Quantitative uniqueness estimates for the shallow shell system and their application to an inverse problem

Quantitative uniqueness estimates for the shallow shell system and their application to an inverse problem

MICHELEDICRISTO, CHING-LUNGLIN ANDJENN-NANWANG

Abstract. In this paper we derive some quantitative uniqueness estimates for the shallow shell equations. Our proof relies on appropriate Carleman estimates. For applications, we consider the size estimate inverse problem.

Mathematics Subject Classification (2010): 53J58 (primary); 35R30 (sec- ondary).

1. Introduction

In this work we study a quantitative uniqueness for the shallow shell system and its application to the inverse problem of estimating the size of an embedded inclusion by boundary measurements. To begin, we letbe a bounded domain inR^{2. With-} out loss of generality, we assume 0 2 . Let✓¯ :  ! Rsatisfy an appropriate regularity assumption which will be specified later. For a shallow shell, its middle surface is described by {(x1,x2,"⇢₀✓(x¯ 1,x2)) : (x1,x2) 2 } for" > 0, where

⇢₀ > 0 is the characteristic length of (see Section 3.1). From now on, we set

✓ = ⇢₀✓¯.Letu = (u1,u2,u3) = (u⁰,u3) :  ! R³represent the displacement vector of the middle surface. Thenusatisfies the following equations:

( @_jn^✓_{i j}(u)=0 in ,

@_{i j}²mi j(u3) @_j(n^✓_{i j}(u)@i✓)=0 in , (1.1) where

mi j(u3) = ⇢²₀

⇢ 4 µ

3( +2µ)(1u3) _{i j}+ 4µ

3 @_{i j}²u3 , n^✓_{i j}(u) = 4 µ

+2µe^✓_kk(u) i j+4µe^✓_{i j}(u), (1.2) e^✓_{i j}(u) = 1

2(@_iu_j +@_ju_i+(@_i✓)@_ju₃+(@_j✓)@_iu₃), Lin and Wang’s work was supported in part by the National Science Council of Taiwan.

Received December 2, 2010; accepted May 20, 2011.

and , µare Lam´e coefficients. Hereafter, the Roman indices (exceptn) belong to {1,2}and the Einstein summation convention is used for repeated indices.

Assume that D is a measurable subdomain ofwith D ⇢ . We consider Lam´e parameters

e= + D 0 and eµ=µ+ Dµ₀,

where _Dis the characteristic function ofD. The domainDrepresents the inclusion inside of . With such parameterse,eµ, we denote the displacement fieldeu = (eu⁰,eu3)^t satisfying (1.1) and the Neumann boundary conditions on@:

8>

<

>:

en^✓_{i j}⌫_j =⇢₀¹bT_i, emi j⌫_i⌫_j =Mb⌫,

(@_ime_{i j} en^✓_{i j}@_i✓)⌫_j+@_s(me_{i j}⌫_i⌧_j)= @_sMb⌧,

(1.3)

whereme_{i j} = me_{i j}(eu₃)anden^✓_{i j} =en^✓_{i j}(eu)are defined in (1.2) with ,µ,ureplaced bye,eµ,eu. Hereafter,⌫ =(⌫₁,⌫₂),⌧ = (⌧₁,⌧₂)are, respectively, the normal and the tangent vectors along @, and s is the arclength parameter of@. Precisely, the tangent vector⌧ is obtained by rotating⌫counterclockwise of angle⇡/2. The boundary fieldMb= Mb⌧⌫+Mb⌫⌧,i.e., Mb⌧ = Mb·⌫ andMb⌫ = Mb·⌧. We remark that in the plate theory,Mb⌧ andMb⌫ are the twisting and bending moments applied on@. The fieldbT satisfies the compatibility condition which will be specified in the following section. An interesting inverse problem is to determine geometric in- formation onDfrom a pair{bT,Mb;eu⁰|@, (eu3|@,@_⌫eu3|@)},i.e., from the Cauchy data of the solutioneu. Despite its practical value, the fundamental global unique- ness, even for the scalar equation, is yet to be proved. For the development of the uniqueness issue for this kind of inverse problems, we refer to [13] and references therein for details.

