Doubling inequalities for the Lam´e system with rough coefficients

Doubling inequalities for the Lam´e system with rough coefficients

Herbert Koch

Ching-Lung Lin

Jenn-Nan Wang

Abstract

In this paper we study the local behavior of a solution to the Lam´e system when the Lam´e coefficients λ and µ satisfy that µ is Lipschitz and λ is essentially bounded in dimension n ≥ 2. One of the main results is the local doubling inequality for the solution of the Lam´e system. This is a quantitative estimate of the strong unique continu- ation property. Our proof relies on Carleman estimates with carefully chosen weights. Furthermore, we also prove the global doubling in- equality, which is useful in some inverse problems.

1 Introduction

Let Ω be an open connected subset of Rⁿ with n ≥ 2. Without loss of generality, we assume 0 ∈ Ω. Let µ(x) ∈ C^0,1(Ω) and λ(x), ρ(x) ∈ L^∞(Ω) satisfy

(µ(x) ≥ δ₀, λ(x) + 2µ(x) ≥ δ₀ ∀ x ∈ Ω,

kµk_C^0,1_(Ω)+ kλk_L^∞_(Ω) ≤ M₀, kρk_L^∞_(Ω) ≤ M₀ (1.1)

∗Mathematisches Institut, Universit¨at Bonn, Endenicher Allee 60, D-53115 Bonn, Ger- many. Partially supported by the DFG through SFB 1060. Email:[email protected]

†Department of Mathematics and Research Center for Theoretical Sciences, NCTS, National Cheng-Kung University, Tainan 701, Taiwan. Partially supported by the Ministry of Science and Technology of Taiwan. Email: [email protected]

‡Institute of Applied Mathematical Sciences, NCTS, National Taiwan Univer- sity, Taipei 106, Taiwan. Partially supported by MOST102-2115-M-002-009-MY3.

Email:[email protected]

with positive constants δ₀, M₀, where we define

kf k_C^0,1_(Ω) = kf k_L^∞_(Ω)+ k∇f k_L^∞_(Ω). The isotropic elasticity system is given by

div(µ(∇u + (∇u)^t)) + ∇(λdivu) + ρu = 0 in Ω, (1.2) where u = (u₁, u₂, · · · , u_n)^t is the displacement vector and (∇u)_jk = ∂_ku_j for j, k = 1, 2, · · · , n. If ρ = 0, (1.2) represents the displacement equation of equilibrium.

Under the assumptions (1.1), the qualitative strong unique continuation property for (1.2) were recently proved by Nakamura, Uhlmann and the second and third authors [12], i.e., if u ∈ H¹(Ω) solves (1.2) and satisfies that for any N ∈ N, there exists a constant CN such that

Z

Br

|u|² ≤ C_Nr^N ∀ r sufficiently small,

then u ≡ 0 in Ω. In fact, in [12], we derived a quantitative estimate on the vanishing order of any nontrivial solution to (1.2). The derivation relies on the optimal three-ball inequalities (see [12] for details).

Another quantitative estimate of the strong unique continuation property is the doubling inequality. When λ, µ ∈ C^1,1 and ρ = 0, doubling inequalities for (1.2) in the form

Z

B2r

|u|²+ |divu|² ≤ K Z

Br

|u|²+ |divu|².

were derived in [1] based on the frequency function method developed in [5]

and [6]. To apply quantitative estimates of the strong unique continuation property to certain inverse problems for the elasticity, it is desirable to derive a doubling inequality containing |u|² only [2], i.e.,

Z

B2r

|u|² ≤ K Z

Br

|u|². (1.3)

Indeed, (1.3) for the Lam´e system with C^1,1 coefficients was proved in [2].

However, as mentioned in [3], the proof given there contains a gap.

In [3], the authors proved doubling inequalities of the form (1.3) when λ, µ ∈ C^2,1 (also ρ = 0). Moreover, these inequalities depend on global

properties of the solution. A key observation in [3] is that for λ, µ ∈ C^2,1, the Lam´e system can be transformed into a fourth order system for u having ∆² as the leading part and essentially bounded coefficients in the lower orders.

