UTMS 2003–9February 28, 2003

Unique determination of inhomogeneity
in a stationary isotropic Lamé systemwith varible coeﬃcients
bySungwhan Kim and Masahiro Yamamoto

TUNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANUNIQUE DETERMINATION OF INHOMOGENEITY IN A
STATIONARY ISOTROPIC LAMÉ SYSTEM WITH VARIBLE
COEFFICIENTSSUNGWHAN KIM† AND MASAHIRO YAMAMOTO‡

Department of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba, Meguro, Tokyo 153-8914 Japane-mail : † sungwhan@ms.u-tokyo.ac.jp‡ myama@ms.u-tokyo.ac.jp
Abstract. We consider a two dimensional stationary Lamé system with variable coeﬃcients. We prove the uniqueness in the inverse source problem ofdetermining polygonal supports of distributing force by observations of surface
displacement and stress. Our method is based on the regularity property ofsolutions to the Poisson equation in a polygonal domain.

1. Introduction
Let Ω be a bounded domain with C 2 -boundary ∂Ω. We assume that Ω isoccupied by a nonhomogeneous isotropic elastic medium and consider a two dimensional stationary isotropic Lamé system with variable coeﬃcients(Lu)(x) ≡ µ(x)∆u(x) + (λ(x) + µ(x))∇(∇ · u(x))+(∇ · u(x))∇λ(x) + (∇u(x) + (∇u(x))T )∇µ(x)= F(x),x∈Ω(1.1)

with the boundary conditionu(x) = f(x),x ∈ ∂Ω.

Here and henceforth,x = (x1 , x2 ) ∈ R2u(x) = (u1 (x), u2 (x))T·T : the transpose of a vector or a matrix under consideration∂u2∂u1(x) +(x)∇ · u(x) =∂x1∂x2
1(1.2)

2SUNGWHAN KIM† AND MASAHIRO YAMAMOTO‡∇u(x) =∂uk∂xl

: 2 × 2 − matrix
1≤k,l≤2λ, µ ∈ C 2 (Ω) : the Lamé coeﬃcients depending on x.
We assume thatµ(x) > 0,λ(x) + 2µ(x) > 0,x ∈ Ω.

(1.3)

System (1.1) with boundary input (1.2) describes the elastic displacement by anexterior force F(x) = (F1 (x), F2 (x))T . In practise, we often need determine theacting distributing force by the resulting surface stress σ(u)ν on a subboundaryΓ ⊂ Ω. Here and henceforth, the surface stress σ(u)ν is deﬁned as follows: letν = ν(x) be the unit outward normal vector to ∂Ω at x, and we deﬁne a 2 × 2matrix σ(u) by
1 0+ µ(x)(∇u(x) + (∇u(x))T )σ(u) = λ(x)(∇ · u)
0 1(1.4)

for u(x) = (u1 (x), u2 (x))T .
In particular, we consider a force F in the form
F(x) = q(x)(χD (x), χE (x))T ,(1.5)

where q is a given and χD denotes the characteristic function of a set D ⊂ Ω.
Force (1.5) describes that the x1 - and x2 -components of the force distribute onlyin D and E respectively with the strength q(x). In this paper, we will discuss
Inverse source problem.

Let Γ be an arbitrary relatively open subset of∂Ω , f in (1.2) be ﬁxed and q ∈ C 2 (Ω) be given such that q > 0 on Ω. Then
determine D and E in (1.1) with (1.5) by σ(u)ν on Γ.
Our main concern is the uniqueness: is the correspondenceσ(u)ν|Γ ←→ (D, E)
one to one?
A similar problem for the Laplacian is called an inverse gravimetry problem:
In −∆u(x) = q(x)χD (x) in R3 and lim|x|→∞ u(x) = 0, determine D by ∇u|∂Ωwith a bounded domain Ω.

