Uniqueness and stability of determining the residual stress by one measurement

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Communications in Partial Differential Equations

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Uniqueness and Stability of Determining the Residual Stress by One

Measurement

Victor Isakov a; Jenn-Nan Wang b; Masahiro Yamamoto c

a Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas, USA b Department of

Mathematics, National Taiwan University, Taipei, Taiwan c Department of Mathematical Sciences, University

of Tokyo, Meguro, Tokyo, Japan Online Publication Date: 01 May 2007

To cite this Article Isakov, Victor, Wang, Jenn-Nan and Yamamoto, Masahiro(2007)'Uniqueness and Stability of Determining the Residual Stress by One Measurement',Communications in Partial Differential Equations,32:5,833 — 848

To link to this Article: DOI: 10.1080/03605300600718453

URL: http://dx.doi.org/10.1080/03605300600718453

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ISSN 0360-5302 print/1532-4133 online DOI: 10.1080/03605300600718453

Uniqueness and Stability of Determining

the Residual Stress by One Measurement

VICTOR ISAKOV

, JENN-NAN WANG

,

AND MASAHIRO YAMAMOTO

1_{Department of Mathematics and Statistics, Wichita State University,}

Wichita, Kansas, USA

2_{Department of Mathematics, National Taiwan University, Taipei, Taiwan}

3_{Department of Mathematical Sciences, University of Tokyo,}

Meguro, Tokyo, Japan

In this paper we prove a Hölder and Lipschitz stability estimates of determining the residual stress by a single pair of observations from a part of the lateral boundary or from the whole boundary. These estimates imply first uniqueness results for determination of residual stress from few boundary measurements.

Keywords Continuation of solutions; Elasticity theory; Inverse problems. Mathematics Subject Classification 35R30; 74B10; 35B60.

1. Introduction

We consider an elasticity system with residual stress. Let
be an open bounded

domain in
3 _{with smooth boundary 
. The residual stress is modelled by a}

symmetric second-rank tensor Rx= rjkx3jk=1 ∈ C7
which is divergence free

· R = 0 in
(1.1)

and satisfies the boundary condition

R= 0 on 
 (1.2)

Received November 16, 2005; Accepted March 7, 2006

Address correspondence to Victor Isakov, Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67206-0033, USA; E-mail: victor.isakov@ wichita.edu

833

where · R is a vector-valued function with components given by · R_j= 3
k=1 _kr_jk 1≤ j ≤ 3

In this paper x∈ R3 _{and = }

1 2 3 is the unit outer normal vector to 
.

Here and below, differential operators  and without subscript are with respect to x variables. Let ux t= u1 u2 u3 Q→
3 be the displacement vector in

Q =
× −T T. We assume that ux t solves the initial boundary value problem:

A_Ru = 2_tu−  +  · u − u −  · uR = 0 in Q (1.3)

u= u₀ _tu= u₁ on
× 0 (1.4)

u= g0 on 
× −T T (1.5)

where  is density and  and are Lamé constants satisfying

0 <  0 <  0 < +  (1.6)

The system (1.3) can be written as

2_tu−  · u = 0

where u= tr I + 2  + R + uR is the stress tensor and  = u + u/2

is the strain tensor. Note that the term · R does not appear in (1.3) due to (1.1).

Also, by the same condition, we can see that · uR_i=

3

jk=1

r_jk_j_ku_i 1≤ i ≤ 3

Since we are only concerned with the residual stress and we are motivated by applications to the material science we suppose that density  and Lamé coefficients

 and are constants. To make sure that the problem (1.3) with (1.4), (1.5) is

well-posed, we assume that

RC1_
< ₀ (1.7)

for some small constant 0>0. The assumption (1.7) is also physically motivated

Man (1998). It is not hard to see that if ₀ is sufficiently small, then the boundary

value problem (1.3), (1.4), (1.5) is hyperbolic, and hence for any initial data u0u1∈

H1
× L2
 and lateral Dirichlet data g0∈ C1−T T H1
, u0= g0 on


× 0, there exists a unique solution u· R u₀u₁g₀∈ C−T T H1_{
 to}

(1.3)–(1.5).