For this fourth order system, three-sphere inequalities and local doubling inequalities were derived in [9]. Using these inequalities, global doubling inequalities for (1.2) were then obtained.

The aim of this paper is to establish doubling inequalities of the form (1.3) for (1.2) when µ ∈ C^0,1 and λ, ρ ∈ L^∞. Our result provides a positive answer to the open problem posed in [3] about the doubling inequality for (1.2) with less regular coefficients. The ideas of our proof originate from our series papers on proving quantitative uniqueness for elliptic equations or systems by the method of Carleman estimates [10], [11], and [12]. In particular, we will use the reduced system derived in [12] (see Section 2 below).

We now state main results of the paper. Their proofs will be given in the subsequent sections. Assume that there exists 0 < ˜R0 ≤ 1 such that BR˜0 ⊂ Ω. Hereafter B_r denotes an open ball of radius r > 0 centered at the origin.

Theorem 1.1 There exists C > 0 depending on n, M₀ and δ₀ so that the following is true. If R > 0 with 3R ≤ ˜R0, u ∈ H_loc¹ (BR) is a nonzero solution to (1.2) and

m = − ln

kuk_L²_(BR\B_R/2)

kuk_L²_(B2R\B_R)

 then

kuk_L²_(Br) ≥ C(r/R)^Cmkuk_L²_(B2R\BR) for all r ≤ R. (1.4) Theorem 1.2 There exists a positive constant ˜C depending only on n, M0

and δ₀ such that the following is true. If u ∈ H_loc¹ (B_R) is a nonzero solution to (1.2) then

kuk_L²_(B2r(x0)) ≤ ˜Ce^Cm˜ kuk_L²_(Br(x0)), (1.5) whenever B_2r(x₀) ⊂ B_R/2. Here m is the constant of Theorem 1.1.

Theorem 1.1 and 1.2 will be proved together. The estimate (1.5) is called local doubling inequalities. Global doubling inequalities in which constants depend on the global property of solution will be proved in Section 4.

2 Reduced system

We now recall the reduced system derived in (1.2). This is a crucial step in our approach. Let us write (1.2) into a non-divergence form:

µ∆u + ∇((λ + µ) divu) + (∇u + (∇u)^t)∇µ − divu∇µ + ρu = 0. (2.1) Dividing (2.1) by µ yields

∆u + 1

µ∇((λ + µ) divu) + (∇u + (∇u)^t)∇µ

µ − divu∇µ µ + ρ

µu

= ∆u + ∇(a(x)p) + G

= 0, (2.2)

where

a(x) = λ + µ

λ + 2µ ∈ L^∞(Ω), p = λ + 2µ µ divu and

G = (∇u + (∇u)^t)∇µ

µ − divu(∇µ

µ + (λ + µ)∇(1 µ)) + ρ

µu.

Taking the divergence on (2.2) gives

∆p + divG = 0. (2.3)

Our reduced system now consists of (2.2) and (2.3). It follows easily from (2.3) that if u ∈ H_loc¹ (Ω), then p ∈ H_loc¹ (Ω).

(∆u + ∇(a(x)p) + G(x, u) = 0,

∆p + divG(x, u) = 0. (2.4)

Note that system (2.4) is not decoupled. We will use (2.4) to prove our theorems.

3 Proofs of Theorem 1.1 and 1.2

This section is devoted to the proofs of Theorem 1.1 and 1.2. The proofs rely on a suitable Carleman estimate proved in [8]. To state the estimate, we consider the equation

∆u + ∇f = g in Rⁿ. (3.1)

We consider t > 0. Given τ  1, let h(t) be a convex function satisfying (h⁰ ∼ τ, i.e., ∃ C > 1, C⁻¹τ ≤ h⁰ ≤ Cτ,

dist(2h⁰, Z) + h⁰⁰ & 1. (3.2) Here and in the sequel, the notation X . Y or X & Y means that X ≤ CY or X ≥ CY with some constant C depending only on n, M₀ and δ₀. We further assume that h satisfies that for any C > 0 there exists R₀ > 0 such that

C|x|τ ≤ (1 + h⁰⁰(− ln |x|)) (3.3) for all τ and |x| ≤ R₀. Given R > 0 h(− ln(R₀x/R)) satisfies 3.3 for |x| ≤ R.

