. Then for 0 < ǫ < 1/2, we have
∇v0 ∞ e ≤ C; (2.6)
α L (Ω)α
∇(v + v ) ≤ C, α = 1, 2, 3; (2.7) 1
2
α ∇vi (x) ≤
and
α ∇vi (x) ≤
e L∞ (Ω)
C
ǫ + dist2 (x, P1 P2 ) Cdist(x, P1 P2 )
e i = 1, 2, α = 1, 2, 3, x ∈ Ω;
,
ǫ + dist2 (x, P1 P2 )
+ C,
e i = 1, 2, α = 4, 5, 6, x ∈ Ω;
α Ci ≤ C, i = 1, 2, α = 1, 2, · · · , 6; α C C1 − C2α ≤ , α = 1, 2, 3. | ln ǫ|
(2.8)
(2.9)
(2.10) (2.11)
Proof of Theorem 1.1 by using Proposition 2.1. Clearly, we only need to prove the theorem under the normalization kϕkC2 (∂Ω) = 1. Since Ci4 Ci5 0 4 6 0 Ci , in Di , i = 1, 2, ∇u = −Ci −Ci5 −Ci6 0
estimate (1.13) follows from (2.10). By (2.5) and Proposition 2.1, we have
3 2 X 6 X α X C α ∇vα + C α α ∇u(x) ≤ C1 − C2 ∇v1 (x) + i i α=1
≤
i=1 α=4
Cdist(x, P1 P2 ) C + . ǫ + dist2 (x, P1 P2 ) | ln ǫ| ǫ + dist2 (x, P1 P2 )
Theorem 1.1 follows immediately.
(2.12)
To complete this section, we recall some properties of the tensor C. For the isotropic elastic material, let C := (Ci j kl ) = λδi j δkl + µ δik δ jl + δil δ jk , µ > 0, dλ + 2µ > 0. (2.13) 7
The components Ci j kl satisfy the following symmetric condition: Ci j kl = Ckl i j = Ckl j i ,
i, j, k, l = 1, 2, · · · , d.
(2.14)
We will use the following notations: (CA)i j =
d X
Ci j kl Akl ,
and
(A, B) := A : B =
d X
Ai j Bi j ,
i, j=1
k,l=1
for every pair of d ×d matrices A = (Ai j ), B = (Bi j ). By the symmetric condition (2.14), we have (CA, B) = (A, CB), (2.15) (CA, B) = (CAT , B) = (CA, C) = (CAT , C). For an arbitrary d × d real symmetric matrix η = (ηi j ), we have Ci j kl ηkl ηi j = λ ηii ηkk + 2µ ηk j ηk j . It follows from (2.13) that C satisfies the ellipticity condition n o n o min 2µ, dλ + 2µ |η|2 ≤ Ci j kl ηkl ηi j ≤ max 2µ, dλ + 2µ |η|2 , where |η|2 =
d P
i, j=1
(2.16)
η2i j . In particular,
n o 2 min 2µ, dλ + 2µ A + AT ≤ C A + AT , A + AT .
(2.17)
3 Estimates of |∇v0|, |∇vαi |, and |∇(vα1 + vα2 )|
We first fix notations. Use (x1 , x2 , x3 ) to denote a point in R3 and x′ = (x1 , x2 ). By a translation and rotation if necessary, we may assume without loss of generality that the points P1 and P2 in (1.10) satisfy ǫ ǫ P1 = 0 ′ , ∈ ∂D1 , and P2 = 0′ , − ∈ ∂D2 . 2 2 Fix a small universal constant R, such that the portion of ∂D1 and ∂D2 near P1 and P2 , respectively, can be represented by x3 =
ǫ + h1 (x′ ), 2
and
ǫ x3 = − + h2 (x′ ), 2
for |x′ | < 2R.
(3.1)
Then by the smoothness assumptions on ∂D1 and ∂D2 , the functions h1 (x′ ) and h2 (x′ ) are of class C 2,γ (BR (0′ )), satisfying ǫ ǫ + h1 (x′ ) > − + h2 (x′ ), 2 2 h1 (0′ ) = h2 (0′ ) = 0,
for |x′ | < 2R,
∇h1 (0′ ) = ∇h2 (0′ ) = 0, 8
(3.2)
∇2 h1 (0′ ) ≥ κ0 I,
∇2 h2 (0′ ) ≤ −κ0 I,
(3.3)
and kh1 kC2,γ (B′2R ) + kh2 kC2,γ (B′2R ) ≤ C.