In this paper we are interested in the following inverse problem:

Determine the residual stress R by a single pair of Cauchy data u u on × −T T, where u = u· R u0u1g0and ⊂ 
.

We will address uniqueness and stability issues. The focus is on the stability since the uniqueness follows immediately from it. Our method is based on Carleman estimates techniques initiated by Bukhgeim and Klibanov (1981). For works on

Carleman estimates and related inverse problems for scalar equations, we refer to books Bukhgeim (2000) and Klibanov and Timonov (2004) for further details and references. Here we only want to mention some related results for the dynamical Lamé system and the residual stress system (1.3). For the Lamé system, the first step has been made by Isakov (1986) where he proved the Carleman estimate and established the uniqueness for the inverse source problem. It should be noted that Isakov (1986) transformed the principal part of the system into a composition of two scalar wave operators. It is well-known that the Lamé system is principally

diagonalized as a system of equations for u and div u. Based on this fact, L2

-Carleman estimates were derived in Eller et al. (2002) and Ikehata et al. (1998) for the Lamé system and applications of to the Cauchy problem and the inverse problem were given. Recently, Imanuvilov et al. (2003) obtained a Carleman estimate for the Lamé system by considering a new principally diagonalized system for u div u curl u. In Imanuvilov et al. (2003), they used this estimate to study the problem of identifying the density and Lamé coefficients by two sets of data measured in a boundary layer and a Hölder-type stability estimate. The continuation of this work is in Imanuvilov and Yamamoto (2005).

For the dynamical residual stress model (1.3), an L2_{-Carleman estimate has been}

proved when the residual stress is small (Isakov et al., 2003). The system with the residual stress is no longer isotropic. In other words, this system is strongly coupled, and it is not possible to decouple the leading part without increasing the order of the system. There are almost no results on Carleman estimates and inverse problems for anisotropic systems which are very important in applications. In Isakov et al. (2003), we used the standard substitution u div u curl u and reduced (1.3) to a new system where the leading part is a special lower triangular matrix differential operator with the wave operators in the diagonal. The key point is that the coupled terms in the leading part contain only the second derivatives of u with respect to x variables and they can be absorbed by div u curl u when the residual stress is small. Using similar Carleman estimates, Lin and Wang (2003) studied the problem of uniquely determining the density function by a single set of boundary data. The unique continuation property for the stationary case of (1.3) was proved in Nakamura and Wang (2003).

In this work we study the problem of recovering the (small) residual stress in(1.3)–(1.5) by single set of Cauchy data. We will derive a Hölder stability estimate in convex hull of the observation surface  and a Lipschitz stability estimate for R in

when = 
and observation time T is large. There are other results concerning

the determination of the residual stress by infinitely many boundary measurements, i.e. by the Dirichlet-to-Neumann map, we refer to Hansen and Uhlmann (2003), Rachele (2003), and Robertson (1977, 1988).

We are now ready to state the main results of the paper. Denote d= inf x and

D= sup x over x ∈
. We assume that

0 < d (1.8)

Let0 Ebe the class of residual stresses defined by

0 E= RC6_
< E Ris symmetric and satisfies (1.1), (1.2), and (1.7)

To study the inverse problem, we need not only the well-posedness of (1.3)–(1.5) but also some extra regularity of the solution u. To achieve the latter property,

the initial data u0u1 and the Dirichlet data g0 are required to satisfy some

smoothness and compatibility conditions). More precisely we will assume that u₀∈ H9_
u

1∈ H8
 and g0∈ C8−T T H1
∩ C5−T T H4
 and

they satisfy standard compatibility assumptions of order 8 at 
× 0. By using

energy estimates (Duvaut and Lions, 1976) and Sobolev embedding theorems as in Imanuvilov et al. (2003) one can show that

 x



tuC0_
≤ C (1.9)

for ≤ 2 and 0 ≤  ≤ 5.