UNIQUE DETERMINATION OF INHOMOGENEITY IN LAMÉ SYSTEM

3

As for the inverse gravimetry, there are many papers and we can consult Anger[2], Isakov [5], [6]. However there seems no papers treating the determination ofsupports of right hand sides in elliptic systems with variable coeﬃcients. Kim
and Yamamoto [8] proposed a proof for the uniqueness within polygonal D’s andwe apply the argument in [7], [8] to prove the uniqueness in determining (D, E)
in (1.1) and (1.5) by f on ∂Ω and σ(u)ν on Γ. Our argument is applicable to thehigher dimensional cases and to other systems such as the stationary Maxwell’sequations, but for conciseness, we restrict ourselves to the two dimensional stationary isotropic Lamé system.
The paper is organized as follows:
Section 2: Main results
Section 3: Preliminaries
Section 4: Proof of the main results.
2. Main results
Let Ω ⊂ R2 be a bounded domain with C 2 boundary ∂Ω and let Dj , Ej , j = 1or 2, be sums of a ﬁnite number of polygons such that Dj , Ej ⊂ Ω :MjDj = ∪n=1Dn,jandNj
Ej = ∪n=1
En,j ,

where Dn,j , En,j = ∅ are open polygons and the Dn,j , En,j ⊂ Ω, Dn,j ∩ Dm,j =En,j ∩ Em,j = ∅ if n = m.
Let uj = (uj,1 , uj,2 ) ∈ H 1 (Ω) × H 1 (Ω), j = 1 or 2, be the weak solution to theLamé system(Luj )(x) = q(x)(χDj (x), χEj (x))T ,x∈Ω

(2.1)

with the Dirichlet boundary datauj (x) = f(x),3x ∈ Γ ⊂ ∂Ω.

(2.2)

3Here f ∈ H 2 (Ω) × H 2 (Ω), q ∈ C 2 (Ω) and q > 0 on Ω.
Then it is well-known thatuj ∈ H 2 (Ω) × H 2 (Ω).

(2.3)

4

SUNGWHAN KIM† AND MASAHIRO YAMAMOTO‡

Henceforth, for D ⊂ R2 , we denote the convex hull (i,e., the smallest convex set
containing D) by co(D). Now we are ready to state our main results. First weconsider the case where D1 = E1 and D2 = E2 . That is, let uj , j = 1 or 2, satisfy
the Lamé system(Luj )(x) = q(x)(χDj (x), χDj (x))T ,x ∈ Ω.

(2.4)

Theorem 2.1. We assume that D1 = E1 and D2 = E2 . Thenσ(u1 (x))ν(x) = σ(u2 (x))ν(x),

x∈Γ

(2.5)

implies that co(D1 ∪ E1 ) = co(D2 ∪ E2 ), where ν = ν(x) = (ν1 (x), ν2 (x))T is theoutward unit normal vector to ∂Ω.
Corollary 2.2. Under the conditions in Theorem 2.1, we further assume thatD1 , D2 are convex. Then (2.5) implies D1 ∪ E1 = D2 ∪ E2 .
In the case where D1 = E1 or D2 = E2 , we do not know whether co(D1 ∪ E1 ) =co(D2 ∪ E2 ), in general. However, if D1 ∩ E1 = D2 ∩ E2 = ∅, we can obtainsome global uniqueness result through the same argument as one of the proof ofTheorem 2.1.
Theorem 2.3. We assume that D1 ∩ E1 = D2 ∩ E2 = ∅. Thenσ(u1 (x))ν(x) = σ(u2 (x))ν(x),

x∈Γ

(2.6)

implies that co(D1 ∪ E1 ) = co(D2 ∪ E2 ), where ν = ν(x) = (ν1 (x), ν2 (x))T is theoutward unit normal vector to ∂Ω.
In a succeeding paper, we will consider a more general case.
3. Preliminaries
In this section, we will give preliminary results for the proof of our main theorems. Henceforth let AB denote the line segment including A and B. The ﬁrst
lemma shows the regularity of an H 1 -solution to an elliptic equation, which plays
an important role in proving our main theorem. The proof is essentially basedon [4]. For completeness, we will give the proof.

