Inverse boundary spectral problem for Riemannian polyhedra

0.31 MB·English

by  Anna Kirpichnikova

7 Inverse Boundary Spectral Problem for
0
0
Riemannian Polyhedra
2

n
a
J Kirpichnikova A., Kurylev Ya.

1
3
February 2, 2008

]
P
A
. We consider an admissible Riemannian polyhedron with piece-wise smooth
h
t boundary. The associated Laplace defines the boundaryspectral data as the set of
a
m eigenvalues and restrictions to the boundary of the corresponding eigenfunctions.
[ Inthis paperweprove thattheboundaryspectral dataprescribedonan opensub-

1 set of the polyhedron boundary determine the admissible Riemannian polyhedron
v uniquely.
1
1
9
1 1 Introduction
0
7
0 Recent years have seen some very significant achievements in the study of
/ inverse boundary-value problems in a single component body. Mathemati-
h
t cally such body is described by a PDE or a system of PDE’s with relatively
a
m smooth coefficients. Starting from the pioneering works [7] and [42], inverse
: boundary-value problems were solved, at least on the level of uniqueness and
v
i sometimes conditional stability, for a wide range of scalar inverse problems,
X
both isotropic and anisotropic, see e.g. [4], [8], [9], [20], [23], [29], [30], [31],
r
a [32], [35], [38], [41] for a far from complete list of references, with further ref-
erences in monographs [17] or [21]. Moreover, for such media there appeared
a number of important results in the study of the inverse boundary-value
problems for systems of PDE’s corresponding to physically important mod-
els of electromagnetism, elasticity and Dirac equations, see e.g. [18], [27],
[28], [33], [34], [36], [37].
Much less is known, however, about the inverse boundary-value problems
for a multicomponent medium. Mathematically, such medium is described
1
1 INTRODUCTION 2
by PDE or system of PDE’s with piece-wise smooth coefficients with dif-
ferent subdomains of the regularity of coefficients corresponding to different
components of the medium. Clearly, the study of inverse problems for the
multicomponentmediaisofsubstantialimportanceforpracticalapplications.
Imagine, for example, a human body with bones, muscle tissue, lungs, etc.
each of those having distinctive values of material parameters, or an upper
crust of the Earth which is a composition of clay, sand, rock, oil, water, etc.
A complete answer to the inverse boundary problems in a multicomponent
medium, at least when the data are measured on the whole boundary, is ob-
tainedonlyforthetwo-dimensionalcase. Namely, itwasshownin[2], [3]that
the Calderon inverse boundary problem in the 2D case has a unique solution
in the class of L∞ coefficients. Clearly, these results cover also the case of
−
a multicomponent medium. In higher dimensions, the results are restricted
mainly to the inverse obstacle problem. In these problems the goal is to find
a shape of an inclusion inside a given medium which parameters are known
a priori. In the case when parameters of a medium and/or inclusion are
unknown they are assumed to be homogeneous throughout each component,
see e.g. [1], [14], [15], [16], [26]. Having said so, we should note that there
exist powerful methods to find singularities for coefficients of lower order, see
e.g. [11].
This paper is devoted to the study of the inverse boundary spectral prob-
lem for the Laplace operator in a multicomponent medium. To be more
precise, we assume that the domain occupied by the medium consists of
a finite number of subdomains with piece-wise smooth boundaries between
them. The metric tensor in each subdomain is smooth but does have jump
singularity across the interfaces, i.e. the boundaries between adjacent sub-
domains. Adding proper transmission conditions across the interfaces and
boundary conditions on the domain’s boundary, defines a Laplace operator
which, from the spectral point of view, has effectively the same properties
as the Laplace operator in a single component medium. Mathematically, the
considered medium may be described as a Riemannian polyhedron. Leav-
ing exact definitions of an appropriate Riemannian polyhedron to the next
section, imagine an n dimensional simplicial complex where simpleces
− M
can be glued together, pairwise, along their (n 1) dimensional faces which
− −