By examining the equation (1.3), we can see that the residual stress tensor appears in the equation without first derivatives because of (1.1). It turns out that a single set of Cauchy data is sufficient to recover the residual stress. To guarantee the

uniqueness, we impose some non-degeneracy condition on the initial data u0u1.

More precisely, we assume that

det M= det  2₁u0 212u0 213u0 22u0 223u0 23u0 2 1u1 212u1 213u1 22u1 223u1 23u1  > E−1 on
 (1.10)

Note that Mx is a 6× 6 matrix-valued function. For example, one can check that

u₀x= x2

1 x22 x32and u1x= x2x3 x1x3 x1x2satisfy (1.10).

We will use the following notation: C  are generic constants depending only on
 T    0 Eu0u1g0, any other dependence is indicated, · kQis the

norm in the Sobolev space Hk_{Q. Q= Q ∩  < x}2_{− }2_t2_{− d}2

1 and
=

∩  < x2− d2₁. Here d1 is some positive constant. u  1 and u  2 denote

solutions of the initial boundary value problems (1.3), (1.4) associated with R  1 and R  2. Finally, we introduce the norm of the differences of the data

F = 4
=2 _tu  2− u  1_5 2× −T T +  tu  2− u  13₂× −T T

Due to (1.6) we can choose positive  so that 2_<   4_<  d2 T2 (1.11)

Theorem 1.1. Assume that the domain
satisfies (1.8),  satisfies (1.11), and for

some d1

x2_{− d}2

1<0 when x∈ 
\ and D

2_{− }2_T2_{− d}2

1<0 (1.12)

Let the initial data u₀u₁ satisfy (1.10).

Then there exist ₀ and constants C  <1, depending on , such that for R 1 R  2∈ 0 E one has

R  2 − R  10
≤ CF (1.13)

The domain
 is discussed in Isakov (2006, Sec. 3.4).

If  is the whole lateral boundary and T is sufficiently large, then a much stronger (and in a certain sense best possible) Lipschitz stability estimate holds.

Theorem 1.2. Let d₁= d. Assume that the domain
satisfies (1.8),

D2<2d2_{ (1.14)}

and

D2_{− d}2

2 < T

2_{ (1.15)}

Let the initial data u₀u₁ satisfy (1.10). Let = 
.

Then there exist an 0 and C such that for R 1 R  2∈ 0 E satisfying the

condition

R 1= R  2 on  × −T T (1.16)

one has

R  2 − R  10
≤ CF (1.17)

Let us show compatibility of conditions (1.15) and (1.11). From conditions (1.11) and (1.14) we have D2_{− d}2 2 <  d2 4

and hence we can find T2 _{between these two numbers.}

As mentioned previously, the proofs of these theorems rely on Carleman estimates. Using the results of Isakov et al. (2003) we will derive needed Carleman estimates in Section 2. Using this estimate we will prove in Section 3 the Hölder stability estimate (1.13). In Section 4, we demonstrate the Lipschitz stability of the Cauchy problem for the residual stress model. This estimate is one of key ingredients to derive the Lipschitz stability estimate for our inverse problem in Section 5.

2. Carleman Estimate

In this section we will describe Carleman estimates needed to solve our inverse

problem. Their proofs can be found in Isakov et al. (2003). Let x t= x2₋

2_t2_{− d}2

1 and x t= exp

2x t, where  is choosen in (1.11) and  < C is a

large constant to be fixed later.

Theorem 2.1. There are constants ₀and C such that for R satisfying (1.7)

 Q _xtu2+ _xtv2+ _xtw2+ 3u2+ 3v2+ 3w2e2 ≤ C Q A_Ru2+ A_Ru2e2 (2.1)

for allu∈ H3

0Q and



Q

2u2+ div u2+ curl u2+ −1ue2≤ C 

QAR

u2e2 (2.2)

for allu∈ H2 0Q.