(3.4)
In particular, we only use a weaker relative strict convexity assumption of ∂D1 and ∂D2 , that is h1 (x′ ) − h2 (x′ ) ≥ κ0 |x′ |2 , if |x′ | < 2R. (3.5) For 0 ≤ r ≤ 2R, denote ǫ ǫ Ωr := (x′ , x3 ) ∈ R3 − + h2 (x′ ) < x3 < + h1 (x′ ), |x′ | < r . 2 2
For 0 ≤ |z′ | < R, let b s (z′ ) := (x′ , x3 ) ∈ R3 − ǫ + h2 (x′ ) < x3 < ǫ + h1 (x′ ), |x′ − z′ | < s . Ω 2 2
(3.6)
3.1 Estimates of |∇v0 |, |∇vαi | for α = 1, 2, 3, and |∇(vα1 + vα2 )|
Lemma 3.1. kvα1
+
vα2 kL∞ (Ω) e
kv0 kL∞ (Ω) e + k∇v0 k L∞ (Ω) e ≤ C.
+
k∇(vα1
+
vα2 )kL∞ (Ω) e
≤ C,
α = 1, 2, · · · , 6.
(3.7) (3.8)
The proof of Lemma 3.1 is essentially the same as in [13] for dimension two. We omit it here. By Lemma 3.1, (2.6) and (2.7) is proved. To estimate (2.8), we introduce a scalar function u¯ ∈ C 2 (R3 ), such that u¯ = 1 on ∂D1 , u¯ = 0 on ∂D2 ∪ ∂Ω, x3 − h2 (x′ ) + 2ǫ , u¯ (x) = ǫ + h1 (x′ ) − h2 (x′ )
in Ω2R ,
(3.9)
and k¯ukC2 (R3 \ΩR ) ≤ C.
(3.10)
Define u¯ α1 = u¯ ψα ,
α = 1, 2, 3,
e in Ω,
(3.11)
u¯ α2 = uψα ,
α = 1, 2, 3,
e in Ω,
(3.12)
in Ω2R ,
(3.13)
e then u¯ α1 = vα1 on ∂Ω. Similarly, we define
e where u is a scalar function in C 2 (R3 ) satisfying u = 1 on such that u¯ α2 = vα2 on ∂Ω, ∂D2 , u = 0 on ∂D1 ∪ ∂Ω, u(x) =
−x3 + h1 (x′ ) + 2ǫ , ǫ + h1 (x′ ) − h2 (x′ )
and kukC2 (R3 \ΩR ) ≤ C. In order to prove (2.8), it suffices to prove the following proposition. 9
(3.14)
e R3 ) be the weak solution of (2.2) Proposition 3.2. Assume the above, let vαi ∈ H 1 (Ω; with α = 1, 2, 3. Then for i = 1, 2, α = 1, 2, 3, Z ∇(vα − u¯ α ) 2 dx ≤ C; (3.15) i i e Ω
and
Consequently,
α
∇vi L∞ (Ω\Ω e R ) ≤ C, C √ √ , |x′ | ≤ ǫ, α ǫ ∇(vi − u¯ αi )(x) ≤ √ C ǫ < |x′ | ≤ R, ′, |x | α ∇vi (x) ≤
and
C , ǫ + |x′ |2
(3.16)
∀ x ∈ ΩR .
(3.17)
∀ x ∈ ΩR ,
(3.18)
C √ ′ ǫ, , |x | ≤ √ ǫ ∇x′ vαi (x) ≤ √ C ǫ < |x′ | ≤ R. ′, |x |
(3.19)
A direct calculation gives, in view of (3.2)-(3.5), that |∂ xk u¯ (x)| ≤
C|xk | , k = 1, 2, ǫ + |x′ |2
Thus |∇¯uαi (x)| ≤ For k, l = 1, 2, |∂ xk xl u¯ (x)| ≤
C , ǫ + |x′ |2
C , ǫ + |x′ |2
|∂ xk x3 u(x)| ¯ ≤
|∂ x3 u(x)| ¯ ≤
C , ǫ + |x′ |2
i = 1, 2, α = 1, 2, 3,
C|x′ | , (ǫ + |x′ |2 )2
x ∈ ΩR . x ∈ ΩR .
∂ x3 x3 u¯ (x) = 0,
(3.20)
(3.21)
x ∈ ΩR . (3.22)
For u¯ αi , defined by (3.11) and (3.12), making use of (1.8) and (3.22), we have, for i = 1, 2, α = 1, 2, 3, X ∂ u(x) Lλ,µ u¯ αi (x) ≤ C ≤ xk xl ¯ k+l