In this paper we are interested in estimating the size of the area ofDin terms of the Cauchy data ofeu. This type of problem has been studied for the scalar equation and for systems of equations such as the isotropic elasticity and plate. We refer to the survey article [3] for the early developments and [20, 21] for the latest results on the plate equations. Specifically, the size ofDis estimated by the following two quantities:

We= Z

@

⇢₀¹bT ·eu⁰+Mb⌫@_⌫eu3+@_sMb⌧eu3

and

W = Z

@

⇢₀¹Tb·u⁰+Mb⌫@_⌫u3+@_sMb⌧u3,

whereu=(u⁰,u3)^t is the displacement vector satisfing (1.1) and (1.3) withD=;, i.e.,e= andeµ=µ. Here we assume that ,µare givena priori, thus, bothWe andW are known. To be more precise, in this paper, we will show that under some

a priori assumptions, there exist positive constantsC1,C2such that C1

We W

W area(D)C2

We W

W , (1.4)

whereC1,C2depend on thea prioridata.

The derivation of the volume bounds on D relies on the following integral inequalities

1 K

Z

D

X

i j

|e^✓_{i j}(u)|²+⇢₀²|@_{i j}²u3|²|W We|

K Z

D

X

i j

|e_{i j}^✓(u)|²+⇢₀²|@_{i j}²u3|²,

(1.5)

where the constant K depends on thea prioridata. The lower bound for area(D) is a consequence of the second inequality of (1.5) and the elliptic regularity es- timate for u. To derive the upper bound for area(D), we shall use the first in- equality of (1.5). As indicated in all previous related results, we need to estimate R

D

P

i j|e^✓_{i j}(u)|²+⇢₀²|@_{i j}²u3|²from below. This can be achieved by the quantitative uniqueness estimates of solutionsusolving (1.1), which is one of the themes of the paper.

For the second order elliptic operator, using the Carleman or the frequency functions methods, quantitative estimates for the strong unique continuation under different assumptions on coefficients were derived in [8–11, 14, 16, 18]. For the isotropic elasticity, similar estimates can be found in [1, 4, 19]. Further, for the elastic plate, quantitative uniqueness estimates were derived in [20, 21]. Note that global versions of quantitative uniqueness estimates, in the form of doubling in- equality, were given in [4] and [21], where their arguments rely on a local version for the power of Laplacian derived in [17].

In this paper, we will derive three-ball inequalities and doubling inequalities for the shallow shell system (1.1) with , µ 2 C^1,1(). Since the first and the second equations in (1.1) have different orders, it seems that the Carleman method is the most efficient way to derive those quantitative uniqueness estimates for (1.1).

We will give detailed derivations of quantitative uniqueness estimates based on the Carleman estimates in Section 4. The investigation of the inverse problem is given in Section 5. Since the Neumann boundary value problem for (1.1) is not standard, we will first study this forward problem in Section 3.

2. Notation

Definition 2.1. Letbe a bounded domain inR^{nwithn 2. Givenk} 2Z^{+, we} say that@is of classC^k,1with constants⇢₀, A0, if, for any pointz 2 @, there

exists a rigid coordinate transformation under whichz=0 and

\B⇢0(0)= {x =(x1,· · ·,xn 1,xn)=(x⁰,xn)2B⇢0(0):xn >'(x⁰)}, where'(x⁰)is aC^k,1function onB_⇢⁰₀(0)= B⇢0(0)\{xn=0}satisfying'(0)=0 andr'(0)=0 ifk 1 and

k'kC^k,1(B⁰_⇢0(0)) A0⇢₀.

Throughout the paper, we will normalize all norms such that they are dimension- ally homogeneous and coincide with the standard definitions when the dimensional parameter is one. With this in mind, we define

k'kC^k,1(B⁰⇢0(0))= Xk

j=0

⇢₀^jkr^j'kL¹(B_⇢⁰₀(0))+⇢₀^k+1kr^k+1'kL¹(B_⇢⁰₀(0)).

Similarly, whenwith@defined above andw:!R, we define kwkC^k,1()=

Xk

j=0

⇢₀^jkr^jwkL¹()+⇢₀^k+1kr^k+1wkL¹(),

kwk²_L²_()=⇢₀ⁿ Z



w²,

kwk²_H^k_()=⇢₀ⁿ Xk

j=0

⇢₀^2j Z

|r^jw|², k 1.