For our purpose, in addition to (3.2), we also require h − t −¹₂ ln(1 + h⁰⁰) to satisfy (3.2). The existence of such weight function h can be found in [8, Section 6]. We will give a more explicit construction of h in appendix.

Theorem 3.1 Assume that a convex h satisfies (3.2) and is evaluated at

− ln |x|. For smooth functions u, f , g satisfying (3.1) and are supported in B₁(0) \ {0}, we have that

τ k|x|⁻²(1 + h⁰⁰)¹²e^huk + k|x|⁻¹(1 + h⁰⁰)¹²e^h∇uk

. τ k|x|⁻¹e^hf k + ke^hgk, (3.4)

where k · k = k · k_L²_(Rⁿ₎.

Theorem 3.1 can be proved by adopting arguments of Proposition 4.1 and 4.2 in [8] (see also [4, Proposition 5.1]) . It can be also proved by modifying the method in [10]. Here we give a sketch of proof.

Proof. We first observe that the estimate is equivalent to the some estimates for functions on Rⁿ\{0} under appropriate decay conditions of the solutions at 0 and ∞. This is seen by truncating, and taking an obvious limit.

We begin with an elliptic reduction and consider the equation

−∆w + Kτ²|x|⁻²w = ∇f

with a fixed large positive constant K. The quadratic form Z

|∇w|²dx + Kτ² Z

|w|²|x|⁻²dx

is an inner product and the Riesz representation theorem ensures that there is a unique solution. We claim that

Z

e2h(− ln |x|)(|∇w|²+ τ²|w|²/|x|²)dx ≤ C Z

e2h(− ln |x|)|f |²dx (3.5) for all h satisfying the first condition of (3.2). It suffices to consider bounded functions h and the inequality follows by multiplying by e2h(− ln |x|)w and integrating by parts. Moreover w decays fast as x → 0 or |x| → ∞ which we see by choosing h growing fast and linearly at ±∞.

We make the ansatz

u = v + w where

∆v = −Kτ²|x|⁻²w + g

and the full estimate (3.4) follows once we prove the estimate for f = 0 and apply it to v. Without loss of generality we assume f = 0 in the sequel and prove (3.4).

To prove it in this case , we introduce polar coordinates in Rⁿ\{0} by setting x = rω, with r = |x|, ω = (ω₁, · · · , ω_n) ∈ Sⁿ⁻¹. Using the new coordinate t = − log r, we obtain that

|x|²⁺ⁿ² ∆(|x|²⁻ⁿ² u) = utt+ ∆_Sⁿ⁻¹u − n − 2 2

2

u = e^{−n+22 t}g(e^−tω).

We can diagonalize −∆_Sⁿ⁻¹ + ⁿ⁻²₂
2

. Its spectrum is {(n − 2

2 + k)² := σ_k² : k = 0, 1, . . . }

and the corresponding eigenspace is spanned by harmonic polynomials. The equation becomes

u^k_tt− σ_k²u^k= e^{−n+22 t}g^k

and the estimate (3.4) follows from (including an additional linear term into h without changing the notation)

Z

e^2h(1 + h⁰⁰)(|u^k_t|²+ (1 + k²)|u^k|² + |h⁰|²|u^k|²)dt ≤ C Z

e^2h−4t|g^k|²dt

Since ∂_t²− σ²_k= (∂_t− σ_k)(∂_t+ σ_k), the claim follows once we prove the elliptic estimate

Z

e^2h(|u⁰|²+ (τ² + σ²)|u|²)dt ≤ C Z

e^2h|(∂_t− σ)u|²dt (3.6) and the commutator type estimate

Z

e^2h(1 + h⁰⁰)|u|²dt ≤ c Z

e^2h|(∂_t+ σ)u|²dt (3.7) In the first case we multiply

u_t− σu = g by e^2hu and integrate. Then

1

2(τ + σ)ke^huk² ≤ Z

e^2h(h⁰+ σ)u²dt = − Z

e^2hugdt ≤ ke^hgk ke^huk together with using the equation to bound ut implies (3.6). For the second estimate we define v = e^hu, multiply

v⁰− (h⁰− σ)v = e^hg by (h⁰− σ)v and obtain

k(h⁰− σ)vk²+1 2

Z

h⁰⁰v²dt = − Z

e^hg(h⁰− σ)vdt.