Carleman estimates of Theorem 2.1 is our basic tool for treating the inverse problem. Lemma 2.2. For u∈ H2 0
 satisfying u= f0+ 3 j=1jfj with f0 fj 1≤ j ≤ 3 belonging to L2, we have 1  
u 2_e2_{dx≤ C}
 f2 0 2 +
f_j2  e2dx

Proof. For the weight function  with large , we can use Theorem 3.1 of Fabre and Lebeaux (1996) to get


u2e2dx≤ C 
 f2 0 2 +
f_j2  e2dx (2.3)

Now we will bound u. Observe that ue_{= ue− ueand hence}

1  
u 2_e2_dx≤ 2  
ue 2_2_{dx+ C}
u 2_e2_{dx (2.4)} We have

ue=  ue+ 2u · e+ 22+  ue = f0+

_jf_je_{+ 2u · e+ }2_2_{+  ue}

Multiplying this equality by −1

ue

_{, integrating by parts, and using the}

Cauchy-Schwarz inequality, we obtain 1  
ue2_2_dx = −
1 f0+
_jf_jue2dx− 2 
u· ue2dx− 
2+ u2e2dx ≤ 1 2 
f₀2 2e 2_dx+1 2 
u 2_e2_dx+1  

f_j_ju+ 2u_je2dx + 2
u· ue2dx+ C 
u2e2dx ≤ 1 2 
f2 0 2e 2_dx+1 2 
u 2_e2_dx+ 1  

f_j2e2dx +  
u 2_e2_dxC  
u 2_e2_dx

where  > 0 is arbitrary and we used thatab ≤ a2₊ 1 4b

2_{. Choosing sufficiently}

small  > 1

C to absorb the term with u in the right side by the left side and using

(2.3), (2.4) we yield the bound of Lemma 2.2.

Now we are ready to prove Theorem 2.1.

Proof of Theorem2.1. We consider (1.3) with a source term,

A_Ru= f in Q (2.5)

Using that   are constants we have from Isakov et al. (2003), Section 2, the new system of equations

                   P₁u=+  v+ f  P2v= 3
jk=1 rjk  · jku+ div f  P₁w= 3
jk=1 rjk  × jku+ curl f  (2.6) where P1= 2t− 3 jk=1−1 jk+ rjkjk, P2= 2t − 3 jk=1−1+ 2 jk+

r_jk_j_k, where jk is the Kronecker delta. Due to (1.6) and (1.7) with small 0,

P₁ and P2 are hyperbolic operators. Using (1.6), (1.11) by standard calculations

one can show that  is strongly pseudo-convex in Q provided  < C is sufficiently large and 0 is sufficiently small (see Isakov et al., 2003, or Isakov, 2006). Observe

that according to the first condition (1.11) the function x t= x2_{− }2_t2_{− d}2 _is

pseudo-convex on Q and according to the second condition (1.11) the gradient of

 is non-characteristic on Q with respect to operators 2

t − , 

2

t −  + 2  ,

and hence with respect to P1 P2 provided 0 is sufficiently small. We fix such 

observing that it depends only on Q   and . It follows from Theorem 3.1 in Isakov et al. (2003) that there exists a constant C > 0 such that for all  > C we have  Q _xtu2+ _xtv2+ _xtw2+ 3u2+ 3v2+ 3w2e2 ≤ C Q f2+ f2e2+ C₀  Q 3
jk=1 jku 2_e2 (2.7) for all u∈ H3

0Q. As well known, u= v − curl w. Therefore, by Theorem 3.2 in

Isakov et al. (2003) and by (2.7),  Q 3
jk=1 jku 2_e2 _{≤ C} Q  u2e2 = C Q v − curl w2e2 ≤ C Q f2+ f2e2+ C₀  Q 3
jk=1 jku 2_e2_

Thus for small 0, we yield  Q 3
jk=1 jku 2_e2 _{≤ C} Q f2+ f2e2

and the estimate (2.7) leads to the first Carleman estimate (2.1).