In particular, if= B⇢(0), thensatisfies Definition 2.1 with⇢₀=⇢.

LetAbe an open connected component of@. For any given pointz₀ 2A^, we define the positive orientation ofAassociated with an arclength parametrization

⇣(s)=(x1(s),x2(s)),s2[0,length(A⁾]such that⇣(0)=z0and⇣⁰(s)=⌧(⇣(s)).

Finally, we define for anyh>0

_h = {x 2|dist(x,@) >h}. 3. The forward problem

3.1. The Neumann boundary value problem for the shallow shell equation At this moment, we assume@2C^1,1with constants A0,⇢₀. Also, letsatisfy

|| A1⇢₀² (3.1)

throughout the article, and

kr✓kL¹()=⇢₀kr✓¯kL¹() A2 (3.2)

for some positive constants A1andA2. We will investigate the Neumann boundary value problem, the forward problem, for the shallow shell system. To begin, let us assume that Lam´e coefficients , µ2L¹()satisfying

0< ₀µ(x), 0 (x), 8x 2. (3.3)

We aim to findu=(u1,u2,u3)=(u⁰,u3)satisfying ( @_jn^✓_{i j}(u)=0 in ,

@_{i j}²mi j(u3) @_j(n^✓_{i j}(u)@i✓)=0 in , (3.4) with boundary conditions

8>

<

>:

n^✓_{i j}(u)⌫j =⇢₀¹Tbi, mi j(u3)⌫_i⌫_j = Mb⌫,

(@_imi j(u3) n^✓_{i j}(u)@i✓)⌫_j +@_s(mi j(u3)⌫_i⌧_j)= @_sMb⌧.

(3.5)

Now assume thatu=(u⁰,u3)satisfies (3.4)-(3.5). Letv=(v⁰, v₃)2(H¹())²⇥ H²(), then multiplying the first and second equations of (3.4) byv⁰ andv₃, re- spectively, and using the standard integration by parts, we can obtain that

Z



X

i j

(n^✓_{i j}(u)e_{i j}^✓(v)+mi j(u3)@_{i j}²v₃)= Z

@

⇢₀¹bT·v⁰+@_sMb⌧v₃+Mb⌫@_⌫v₃. (3.6)

The boundary field Mb=Mb⌧⌫+Mb⌫⌧in the cartesian coordinates is written as Mb=Mb1e2+Mb2e1.

In view of the relation

@_sMb_⌧v₃=@_s(Mb_⌧v₃) Mb_⌧@_sv₃, one can see that the right-hand side of (3.6) becomes

Z

@

⇢₀¹bT·v⁰ Mb⌧@_sv₃+Mb⌫@_⌫v₃.

Recall that@_jv₃=@_sv₃⌧_j +@_⌫v₃⌫_j for j =1,2. Using the relation⌧ =( ⌫₂,⌫₁) if⌫ =(⌫₁,⌫₂), we get that

Mb1@₁v₃ Mb2@₂v₃ = Mb1(@_sv₃⌧₁+@_⌫v₃⌫₁) Mb2(@_sv₃⌧₂+@_⌫v₃⌫₂)

= (Mb1⌧₁ Mb2⌧₂)@_sv₃+(Mb1⌫₁ Mb2⌫₂)@_⌫v₃

= Mb⌧@_sv₃+Mb⌫@_⌫v₃

In view of the above computations, we deduce that Z



X

i j

(n^✓_{i j}(u)e_{i j}^✓(v)+m_{i j}(u₃)@_{i j}²v₃)= Z

@

⇢₀¹bT·v⁰+Mb₁@₁v₃ Mb₂@₂v₃ (3.7) (see the similar derivation for the plate equation in [20]). Letv⁰ =a+W ·x+b✓

andv₃=c b·x, wherea=(a1,a2),b=(b1,b2)are two-dimensional vectors, W is a 2⇥2 skew-symmetric matrix, andcis a scalar. Thene_{i j}^✓(v)=@_{i j}²v₃=0 for alli, j. Thus, to solve (3.4) and (3.5), the pair(bT,M)b must satisfy the compatibility condition Z

@

⇢₀¹bT ·(a+W·x+b✓) b₁Mb₁+b₂Mb₂=0. (3.8) Note that takingb = 0, we have the usual compatibility condition for the traction of the elasticity equation,i.e.,

Z

@

bT ·(a+W ·x)=0.