The estimate follows by an application of the Cauchy-Schwarz inequality.

2

Besides the Carleman estimate, we also need an interior estimate (Caccioppoli- type estimate) for the Lam´e system (1.2). For fixed a₃ < a₁ < a₂ < a₄, there exists a constant C1 such that

Z

a1r<|x|<a2r

||x|^|α|D^αu|²+ ||x|^|α|+1D^αp|²dx ≤ C₁ Z

a3r<|x|<a4r

|u|²dx, |α| ≤ 1 (3.8) for all sufficiently small r. Estimate (3.8) can be found in Lemma 3.1 of [12].

We are now ready to prove Theorem 1.1 and 1.2. Let us define the cut-off function χ(x) ∈ C₀^∞(Rⁿ\{0}) such that

χ(x) =







0 if |x| ≤ r/3,

1 in 5r/12 ≤ |x| ≤ 5 ˜R/4, 0 if 3 ˜R/2 ≤ |x|,

where ˜R is a small number that will be chosen later and r  ˜R. Denote

˜

u = χu and ˜p = χp. Then it follows from (2.4) that ˜u and ˜p satisfy

∆˜u + ∇(a˜p) = (∇²χ)u + 2∇χ · ∇u + (∇χ)ap − χG := F (3.9) and

∆˜p + div(χG) = (∇²χ)p + 2∇χ · ∇p + (∇χ)G := H. (3.10) Applying (3.4) to (3.9) with u = ˜u, f = a˜p, g = F yields

τ k|x|⁻²(1 + h⁰⁰)¹²e^huk + k|x|˜ ⁻¹(1 + h⁰⁰)¹²e^h∇˜uk . τ k|x|⁻¹e^ha˜pk + ke^hF k

≤ C(τ k|x|⁻¹e^hpk + ke˜ ^hF k),

(3.11)

where C = C(n, M0, δ0). Replacing h by h − t − ¹₂ln(1 + h⁰⁰) in (3.4) and applying the new estimate to (3.10), we have that

τ k|x|⁻¹e^hpk + ke˜ ^h∇˜pk . τ k(1 + h⁰⁰)⁻¹²e^hχGk + k|x|(1 + h⁰⁰)⁻¹²e^hHk. (3.12) Now, K×(3.12)+(3.11) gives

τ k|x|⁻²(1 + h⁰⁰)¹²e^huk + k|x|˜ ⁻¹(1 + h⁰⁰)¹²e^h∇˜uk + Kτ k|x|⁻¹e^hpk˜

≤ C(τ k|x|⁻¹e^hpk + ke˜ ^hF k + Kτ k(1 + h⁰⁰)⁻¹²e^hχGk + Kk|x|(1 + h⁰⁰)⁻¹²e^hHk).

(3.13) We then choose K ≥ C and ˜R = R(n, M₀, δ₀) satisfying

CKτ (1 + h⁰⁰)⁻¹² ≤ |x|⁻¹(1 + h⁰⁰)¹²

for all |x| ≤ 3R/2 since (3.3) holds. Consequently, we obtain from (3.13) that

τ k|x|⁻²(1 + h⁰⁰)¹²e^huk{5r/12≤|x|≤5R/4}≤ Cke^hF k{r/3≤|x|≤5r/12}∪{5R/4≤|x|≤3R/2}

+ Cτ k(1 + h⁰⁰)⁻¹²e^hGk{r/3≤|x|≤5r/12}∪{5R/4≤|x|≤3R/2}

+ Ck|x|(1 + h⁰⁰)⁻¹²e^hHk{r/3≤|x|≤5r/12}∪{5R/4≤|x|≤3R/2} := RHS.