To prove the second estimate we will use Carleman estimates for elliptic and hyperbolic operators in Sobolev norms of negative order. Applying Theorem 3.2 in Imanuvilov et al. (2003) to each of scalar hyperbolic operators in (2.6) we obtain

  Qu 2_e2 _{≤ C} Q v2+ −2f2e2   Q v2e2 ≤ C  Q ₀u2+ f2e2   Qw 2_e2 _{≤ C} Q ₀u2+ f2e2 Adding these inequalities we arrive at

  Q u2+ v2+ w2e2≤ C  Q 0u2+ f2e2 (2.8)

To eliminate the first term in the right side we use again the known identity u=

v− curl w, apply Lemma 2.2, and integrate with respect to t over −T T to get



Qu

2_e2_{≤ C} Q

v2+ w2e2

Using this estimate and choosing 0small and  > C we complete the proof of (2.2).

In order to use (2.1), it is required that the Cauchy data of the solution and the source term vanish on the lateral boundary. To handle non-vanishing Cauchy data, the following lemma is useful.

Lemma 2.3. For any pair of g₀g₁∈ H52× −T T × H

3

2× −T T, we can find a vector-valued functionu∗∈ H3_{Q such that}

u∗= g₀ _u∗= g₁ A_Ru∗= 0 on  × −T T

and

u∗_

3Q≤ Cg05

2× −T T + g132× −T T (2.9) for some C >0 provided ₀in (1.7) is sufficiently small.

Proof. By standard extensions theorems for any g2∈ H12× −T T we can find

u∗∗∈ H3_{Qso that}

u∗∗= g₀ _u∗∗= g₁ 2_u∗∗= g₂ on × −T T

and u∗∗_

3Q≤ Cg21₂× −T T + g13₂× −T T + g05₂× −T T

Since × −T T is non-characteristic with respect to A_Rprovided 0 is small, the

condition ARu∗∗= 0 on  × −T T is equivalent to the fact that g2 can be written

as a linear combination (with C1 _{coefficients) of }2

tg0 and tangential derivatives of

g₀ (of second order) and of g₁ (of first order) along  . In particular,

g212× −T T ≤ Cg132× −T T + g052× −T T

Choosing g2 as this linear combination we obtain (2.9).

3. Hölder Stability for the Residual Stress

In this section we prove the first main result of the paper, Theorem 1.1. Let u  1 and u  2 satisfy (1.3), (1.4), (1.5) corresponding to R  1 and R  2, respectively. Denote u= u  2 − u  1 and F = R  2 − R  1 = fjk, j k= 1     3. By

subtracting equations (1.3) for u  1 from the equations for u  2 we yield

AR 2u=   u  1F on Q where   u  1F = 3
jk=1 f_jk_j_ku  1 (3.1) and u= _tu= 0 on
× 0 (3.2)

Differentiating (3.1) in t and using time-independence of the coefficients of the system, we get A_{R 2}U=   U  1F on Q (3.3) where U=     2 tu 3 tu 4 tu     and U  1 =     2 tu  1 3 tu  1 4 tu  1     

By extension theorems for Sobolev spaces there exists U∗∈ H2_{Qsuch that}

U∗= U _U∗= _U on × −T T (3.4)

and U∗_

2Q≤ CU32× −T T + U12× −T T ≤ CF (3.5)

due to the definitions of u U, and F . We now introduce V= U − U∗. Then

AR 2V= F − AR 2U∗ on Q (3.6)

and

V= _V= 0 on  × −T T (3.7)

To use the Carleman estimate (2.1), we introduce a cut-off function ∈ C2_R4_

such that 0≤ ≤ 1, = 1 on Q

2and = 0 on Q\Q0. By Leibniz’ formula

A_{R 2} V= A_{R 2}V+ A₁V= F − A_{R 2}U∗+ A₁V

due to (3.6). Here (and below) A1denotes a first order matrix differential operator

with coefficients uniformly bounded by C. By the choice of , A1V= 0 on Q2.