On the other hand, to guarantee uniqueness for the forward problem, we impose the following normalization conditions

Z



u=0, Z

ru3=0, Z



(@₁u2 @₂u1)+(@₁✓ @₂u3 @₂✓ @₁u3)=0. (3.9) To solve the forward problem, the following Poincar´e-Korn inequality is very im- portant.

Proposition 3.1. There exists an absolute constantC>0, depending onA0,A1,A2, such that for allu=(u⁰,u3)2(H¹())²⇥H²()satisfying(3.9)we have

ku⁰k²_H¹_()+ku3k²_H²_()C Z



X

i j

|e^✓_{i j}(u)|²+⇢₀²|@_{i j}²u3|². (3.10) Proof. The inequality (3.10) is a combination of Poincar´e’s and Korn’s inequalities.

By abuse of notation, the variablex in our proof stands for(x₁,x₂,x₃)=(x⁰,x₃).

Now let 0 = ⇥ ( ⇢₀,⇢₀) ⇢ R³ and introduce new variablesex⁰ = x⁰ and ex₃=x₃+✓(x⁰). Denote bye0the domain ofunder the coordinate transformation x 7! ex, i.e., e0 = ⇥( ⇢₀+✓,⇢₀+✓). Both domains0 ande0 are clearly Lipschitz. One0, we have the standard Korn’s inequality: there exists a constant K0>0 such that for any 3 vectorv2 H¹(e0)satisfying

Z

e0

vdex =0, Z

e0

⇣

rexv (rexv)^T⌘

dex=0, (3.11) we have

⇢₀²kvk²_L2(e0)+krexvk²_L2(e0) K0kbrexvk²_L2(e0), (3.12)

where brexv = rexv+(rexv)^t /2 and K0 depends on A0,A1,A2. Let w(x) = w(x₁,x₂,x₃)2H¹(0), thenv(ex):=w(ex₁,ex₂,ex₃ ✓(ex⁰))2H¹(e0). By observing that the Jacobian of the coordinate transformationx 7!exis 1, we can write (3.11), (3.12) in terms ofxand get that for allw2 H¹(0)satisfying

8>

>>

<

>>

>: R

0wdx =0, R

0(@₁w₂ @₁✓ @₃w₂ @₂w₁+@₂✓ @₃w₁)dx =0, R

0(@₁w₃ @₁✓ @₃w₃ @₃w₁)dx =0, R

0(@₂w₃ @₂✓ @₃w₃ @₃w₂)dx =0,

(3.13)

we have

⇢₀²kwk²_L2(0)+krx^✓wk²_L2(0)K0krb^✓xwk²_L2(0), (3.14) where

rx^✓w= 0

@

(@₁ @₁✓ @₃)w₁ (@₁ @₁✓ @₃)w₂ (@₁ @₁✓ @₃)w₃ (@₂ @₂✓ @₃)w₁ (@₂ @₂✓ @₃)w₂ (@₂ @₂✓ @₃)w₃

@₃w₁ @₃w₂ @₃w₃ 1 A

and the symmetric part brx^✓wofrx^✓wis defined similarly. In fact, by the form of rx^✓w, (3.14) can be improved to

⇢₀²kwk²_L²₍₀₎+krxwk²_L²₍₀₎ K1kbrx^✓wk²_L²₍₀₎ (3.15) for some constant K1, also depending on A0,A1,A2. Now let u = (u⁰,u3)^t 2 (H¹())²⇥H²(), we apply (3.13) and (3.15) to

w(x)=(u1(x⁰) x3@₁u3(x⁰),u2(x⁰) x3@₂u3(x⁰),u3(x⁰)),

where(x⁰,x3)2 ⇥( ⇢₀,⇢₀). It is easy to check that the constraints (3.13) are reduced to the normalization conditions (3.9). On the other hand, easy computations show that (3.15) becomes

⇢₀ Z



⇢₀²|u|²+ |ru|²+⇢₀²X

i j

|@_{i j}²u3|²

!