UNIQUE DETERMINATION OF INHOMOGENEITY IN LAMÉ SYSTEM

5Lemma 3.1. Let P1P2 P3 be the interior of a triangle which has three vertices
Pj ∈ R2 , j = 1, 2, 3. Assume that f ∈ Lµ (P1 P2 P3 ) for some µ > 2. Ifv ∈ H 1 (P1 P2 P3 ) is the solution to a Dirichlet problem for the Laplace equation∆v = fin P1 P2 P3(3.1)
v=0
on P1 P2 ∪ P2 P3 ∪ P3 P1 ,then there exists a number p > 2 such thatv ∈ W 2,p (P1 P2 P3 ).

(3.2)

Proof. Let θ1 , θ2 and θ3 be the angles P3 P1 P2 , P1 P2 P3 and P2 P3 P1 , respectively. Since 0 < θj < π for any j = 1, 2, 3, we can take a real number q0 ∈ (1, 2)
so that

Let p := min2< minq0q0,µq0 −1

π π π, ,θ1 θ2 θ3


.

(3.3)

. Clearly the number p is greater than 2. We claim thatv ∈ W 2,p (P1 P2 P3 ).

pLet q := p−1. Then by (3.3) we have2(p − 1)
22=≤< minqpq0π π π, ,θ1 θ2 θ3(3.4)
,2θwhich implies that the number qπj is not an integer for any j = 1, 2, 3. Sincep ≤ µ and f ∈ Lµ (P1 P2 P3 ), we see that f ∈ Lp (P1 P2 P3 ). Therefore it follows
from Theorem 4.4.4.13 in [4] that there exist real numbers cj,m and a function w
such thatw−cj,m Sj,m

∈ W 2,p (P1 P2 P3 )1≤j≤3− 2q < λj,m < 0λj,m = −1and∆w = fin P1 P2 P3
w=0on P1 P2 ∪ P2 P3 ∪ P3 P1 ,where m is a negative integer, λj,m = mπ, and the functions Sj,m are deﬁned
θjin equation (4, 4, 3, 7) in [4]. We note that Sj,m does not necessarily belong to6SUNGWHAN KIM† AND MASAHIRO YAMAMOTO‡

W 2,p (P1 P2 P3 ). The uniqueness of the Dirichlet problem yields
w = v.
Furthermore our choice of constants p, q implies that there are not negative
integers m such that

2mπ− < λj,m =< 0.
qθjHence we can conclude thatv ∈ W 2,p (P1 P2 P3 ).

Applying the above lemma and the Sobolev imbedding theorem (e.g., [1]), wecan prove that an H 1 -solution to a Cauchy problem of the Laplace equation is of
C 2 -class in a neighbourhood of a corner of a triangular domain. This proposition
plays an essential role in proving our theorems.
Lemma 3.2. Let P1P2 P3 be the interior of a triangle which has three vertices
Pj ∈ R2 , j = 1, 2, 3. Let G ∈ W 1,µ (P1 P2 P3 ) for some µ > 2 and let y ∈H 1 (P1 P2 P3 ) be the solution to the Laplace equation∆y = Gin P1 P2 P3y = |∇y| = 0on P1 P2 ∪ P1 P3 .

(3.5)

Then there exists a neighbourhood U of P1 such that the solution y belongs toC 2 (U ∩ P1 P2 P3 ).
The proof is done similarly to Proposition 2.2 in [7] and the details are omittedhere.
4. Proof of Theorem 2.1
Assume contrarily that co(D1 ) = co(D2 ). Then, since co(D1 ) and co(D2 ) areconvex polygons, there exists a vertex O of co(D1 ) such that O ∈ Ω \ co(D2 ) ora vertex O of co(D2 ) such that O ∈ Ω \ co(D1 ). Without loss of generality, wemay assume the former case. Then there exists a polygon Dn,1 for some integer