Because of (3.7) the function V∈ H2

0Q, so we can apply to it the Carleman

estimate (2.2) to get  Q  V2e2≤ C  Q F2+ A_{R 2}U∗2e2+ C  Q\Q 2 A1V 2_e2 ≤ C  QF 2_e2_{+ F}2_e2!_{+ Ce}21  (3.8)

where != sup  over Q and 1= e



4. To get the last inequality we used the bounds

(3.5) and (1.9).

On the other hand, from (1.3), (3.1), (3.2) we have 2_tu=
f_jk_j_ku 3_tu=
f_jk_t_j_ku

on
× 0. So using the definitons of M F we obtain 2

tu  3

tu= MF on
× 0,

and from the condition (1.10) we have

F2_{≤ C}
=23  tu 0 2  (3.9) Since  T= 0, 
  tux 0 2 e2x0dx= −  T 0 _t  
   tux t 2_e2xt  dx dt ≤ Q 2 2_+1 t u  tu + t  tu 2_e2 + 2 Q\Q 2  tu 2 _ t e 2

where = 2 3. The right side does not exceed

C   Q  U2e2+ C  Q\Q 2 U2_e2  ≤ C  Q  V2e2+ C  Q\Q 2 U2_e2₊ QU ∗_2_e2 

because U= V + U∗. Using that = 1 on


2,  < 1 on Q\Q2and  < ! on

Qfrom these inequalities, from (3.8), from (3.5), and from (1.9) we yield


 2  tu 2_ 0e20≤ C   QF 2_e2_{+ Ce}21+ e2!_F2   (3.10) Using that = 1 on


2, from (3.9) and (3.10) we obtain


 2 F2_e20_{≤ C}  Q 2 F2_e2_{+ e}2!_F2_{+ Ce}21  (3.11) where we also split Q in the right side of (2.7) into Q

2 and its complement, and

used thatF ≤ C and  < 1 on the complement. To eliminate the integral in the

right side of (3.11) we observe that  Q 2 F2_xe2xt_{dx dt≤}
 2 F2_xe2x0  T −Te 2xt−x0_dt  dx

Due to our choice of function  we have x t− x 0 < 0 when t = 0. Hence

by the Lebesgue Theorem the inner integral (with respect to t) converges to 0 as

goes to infinity. By reasons of continuity of , this convergence is uniform with

respect to x∈
. Choosing  > C we therefore can absorb the integral over Q 2in

the right side of (3.11) by the left side arriving at the inequality 

F

2_e20_{≤ Ce}2!_F2_{+ Ce}21_

Letting ₂= e2 ≤  on
 and dividing the both parts by e22 we yield



F

2_{≤ Ce}2!−2_F2+ e−22−1_≤ Ce2!_F2+ e−22−1_{ (3.12)}

since e−22_{< C. To prove (1.13) it suffices to assume that F <} 1

C. Then

= _!+−log F

2−1 > C and we can use this  in (3.12). Due to the choice of ,

e−22−1= e2!_F2= F2!2−1+2−1

and from (3.12) we obtain (1.13) with = 2−1

!+2−1. The proof of Theorem 1.1 is now

complete.

4. Lipschitz Stability in the Cauchy Problem

Now we will prove a Lipschitz stability estimate for the Cauchy problem for the system (2.5). This estimate is a key to prove the estimate (1.17) in the inverse problem. Before going to the main result of this section, we state a lemma concerning the boundary condition for auxiliary functions v and w. We refer to Lin and Wang (2003) for the proof.

Lemma 4.1. Let a solution u∈ H3_{Q to systemA}

Ru= f in Q satisfy

f= u = _u= 0 on  × −T T

and let R satisfy (1.7) with ₀sufficiently small.