C Z



X

i j

⇣

⇢₀|e^✓_{i j}(u)|²+⇢₀³|@_{i j}²u3|²

⌘

withConly depending onA0,A1,A2. 3.2. Existence and uniqueness

We will use the variational method to solve the forward problem. This seems to be standard. But we could not find any literature discussing the Neumann boundary value problem for the shallow shell. For the sake of completeness, we give a proof

of this forward problem. The arguments used here are adapted from [20]. To begin, let us introduce

H(u, v) = Z



X

i j

n^✓_{i j}(u)@jv_i +mi j(u3)@_{i j}²v₃+n^✓_{i j}(u)@i✓ @_jv₃

= Z



X

i j

n^✓_{i j}(u)e_{i j}^✓(v)+m_{i j}(u₃)@_{i j}²v₃

and

L(v)= Z

@

⇢₀¹bT·v⁰+@_sMb⌧v₃+Mb⌫@_⌫v₃.

We now give a weak formulation of the Neumann boundary value problem (3.4)- (3.5).

Definition 3.2. A vector valued functionu =(u⁰,u3)^t 2(H¹())²⇥H²()is a weak solution to (3.4)-(3.5) if and only if

H(u, v)= L(v) for allv=(v⁰, v₃)^t 2 H¹()⇥H²(). (3.16) From the above computations, we know that

L(v)= Z

@

⇢₀¹bT ·v⁰+Mb1@₁v₃ Mb2@₂v₃:=eL(v).

In other words, (3.16) is equivalent to

H(u, v)=eL(v) for allv=(v⁰, v₃)^t 2(H¹())²⇥H²(). (3.17) Theorem 3.3. Assume that✓satisfies(3.2)and , µ2L¹()satisfy(3.3). Given any boundary field (bT,M)b 2 H ^1/2(@)and the compatibility condition (3.8) holds. Then(3.4)-(3.5)admits a unique weak solutionu = (u⁰,u₃)^t satisfying the conditions(3.9)and

ku⁰kH¹()+ku3kH²()Ck(bT,Mb)k(H ^1/2(@))³, (3.18)

whereCdepends onA0,A1,A2, ₀.

Proof. LetV be the subspace of(H¹())²⇥H²()characterized by V = {w=(w⁰, w₃)2(H¹())²⇥H²(): wsatisfies (3.9)}. In view of (3.10), we have that

Z



X

i j

|e_{i j}^✓(u)|²+⇢₀²|@_{i j}²u3|² ku⁰k²_H¹_()+ku3k²_H²_()

C Z



X

i j

|e_{i j}^✓(u)|²+⇢₀²|@_{i j}²u3|²

(3.19)

for allu2V. We now define a functionalJ :V !R^by J(u)= 1

2H(u,u) eL(u).

We first want to prove that J has a unique minimizer onV. To this end, it suffices to show that J is coercive and strictly convex onV. It is easy to see that

H(u,u) = Z



n_{i j}^✓(u)e_{i j}^✓(u)+mi j(u3)@_{i j}²u3

= Z



4 µ +2µ

X

k

e_kk^✓ (u)²+4µX

i j

|e^✓_{i j}(u)|²

+ 4 µ⇢₀²

3( +2µ)|1u3|²+4µ⇢₀² 3

X

i j

|@_{i j}²u3|² 4 0

3 Z



X

i j

|e^✓_{i j}(u)|²+⇢²₀|@_{i j}²u3|².

Thus, (3.19) implies

H(u,u) C(ku⁰k²_H¹_()+ku3k²_H²_()) (3.20) withCdepending only on A0,A1,A2, ₀. On the other hand, the trace inequality leads to

eL(u)Ck(bT,Mb)k(H ^1/2(@))³(ku⁰kH¹()+ku3kH²()).

Consequently, we obtain that J(u) C⇣

ku⁰k²_H¹_()+ku3k²_H²_()

k(bT,Mb)k(H ^1/2(@))³(ku⁰kH¹()+ku₃kH²())⌘ , which shows thatJ is coercive and bounded from below onV.