(3.14)

Here and after, we use k · k_A to denote the L² norm over the region A.

In view of (3.8), we can easily derive that

RHS ≤ Cτ e^˜h(r/3)(r/3)⁻²kuk{r/4≤|x|≤r/2}+Cτ e^˜h(5R/4)(5R/4)⁻²kuk{R/2≤|x|≤2R}, (3.15) where we denote ˜h(a) = h(− ln a). Now we choose τ = τ₀ such that

Ce^˜h(5R/4)(5R/4)⁻²kuk{R/2≤|x|≤2R} ≤ 1

2R⁻²e^˜h(R)kuk{2R/3≤|x|≤R}. (3.16) More precisely we choose from now on

τ₀ ∼ lnkuk{2R/3≤|x|≤R}

kuk{R/2≤|x|≤2R}

 .

so that (3.16) is satisfied. Combining (3.14), (3.15), (3.16) yields

k|x|⁻²e^huk{5r/12≤|x|≤5R/4} ≤ Ce^˜h(r/3)(r/3)⁻²kuk{r/4≤|x|≤r/2}. (3.17) The estimate implies that

kuk{|x|≤r}≥ Ce^˜h(R)kuk{2R/3≤|x|≤R}(r/R)²e^−˜h(r/3) ≥ Cr^m,

which establishes Theorem 1.1. Next, adding e^h(r/2)(r/2)⁻²kuk{|x|≤r/2} to both sides of (3.17) gives

e^˜h(r)r⁻²kuk{|x|≤r}≤ e^˜h(r)r⁻²kuk{|x|≤r/2}+ e^˜h(r)r⁻²kuk{r/2≤|x|≤r}

≤ Ce^˜h(r/3)(r/3)⁻²kuk{|x|≤r/2}, which leads to Theorem 1.2.

4 Global doubling inequalities

In the previous section, we have proved local doubling inequalities. Nonethe- less, global doubling inequalities are more suitable for inverse problems (for example, see [2]). In this section we derive global doubling inequalities along the lines in [3]. For brevity, we will not give detailed arguments here. We refer to [3] for detailed proofs. To begin, we give the definition of Lipschitz boundary.

Definition 4.1 We say that the boundary ∂Ω is of Lipschitz class with con- stants r₀ and L₀, if, for any x₀ ∈ ∂Ω, there exists a rigid transformation of coordinates under which x0 = 0 and

Ω ∩ B_r0(0) = {x ∈ B_r0(0) : x_n > ψ(x⁰)},

where x = (x⁰, x_n) with x⁰ ∈ Rⁿ⁻¹, x_n ∈ R and ψ is a Lipschitz continuous function on B_r0(0) ⊂ Rⁿ⁻¹ satisfying ψ(0) = 0 and

kψk_C^0,1_(Br0(0)) ≤ L₀r₀.

Let us denote Ω_d = {x ∈ Ω : dist(x, ∂Ω) > d}. Using three-ball in- equalities proved in [11] or [12], one can prove the following theorem (see [2], [3]).

Theorem 4.1 [3, Theorem 3.2] Let ∂Ω be of Lipschitz class with constants r₀, L₀, and λ, µ satisfy (1.1) , u ∈ H_loc¹ (Ω) be a nontrivial solution to (1.2).

Then for every σ > 0 and for every x ∈ Ω^4σ

θ , we have Z

Bσ(x)

|u|²dx ≥ C_σ Z

Ω

|u|²dx,

where 0 < θ < 1 depends on n, δ₀, M₀ only and C_σ depends on n, δ₀, M₀, r₀, L₀, |Ω|, kuk_H1/2(Ω)/kuk_L²_(Ω), and σ.