Then

_ku= _j_ku= 0 on  × −T T for 1 ≤ i j k ≤ 3 Now we can prove the following result.

Theorem 4.2. Suppose that
and T satisfy the assumptions of Theorem 1.2. Let u∈

H3Q3solve the Cauchy problem



A_Ru= f in Q

u= _u= 0 on 
× −T T (4.1)

with f∈ L2_{−T T H}1_{
 and f= 0 on 
× −T T. Furthermore, assume that}

(1.7) holds for sufficiently small 0.

Then there exists a constant C >0 such that u2

H1_Q+ v2_H1_Q+ w2_H1_Q≤ Cf2_L2_−TTH1_
 (4.2)

By virtue of (4.2) and equivalence of the normsu1
and of

div u0
+ curl u0


in H1

0
 (e.g., Duvaut and Lions, 1976, pp. 358–369), it is not hard to derive the

following

Corollary 4.1. Under the conditions of Theorem 4.2

u0Q+ xtu0Q+ tu0Q≤ Cf2L2_−TTH1_
 (4.3) Proof of Theorem4.2. By standard energy estimates for the system (2.6) we get

C−1  E0− C 
×0tf 2_{+ f}2_  ≤ Et ≤ C  E0+ 
×0tf 2_{+ f}2_  (4.4) for some C where

Et=



_tu2+ _tv2+ _tw2+ u2+ v2+ w2+ u2+ v2+ w2 t

To use the Carleman estimate (2.1) we need to cut off u near t= T and t = −T.

We first observe that from the definition

1≤ x 0 x ∈


and from the condition (1.15)

x T= x −T < 1 when x ∈


So there exists a  > 1

C such that

1−  <  on
× 0   < 1 − 2 on
× T − 2 T (4.5)

We now choose a smooth cut-off function 0≤ 0t≤ 1 such that 0t= 1 for

−T + 2 < t < T − 2 and t = 0 for t > T − . It is clear that A_R ₀u= ₀f+ 2_{t 0}_tu+ 2_{t 0}u

By Lemma 4.1 and basic facts about Sobolev spaces 0u∈ H03Q, we can use the

Carleman estimate (2.1) to get  Q 3 ₀u2+  ₀v2+  ₀w2+ _xt ₀u2+ _xt ₀v2+ _xt ₀w2e2 ≤ C  Q f2+ f2e2+ 
×T−2<t<Ttu 2_{+ } tu 2_{+ u}2_{+ u}2_e2  

Shrinking the integration domain on the left side to
× 0  where = 1 and

1−  <  and using that  < 1 − 2 on
× T −  T we derive that e21−   0 Etdt ≤ C  Q f2+ f2e2+ Ce21−2  T T−2 
_tu2+ _tu2+ u2+ u2   (4.6) To eliminate the last integral in (4.6) we remind that

curl _tu= _tw div _tu= _tv _tu= _tv− curl_tw and use the standard elliptic L2_-estimate


t u2 t≤ C  
_tv2+ _tw2   t Now using the energy bound (4.4) we derive from (4.6)

e21− CE0− Ce 2! Q f2_{+ f}2_{≤ Ce}2! Q f2_{+ f}2_{+ Ce}21−2_E0

Choosing  so large that e−2< 

C2 and fixing this  we eliminate the term with E0

on the right side. Using again (4.4) we complete the proof.

5. Lipschitz Stability for the Residual Stress

In this section we prove the second main result of the paper, Theorem 1.2. We will use the notation of Section 4.

In view of Lemma 2.3, there exists U∗∈ H3_{Qsuch that}

U∗= U _U∗= _U AR 2U∗= 0 on 
× −T T (5.1)

and U∗_

3Q≤ CU52× −T T + U32
× −T T ≤ CF (5.2)

due to the definition of F . We introduce V= U − U∗. Due to (5.1),

V= _V= 0 A_{R 2}V= 0 on 
× −T T (5.3)

Applying Corollary 4.1 to (3.6), (3.7) and using (5.2) gives V2 0Q+ xtV 2 0Q+ tV 2 0Q≤ C  F2 1
+ F 2_ (5.4) On the other hand, as in the proof of Theorem 1.1 we will bound the right side by V.