Now fort 2[0,1]andu, v 2V, we have that

H(tu+(1 t)v,tu+(1 t)v) t H(u,u) (1 t)H(v, v)

= t(1 t)H(u v,u v)0 and fort 2(0,1)

H(tu+(1 t)v,tu+(1 t)v)=t H(u,u)+(1 t)H(v, v) if and only if

H(u v,u v)=0.

Since u, v 2 V, we see that H(u v,u v) = 0 if and only if u = v in (H¹())²⇥H²(). In other words, we have shown thatH(u,u)is strictly convex onV. Taking into account thateL(u)is linear, we have that J(u)is strictly convex onV. Therefore,J(u)has a unique minimizer, denoted byw, onV. In other words,

J⁰(w)[v] =0 for allv2V,i.e.,

H(w, v)=eL(v) (3.21)

for allv 2 V. Now we need to show that (3.21) is valid for allv 2 (H¹())²⇥ H²(), that is, w indeed a weak solution. Given any z = (z⁰,z3) 2 H¹()⇥ H²(), one can easily check thatezsatisfies (3.9), where

ez⁰ =z⁰ 1

|| Z



z⁰

 1

|| Z



(rz⁰ (rz⁰)^t)

2 (x x)+(✓ ✓_) 1

|| Z

rz3, ez₃ =z₃ 1

|| Z



z₃

✓ 1

|| Z

rz₃

◆

·(x x), and

✓_ = 1

|| Z



✓, x = 1

|| Z



x.

Since(bT,Mb)satisfies the compatibility condition (3.8), we conclude that H(w,z)= H(w,ez)=eL(z)=eL(z) 8z2(H¹())²⇥H²().

The estimate (3.18) is an easy consequence of (3.20) and the trace inequality.

3.3. Global regularity

To study the inverse problem, we also need a global regularity theorem for the shallow shell equations. To simplify our presentation, we impose a technical as- sumption on✓¯(or✓) in this section. Assume that✓¯satisfies

¯

✓=r✓¯=0 on @. (3.22)

We shall prove the following theorem.

Theorem 3.4. Assume that is a bounded domain in R^{2 satisfying (3.1)whose} boundary@is of classC^4,1with constantsA₀and⇢₀. Let , µ2C^1,1()¯ satisfy (3.3)and✓¯ 2C^2,1()¯ satisfy(3.22)and

k kC^1,1()¯ +kµkC^1,1()¯ +k✓¯kC^2,1()¯ A2. (3.23) Let u 2 (H¹())² ⇥ H²()be the weak solution of (3.4),(3.5) with Neumann boundary condition(bT,M)b 2(H^1/2(@))²⇥ H^3/2(@)satisfying(3.8). Assume that u satisfies the normalization conditions(3.9). Then there exists a constant C >0, depending onA0,A1,A2, 0such that

ku⁰kH²()+kukH⁴()Ck(bT,Mb)k(H^1/2(@))²⇥H^3/2(@). (3.24)

Proof. To prove this theorem, it suffices to consider (3.4) with homogeneous Neu- mann boundary conditions. In view of (3.22), the boundary conditions (3.5) are simplified to

8>

<

>:

ni j(u⁰)⌫_j =⇢₀¹Tbi, mi j(u3)⌫_i⌫_j = Mb⌫,

@_imi j(u3)⌫_j +@_s(mi j(u3)⌫_i⌧_j)= @_sMb⌧,

(3.25)

where

ni j(u⁰)= 4 µ

+2µekk(u⁰)⌫_i+4µei j(u⁰)⌫_j =⇢₀¹bTi on @

with ei j(u⁰) = ¹₂(@_iuj + @_jui). It is clear that boundary conditions (3.25) are decoupled. Using the result in [20, Proposition 8.1], one can findw˜₃satisfying

(mi j(w˜₃)⌫_i⌫_j =Mb⌫,

@_imi j(w˜₃)⌫_j +@_s(mi j(w˜₃)⌫_i⌧_j)= @_sMb⌧

on@and the estimate

kw˜₃kH⁴()CkMbkH^3/2(@). (3.26) Similarly, we can choosew˜⁰such that

ni j(w˜⁰)⌫_j =⇢₀¹bTi on @

and

kw˜⁰kH²() CkbTk(H^1/2(@))². (3.27)