UNIQUE DETERMINATION OF INHOMOGENEITY IN LAMÉ SYSTEM

71 ≤ n ≤ M1 such that O is a vertex of Dn,1 . Since O ∈ Dn,1 \ co(D2 ), we cantake a small triangle OAB such thatOA ∪ OB ⊂ ∂Dn,1

OAB ⊂ Dn,1 \ co(D2 ).

and(4.1)

Here and henceforth OAB means the interior of the triangle with the vertices
O, A and B.
Let us deﬁne v = (v1 , v2 ) := u1 − u2 in Ω. Then it follows from (2.2) and (2.4)
that the function v satisﬁes(Lv) = µ∆v + (λ + µ)∇(∇ · v) + (∇ · v)∇λ + (∇v + (∇v)T )∇µin Ω= q(x)((χD1 − χD2 ), (χD1 − χD2 ))T(4.2)

andv = σ(v)ν = 0on Γ.

(4.3)

Let us denote by D the connected component of Ω \ (D1 ∪ D2 ) with ∂Ω as its
boundary portion. Therefore, the unique continuation (e.g., [3]) implies thatv≡0on D.

(4.4)

on OA ∪ OB.

(4.5)

Since OA ∪ OB ⊂ D, we havev = |∇v| = 0
For simplicity, we deﬁne for j = 1, 2
Hj = (Hj,1 , Hj,2 ) := (∇ · uj )∇λ + (∇uj + (∇uj )T )∇µandGj := −2{(∂xi µ)(∆uj,i ) + [∂xi (λ + µ)][∂xi (∇ · uj )] + ∂xi Hj,i }.

i=1

Then by (2.3), (4.1), (4.2) and (4.5), we obtain that the function ∇ · v is an
H 1 -solution to the following elliptic equation
1[(G1 − G2 ) + ∂x1 q + ∂x2 q]∆(∇ · v) = λ+2µ∇·v =0on OA ∪ OB.

in OAB(4.6)

Since µ, λ, q1 , q2 ∈ C 2 (Ω) and λ + 2µ > 0 on Ω, the deﬁnition of Hj and Gjyields
1[(G1 − G2 ) + ∂x1 q + ∂x2 q] ∈ L2 (OAB).
λ + 2µ(4.7)

8

SUNGWHAN KIM† AND MASAHIRO YAMAMOTO‡

By using an appropriate cut-oﬀ function, we can easily deduce from (4.6) and(4.7) that1∇ · v ∈ H 2 ( OAB).
2(4.8)

Here and henceforth, for any s > 0, 1s OAB means the interior of the triangle
with the vertices O, 1s (A − O) + O and 1s (B − O) + O.
On the other hand, we have from (4.1) and (4.2) that for i = 1, 2µ∆vi = −(λ + µ)[∂xi (∇ · v)] − (H1,i − H2,i ) + qin OAB.

(4.9)

Since µ ∈ C 2 (Ω) and µ > 0 on Ω, (4.8) yields−111λ+µ[∂xi (∇ · v)] − (H1,i − H2,i ) + q ∈ H 1 ( OAB).
µµµ2

Then (4.5), (4.8) and the Sobolev imbedding theorem implies that
∆vi ∈ L ( 12 OAB)
for any + ≥ 2vi = 0on OA ∪ OB.

(4.10)(4.11)

By using an appropriate cut-oﬀ function and Lemma 3.1, we obtain that thereexists a number η > 2 such that1vi ∈ W 2,η ( OAB).
4(4.12)

Then (4.12) and the deﬁnition of Hj and Gj imply that1
G1 − G2 ∈ Lη ( OAB),
4(4.13)

1
1[(G1 − G2 ) + ∂x1 q + ∂x2 q] ∈ Lη ( OAB).
λ + 2µ4(4.14)and hence

Applying again Lemma 3.1 to (4.6) and using an appropriate cut-oﬀ function, weobtain that there exists a number ρ > 2 such that1
∇ · v ∈ W 2,ρ ( OAB).
8(4.15)