We will use the cut off function ₀ of Section 4. According to Lemma 4.1 and

(5.3), 0V∈ H03Q. By Leibniz’ formula

A_{R 2} ₀V= ₀  U  1F − ₀A_{R 2}U∗+ 2_{t 0}_tV+ 2_{t 0}V and by the Carleman estimate (2.1)

 Q 2 0 3_V2_{+ V}2_e2 ≤ C  Q F2+ F2+ A_{R 2}U∗2+ A_{R 2}U∗2e2 +
×T−2<t<TV 2_{+ } xtV 2_{+ } tV 2_e2  ≤  Q F2+ F2e2+ e2!F2+ e21−2 
F2+ F2  

where we let != supQand used (4.5) and (5.4). Since U= V + U∗ from (5.2) we

obtain  Q 2 0U 2_{+ U}2_e2 ≤ Ce2!_F2₊
  T −Te 2xt_{dt+ e}21−2  F2+ F2xdx (5.5)

Utilizing (3.2) and (1.10), similarly to deriving (3.9), we get from (3.1) that 2

tu  3

tu= MF on
× 0. Therefore, using (1.10) we will have


F2+ F2e2s0≤ C 

=23k=01  t k_{u 0}2_e20 = −C T 0 _t  

₂ 0t k_u2_e2_dx  dt

≤ C Q 2 0
 t k_u+1 t  k_{u + } tt k_u2_e2 + C
×T−2T 0t 0
t k_u2_e2_

Now as in the proofs of Section 3 the right side is less than C   Q  2₀U2+ U2e2+ 
×T−2TU 2_{+ U}2_e2  ≤ C  Q  2₀U2+ U2e2+ e21−2F2_1
+ F2  

where we used equality U= U∗+ V and (5.2), (5.4). From two previous bounds we

conclude that 
F2+ F2e20≤ C  e2!F2+ 
  T −Te 2t_{dt+ e}21−2  F2+ F2   (5.6)

Due to our choice of , 1≤  0  t −  0 < 0 when t = 0. Thus by the

Lebesgue theorem as in the proofs of Section 3, we have 2C   T −Te 2t_{dt+ e}21−  ≤ e20

uniformly on
when  > C. Hence choosing and fixing such large  we eliminate the second term on the right side of (5.6). The proof of Theorem 1.2 is now complete.

By using Carleman estimates on functions satisfying the homogeneous zero

boundary conditions (g0= 0 or zero stress boundary condition) one can replace 

in Theorem 1.2 by its “large” part  (Isakov, 2006, Section 4.5, for scalar equations).

6. Conclusion

Using similar methods one can expect to demonstrate uniqueness and stability for both variable   and residual stress most likely from two sets of suitable boundary data. Motivation is coming from geophysical problems. Our assumptions exclude zero initial data. So far it looks like a very difficult question to show uniqueness from few sets of boundary data when the initial data are zero. The stability guaranteed by Theorems 1.1 and especially by Theorem 1.2 indicates a possibility of a very efficient algorithms with high resolution for numerical idenitification of residual stress from single lateral measurements. It would a very good idea to run some numerical experiments to understand possibilities of practical applications of these stability properties.

Acknowledgments

The work of Victor Isakov was in part supported by the NSF grant DMS 04-05976. The work of Jenn-Nan Wang was partially supported by the grant

of National Science Council of Taiwan NSC 94-2115-M-002-003.The work of Masahiro Yamamoto was partly supported by Grant 15340027 from the Japan Society for the Promotion of Science and Grant 15654015 from the Ministry of Education, Cultures, Sports and Technology.

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