We now ready to state global doubling inequalities. To describe the the- orem, we introduce more notations. Instead of the strong ellipticity, we say that Lam´e coefficients λ, µ satisfy the strong convexity condition if

µ(x) ≥ ˜δ₀ > 0, 2µ(x) + nλ(x) ≥ ˜δ₀ ∀ x ∈ Ω. (4.1) It is known that the strong convexity implies the strong ellipticity. Let ϕ ∈ L²(∂Ω, Rⁿ) be a vector field satisfying the compatibility condition

Z

∂Ω

ϕ · rds = 0

for every infinitesimal rigid displacement r, that is, r = c + W x, where c is a constant vector and W is a skew n × n matrix. Consider the boundary value problem:

(div(µ(∇u + (∇u)^T)) + ∇(λdivu) = 0 in Ω,

(µ(∇u + (∇u)^T) + (λdivuI_n))ν = ϕ on ∂Ω, (4.2)

where I_n is the n × n identity matrix, ν is the unit outer normal to ∂Ω, and ϕ satisfies the compatibility condition. In order to ensure the uniqueness of the solution to (4.2), we assume the following normalization conditions:

Z

Ω

udx = 0, Z

Ω

(∇u − (∇u)^T)dx = 0. (4.3) Theorem 4.2 [3, Theorem 3.7] Let ∂Ω be of Lipschitz class with constants r0, L0, and λ, µ satisfy (4.1) , the second condition of (1.1). If u ∈ H¹(Ω, Rⁿ) is the weak solution to (4.2) satisfying the normalization condition (4.3).

Then there exists a constant 0 < ϑ < 1, only depending on n, ˜δ₀, M₀, such that for every ¯r > 0 and for every x0 ∈ Ω¯r, we have

Z

B2r(x0)

|u|²dx ≤ C Z

Br(x0)

|u|²dx

for every r with 0 < r ≤ ^ϑ₂¯r, where C depends on n, ˜δ₀, r₀, L₀, |Ω|, ¯r, and kϕk_H^−1/2_(∂Ω)/kϕk_H⁻¹_(∂Ω).

Appendix

In this appendix, we would like to construct a weight function h satisfying the conditions described in Section 3. Let τ ∈ N + ⁵₄  1 and define a = 2 ln τ . We choose

h⁰⁰(t) = δτ e^−t/2, where δ > 0 is sufficiently small. We then set

h⁰(t) = τ − 2δτ e^−t/2 and

h(t) = τ t + 4δτ e^−t/2.

It is clear that h is convex and h⁰ satisfies the first condition of (3.2).

To verify the second condition of (3.2), we observe that τ e^−t/2 ≤ 1 if t ≥ 2 ln τ (= a) and τ e^−t/2 ≥ 1 if t ≤ 2 ln τ . So, for t ≤ a, we have h⁰⁰(t) ≥ δτ e^−t/2 ≥ C_δ(1 + τ e^−t/2) for some C_δ > 0. Next, for a < t, we can see that τ − 2δ ≤ h⁰(t) ≤ τ , then dist(2h⁰, Z) ≥ ¹₂− 4δ ≥ C(1 + τ e^−t/2) holds for some absolute constant C > 0 provided δ ≤ ₁₆¹.

To check (3.3), as we noted above, if t ≤ 2 ln τ , then 1 + h⁰⁰(− ln |x|) ≥ 1 + δτp|x| ≥ δτ|x|

for |x| < 1. On the other hand, for t ≥ 2 ln τ , we have

1 + h⁰⁰(− ln |x|) ≥ 1 ≥ τ e^{ln |x|/2}= τp|x| ≥ τ|x|.

Finally, let us define ˜h = h − t − ¹₂ln(1 + h⁰⁰), then we have

˜h⁰⁰(t) = δτ e^−t/2− 1

8δτ e^−t/2(1 + δτ e^−t/2)⁻¹+ 1

8δ²τ²e^−t(1 + δτ e^−t/2)⁻². We choose

˜h⁰(t) = τ − 2δτ e^−t/2− 1 + 1

4δτ e^−t/2(1 + δτ e^−t/2)⁻¹ and

˜h(t) = τ t + 4δτ e^−t/2− t − 1

2ln(1 + δτ e^−t/2).

The same arguments imply that ˜h satisfies the required conditions provided δ is small.

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