we continue to call interfaces (sometimes (n 1) interfaces). Imagine now
− −
that each simplex has its own smooth metric g which, in principle, may have
jumpsacrossinterfacesbetweenadjacentsimpleces. This, togetherwithsome
additional geometric/combinatoric conditions described in section 2, defines
1 INTRODUCTION 3
a Riemannian polyhedron ( ,g). Starting from the corresponding Dirichlet
M
form on H1( ,g) functions and using standard methods of spectral the-
M −
ory, the Laplace operator with Neumann boundary conditions, ∆, is then
well-defined in L2( ,g). Denote by λ , ϕ ∞ the set of all eigenvalues,
M { k k}k=1
counting multiplicity, and corresponding orthonormal eigenfunctions of ∆.
Let Γ ∂ be open.
⊂ M
Definition 1.1 The collection (Γ, λ , ϕ ∞ ) is called the (local) bound-
{ k k|Γ}k=1
ary spectral data (LBSD) of the Riemannian polyhedron ( ,g).
M
Let now ( ,g) and ( ,g) be two Riemannian polyhedra with LBSD
M M
(Γ, {λk, ϕk|Γ}∞k=1) and Γ, {λk, ϕk|Γe}∞k=1 , correspondingly.
f e
(cid:16) (cid:17)
Definition 1.2 LBSD foer ( e ,ge) and ( ,g) are equivalent if
M M
1. Γ and Γ are homeomorphic, κ : Γ Γ;
→f e
2. λ = λ , k = 1,2,...;
k ke e
3. If λ has multiplicity m+1, m = 0,1,..., i.e. λ = λ = λ , then
k e k k+1 k+m
there is an (m+1) (m+1) unitary matrix U such that
k
×
(ϕk|Γ,...,ϕk+m|Γ) = Uk(κ∗ϕk|Γe,...,κ∗ϕk+m|Γe).
We can now formulate the main result oef the papere:
Theorem 1.3 Let ( ,g) and ( ,g) be two admissible Riemannian poly-
M M
hedra. Let, in addition, the metric tensors g and g do have jumps across
all (n 1) interfaces in andf e, correspondingly. Assume that LBSD
− − M M
(Γ, {λk, ϕk|Γ}∞k=1) and Γ, {λk, ϕk|Γe}∞k=1 are equivealent. Then (M,g) and
f
( ,g) are isometric. (cid:16) (cid:17)
M e e e
Let us make some comments on this theorem:
f e
1. If Ω is an n dimensional simplex of with a smooth metric g , then
i i
− M
g determines an inner metric on any l dimensional, l < n, simplex
i
of which lies in Ω (here and later we assume each simplex to be
i
M
close). In particular, any (n 1) interface γ of belongs to two
− − M
adjacent n simpleces, which we often denote in such case Ω and Ω ,
− +
−
2 PRELIMINARY CONSTRUCTIONS 4
and therefore, has two different metric tensors g and g . By a
− γ + γ
| |
metric tensor g having a jump singularity across γ we mean that, for
any p γ,
∈
g (p) = g (p).
− +
6
This assumption is of a technical nature and, in section 6 we will sig-
nificantly weaken it.
2. As shown in section 2, any Riemannian polyhedron has a natural struc-
ture of a metric space. The isometry of ( ,g) and ( ,g) is under-
M M
stood with respect to these metric structures.
f e
3. The boundary ∂ of a Riemannian polyhedron ( ,g) is itself a (n
M M −
1) dimensional Riemannian polyhedron, probably disconnected. As
−
Γ, Γ are open subsets of ∂ , ∂ , by reducing them if necessary we
M M
assume that Γ and Γ are open subsets of some (n 1) dimensional
− −
simeplex of ∂ , ∂ , correspondfingly. In the future, we will always
M M
assume this conditioen to be true.
f
The plan of the paper is as follows: In section 2 we provide some prelim-
inary material on geometry of Riemannian polyhedra and properties of the
Laplace operator on them. Section 3 is devoted to the description and some
properties of the non-stationary Gaussian beams on a Riemannian polyhe-
dron. WeproveTheorem 1.3insections 4and5. Thelast section6isdevoted
to some generalizations and open questions.
2 Preliminary constructions
2.1 Admissible Riemannian polyhedron
In this section we will introduce, following mainly [10] and [6], an admissible
Riemannian polyhedron which is the main object of the paper. We start with
a closed n dimensional finite simplicial complex
−
I
= Ω ,
i
M
i=1
[
2 PRELIMINARY CONSTRUCTIONS 5
(A) (B) (C)
Figure1: Case(A)isprohibitedbecauseitsstructureisnot(n 1)-chainable;
−
Case (B) is prohibited as it is not dimensionally homogeneous; Case (C) is
appropriate
where Ω are closed n dimensional simpleces of , with Ωint standing for
i − M i
the interior of Ω which is an open subset of . We assume that is di-
i
M M
mensionally homogeneous, i.e. any k simplex, 0 k n, of is contained
− ≤ ≤ M
in at least one Ω . We assume also that any (n 1) dimensional simplex
i
− −
γ belongs either to two different n simpleces, Ω and Ω , which in this case
i j