The constantCin (3.26) and (3.27) depend onA0,A1,A2, ₀. By setting w⁰= ˜w⁰ 1

|| Z

w˜⁰

 1

|| Z



(rw˜⁰ (rw˜⁰)^t)

2 (x x)+(✓ ✓) 1

|| Z

rw˜₃ and

w₃= ˜w₃ 1

|| Z

w˜₃

✓ 1

|| Z

rw˜₃

◆

·(x x),

we can see that(w⁰, w₃)satisfies the boundary condition (3.25), the normalization conditions (3.9), and the estimate

kw⁰kH²()+kw₃kH⁴()Ck(bT,Mb)k(H^1/2(@))²⇥H^3/2(@), (3.28) whereCdepends on A0,A1,A2, ₀.

So now by lettingu=w+v, we obtain thatvsatisfies ( @_jn^✓_{i j}(v)= fi in ,

@_{i j}²mi j(v₃) @_j(n^✓_{i j}(v)@_i✓)= f3 in , (3.29) with homogeneous Neumann boundary conditions on@

8>

<

>:

ni j(v⁰)⌫_j =0, mi j(v₃)⌫_i⌫_j =0,

@_imi j(v₃)⌫_j +@_s(mi j(v₃)⌫_i⌧_j)=0,

(3.30)

where f =(f1, f2, f3)=(f⁰, f3)is given by (fi =@_jn^✓_{i j}(w),i =1,2,

f3= @_{i j}²mi j(w₃)+@_j(n^✓_{i j}(w)@_i✓).

Using the integration by parts, it is not hard to check that f satisfies the following compatibility conditions

Z



f =0, Z



(f1x2 f2x1)=0, Z



(f1✓+ f3x1)=0, Z



(f2✓+ f3x2)=0.

(3.31) Now to obtain a global estimate forv, we decouple (3.29) as follows

( @_jni j(v⁰)= fi +¹₂@_j(@_i✓ @_jv₃+@_j✓ @_iv₃):= ˜fi in ,

@_{i j}²m_{i j}(v₃)= f₃+@_j(n^✓_{i j}(v)@_i✓):= ˜f₃ in . (3.32) By (3.31) and straightforward computations, we can deduce that f˜=(f˜1, f˜2, f˜3)= (f˜⁰, f˜3)satisfy

Z



f˜=0, Z



(f˜1x2 f˜2x1)=0, Z



f˜3x1= Z



f˜3x2=0, (3.33) which are the compatibility conditions for the existence of the boundary value prob- lem (3.32) and (3.30). Recall the global estimate for the isotropic elasticity with homogeneous Neumann boundary condition, we have

kv⁰kH²() C(⇢₀²kf˜⁰kL²()+kv⁰kH¹())

C(⇢₀²kf⁰kL²()+kv₃kH²()+kv⁰kH¹()), (3.34) whereC depend onA0,A1,A2, ₀. Forv₃, we use [20, Proposition 8.2] to obtain that

kv₃kH⁴()  C(⇢₀²kf˜3kL²()+kv₃kH²())

 C(⇢₀²kf˜3kL²()+kv₃kH²()+kv⁰kH¹()). (3.35)

The dependence ofCis the same as above. Putting (3.34) and (3.35) together yields kv⁰kH²()+kv₃kH⁴()C(⇢₀²kfkL²()+kv₃kH²()+kv⁰kH¹()). (3.36) Now using the weak formulation of the boundary value problem (3.32), (3.30), the Poincar´e-Korn inequality (3.10), (3.28), we get from (3.36) that

kv⁰kH²()+kv₃kH⁴()  C(⇢₀²kfkL²()+kv₃kH²()+kv⁰kH¹())

 C⇢₀²kfkL²()

 Ck(bT,M)b k(H^1/2(@))²⇥H^3/2(@). (3.37)

Finally, combining (3.28) and (3.37) gives (3.24).