Let p = min{η, ρ}. It is clear that p > 2. Hence, it follows from (4.5), (4.9) and(4.15) that the function vi is an H 2 -function satisfying
[∂xi (∇ · v)] − µ1 (H1,i − H2,i ) + µ1 q ∈ W 1,p ( 18 OAB)∆vi = − λ+µµon OA ∪ OB.
vi = |∇vi | = 0(4.16)

UNIQUE DETERMINATION OF INHOMOGENEITY IN LAMÉ SYSTEM

9Then Lemma 3.2 yield that there exists a large number S > 0 such that1vi ∈ C 2 ( OAB),
S
and hence by (4.5), for i = 1, 2, we have ∆vi (O) = 0 andλ(O) + µ(O)1[∂xi (∇ · v(O))] −(H1,i (O) − H2,i (O)) = 0.
−µ(O)
µ(O)

(4.17)

Since q > 0 on Ω, this contradicts (4.16). Therefore the proof of Theorem 2.1 iscomplete.
References[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975

[2]

G. Anger, Inverse Problems in Diﬀerential Equations, Plenum Publ., New York, 1990

[3]

D. D. Ang, M. Ikehata, D. D. Trong and M. Yamamoto, Unique continuation for astationary isotropic Lamé system with variable coeﬃcients, Commun. in Par. Diﬀ. Eq.,
23(1998), 371-385[4]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985

[5]

V. Isakov, Inverse Source Problems, American Mathemathical Society, Providence, Rhode
Island, 1990

[6]

V. Isakov, Inverse Problems for Partial Diﬀerential Equations, Springer-Verleg, Berlin,
1998

[7]

S. Kim and M. Yamamoto, Uniqueness in identiﬁcation of the support of a source termin an elliptic equation, to appear in SIAM J. Math. Anal.

[8]

S. Kim and M. Yamamoto, Uniqueness in the two-dimentional inverse gravimetry problem: case of variable coeﬃcient, to appear in the Proceedings for International Conference on
Inverse Problems (January 9-12, 2002), World Scientiﬁc, Singapore, 2003

Preprint Series, Graduate School of Mathematical Sciences, The University of TokyoUTMS
2002–37 Naoto Kumano-go: Interchange with Feynman path integrals and RiemannStieltjes integrals.
2002–38 Sungwhan Kim and Masahiro Yamamoto: On an inverse problem of determining a support of a source term in an elliptic equation.
2002–39 Sungwhan Kim and Masahiro Yamamoto: Uniqueness in the two-dimentional
inverse conductivity problem of determining convex polygonal supports: Case ofvariable conductivity.
2003–1 Akihiro Shimomura: Scattering theory for Zakharov equations in three spacedimensions with large data.
2003–2 Oleg Yu. Imanuvilov and Masahiro Yamamoto: Carleman estimates for thenon-stationary Lamé system and the application to an inverse problem.
2003–3 Xu Bin: Sobolev norms of eigenfunctions on a closed Riemannian manifold.
2003–4 Xu Bin: Compact Lie group actions on compact manifolds of nonpositive curvature.
2003–5 J. Elschner, G. Schmidt and M. Yamamoto: An inverse problem in periodic
diﬀractive optics: Global nuiqueness with a single wave number.
2003–6 Naoto Kumano-go: New curvilinear integrals along paths of Feynman path integral.
2003–7 Susumu Yamazaki: Microsupport of Whitney solutions to systems with regular
singularities and its applications.
2003–8 Naoto Kumano-go and Daisuke Fujiwara: Smooth functional derivatives inFeynman path integral.
2003–9 Sungwhan Kim and Masahiro Yamamoto : Unique determination of inhomogeneity in a stationary isotropic Lamé system with varible coeﬃcients.

The Graduate School of Mathematical Sciences was established in the University ofTokyo in April, 1992. Formerly there were two departments of mathematics in the University of Tokyo: one in the Faculty of Science and the other in the College of Arts andSciences. All faculty members of these two departments have moved to the new graduate school, as well as several members of the Department of Pure and Applied Sciencesin the College of Arts and Sciences. In January, 1993, the preprint series of the former
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