4. Quantitative uniqueness estimates

4.1. Main theorems

In this section, we would like to derive the three-ball inequalities for (1.1), which is a form of quantitative uniqueness estimate. The regularity of @is irrelevant for the estimates derived here. But to make the paper consistent, we assume that

 is at least a Lipschitz domain with constant A0 and⇢₀. Let (x), µ(x)satisfy (3.3) and , µ,✓¯satisfy estimate (3.23). We now first state the main results of this section. Assume that B_⇢0R¯0 ⇢ with R¯0  1. Let us denoteUr = (ru⁰,u3) = (ru1,ru2,u3). Then the following local estimates hold.

Theorem 4.1. There exists a positive number R1, depending on 0,K1,K2, such that if 0<r1<r2<r3⇢₀R¯0andr1/r3<r2/r3< R1, then

Z

|x|<r2

|Ur2|²dx C1

✓Z

|x|<r1

|Ur1|²dx

◆⌧✓Z

|x|<r3

|Ur3|²dx

◆1 ⌧

(4.1) for(u⁰,u₃)2(H¹(B_⇢0R¯0))²⇥H³(B_⇢0R¯0)satisfying(1.1)inB_⇢0R¯0, whereC₁ >0 and0<⌧ <1depend onr1/r3,r2/r3, 0,A2.

Remark 4.2. The estimate (4.1) is the three-ball inequality. ConstantsC1 and⌧ appeared above can be explicitly written as⌧ = B/(E+B)and

C1=max{C0[(log(r1/r3))²/(log(r2/r3))²](r2/r1)²,exp(B 0)}(r3/r1)^2⌧, whereC₀>1 and ₀are constants depending on ₀,A₂and

E= E(r1/r3,r2/r3)=(log(r1/r3) 1)² (log(r2/r3))², B= B(r2/r3)= 1 2 log(r2/r3).

Remark 4.3. Ifr31, then (4.1) is reduced to Z

|x|<r2

|U|²dx  C1

r₂²

✓Z

|x|<r1

|U|²dx

◆⌧✓Z

|x|<r3

|U|²dx

◆1 ⌧

. (4.2) By abuse of notation, we denoteU =(u⁰,u3).

Using the three-ball inequality, we can prove

Theorem 4.4. If (u⁰,u₃) 2 (H¹(B_⇢0R¯0))² ⇥ H³(B_⇢0R¯0) is a nontrivial solution to (1.1), then we can find a constant R₂ depending on ₀,A₂ and a constant m₁ depending on ₀,A₂andkU_R2

2kL²(|x|<⇢0R₂²)/kU_R4

2kL²(|x|<⇢0R⁴₂)such that Z

|x|<R|U|²dx K R^m1, (4.3)

whereRis sufficiently small and the constantK depends onR₂andU.

In view of the standard unique continuation property for (1.1) in a connected domain containing the origin, ifuvanishes in a neighborhood of the origin then it vanishes identically in. Theorem 4.4 provides an upper bound on the vanishing order of a nontrivial solution to (1.1). The following doubling inequality is another quantitative estimate of the strong unique continuation for (1.1).

Theorem 4.5. Let(u⁰,u₃)2(H¹(B_⇢0R¯0))²⇥H³(B_⇢0R¯0)be a nonzero solution to(1.1).

Then there exist positive constants R₃, depending on ₀,A₂, kU_R2

2kL²(|x|<⇢0R²₂)/ kU_R⁴

2k_L²_(|x|<⇢0R2⁴₎, andC2, depending on 0,A2,m1, such that if0 <r  ⇢₀R3,

then Z

|x|2r|U|²dx C₂ Z

|x|r|U|²dx, (4.4) whereR₂andm₁are the constants obtained in Theorem4.4.

The rest of this section is devoted to the proofs of Theorem 4.1, 4.4, and 4.5.

4.2. Preliminaries

From now on, it suffices to take⇢₀ = 1. The first step is to transform the system (1.1) into a new system with uncoupled principal parts. To simplify the notation in the following proofs, we denoteu = u⁰ = (u1,u2)(suppress the prime),w = u3, and v = divu⁰ = divu. Putting (1.1) and the equation obtained by taking the divergence of the first system of (1.1) together, we come to the following new

system 8

><

>:

1u= P1(Du,Dv)+ P2(D²w,Dw), 1v= P3(Du,Dv)+ P4(D³w,D²w,Dw), 1²w= P5(D³w,D²w,Dw)+P6(Du),

(4.5)