we will often denote by Ω and Ω , or to only one n simplex Ω . In the
− + i
former case we call γ an interface (sometimes (n 1) dimensional interface)
− −
between Ω and Ω , in the latter case we call γ a boundary (n 1) simplex
− +
− −
with (n 1) simpleces having this property forming the boundary ∂ .
− − M
We denote by k, 0 k n the k skeleton of which consists of all
M ≤ ≤ − M
k simpleces of . Clearly, = n. We use notations
− M M M
n−2
int = Ωint, reg = k .
M i M M\ M
!
k=0
[ [
Following [10], we assume that is (n 1) chainable, i.e. reg is path
M − − M
connected, see Fig. 1.
Assume now that each n simplex Ω is equipped with a smooth (up to
i
−
∂Ω ) Riemannian metric g , i.e. (Ω , g ) is a smooth Riemannian manifold
i i i i
with a piecewise smooth boundary. This makes it possible to introduce the
arclength for admissible paths η : [0,a] . We call a path η admissible
→ M
if η−1( int) [0,a] is a (relatively) open subset of [0,a] of full measure
M ⊂
and, if η(α,β) is in some n simplex Ωint, then η : (α,β) Ωint is piecewise
− →
smooth. Naturally, the arclength η(α,β) of the path η between η(α) and
| |
2 PRELIMINARY CONSTRUCTIONS 6
η(β) is taken as
β
η(α,β) = [g (η(t))η˙ (t)η˙ (t)]1/2dt, (1)
mj m j
| |
Zα
where η (t), α < t < β are, for example, baricentric coordinates in Ω. As
j
η−1( int) (0,a) consists of at most a countable number of open intervals
M ∩
(α ,β ) we define
i i
η[0,a] = η(α ,β ) . (2)
i i
| | | |
i
X
Next we introduce, for any p,q , the distance, d(p,q),
∈ M
d(p,q) = inf η ,
η | |
where infenum is taken over all admissible paths connecting p and q. This
makes ( , g) into a metric space with its metric topology being the same
M
as the topology of a simplicial complex, see [10].
Definition 2.1 ( , g) is an admissible Riemannian polyhedron if, for any
M
p,q ,
∈ M
d(p,q) = inf η ,
ηe | |
where η run over the subset of admissibleepaths between p and q such that
n−2
e η−1 k ( 0 1 ) = .
M \ { }∪{ } ∅
!
k=0
[
e
As is finite, the above condition is independent of a particular choice of
M
metric g.
2.2 Boundary normal and interface coordinates
InadditiontothebaricentriccoordinatesinanyΩint,wewilloftenusebound-
ary normal or interface coordinates associated with (n 1) subsimpleces of
− −
Ω.
2 PRELIMINARY CONSTRUCTIONS 7
p p” p p”
(A) (B)
Figure 2: A sample of two topologically different Riemannian polyhedra hav-
ing the same spectral properties. It is stronger than the (n 1) chainability
− −
andisaimedatavoiding topologicallydifferent Riemannianpolyhedra which,
however, can have the same spectral properties
Let first γ Ω ∂ be a boundary (n 1) dimensional simplex with its
∈ ∩ M − −
(n 1) dimensional interior denoted by γint. We introduce boundary normal
− −
coordinates in an relatively open subset U Ω as
⊂
p (s(p),σ(p)), p U.
→ ∈
Here σ(p) = d(p,∂ ), and we assume that there is a unique q γint with
M ∈
d(p,∂ ) = d(p,q),suchthatpliesonthenormalgeodesictoq,ς (τ), ς (0) =
ν ν
M
q, ς (d(p,∂ )) = p and ς (0,d(p,∂ )) Ωint. If s(q) = (s1,...,s(n−1)) are
ν ν
M M ∈
some (local) coordinates on γ, e.g. baricentric coordinates, then s(p) = s(q).
Let now γ Ω Ω be an (n 1) interface between n simpleces Ω
− + −
⊂ ∩ − − −
and Ω . Let U be relatively open subsets of Ω with the nearest point
+ ± ±
on ∂Ω lying on γ such that (U γ) = (U γ). Denote by (s,σ ) the
± − + ±
boundary normal coordinates in U , where s = (s1,...,s(n−1)) are some
±
T T
local coordinates on γ, e.g. baricentric coordinates with respect to Ω or
−
Ω . We introduce the interface coordinates (s,σ) on U U :
+ − +
(s, σ), in U , S
−
(s,σ) = −
((s,σ), in U+.
2 PRELIMINARY CONSTRUCTIONS 8
Then the metric element in these coordinates takes the form,
(dl)2 = (dσ)2 +(g ) (s,σ)dsαdsβ. (3)
± αβ
Throughout the paper we assume that the following condition takes place:
Condition 2.2 For any interface γ and any point q = (s,0) on γ, the metric
tensor g has a jump singularity at q.
αβ
2.3 Laplace operator
Let H1( ) be the Sobolev space of functions u L2( ) such that u =
i
M ∈ M
u H1(Ω ) and, for any interface γ between Ω and Ω ,
|Ωi ∈ i i j
u = u .
i γ j γ
| |
The inner product on H1( ) determines the closed non-negative Dirichlet
M
form,
I
D[u,v] = (u ,v ) , u,v H1( ).
i i H1(Ωi) ∈ M
i=1
X
By the standard technique of the theory of quadratic forms, the form D
determines a self-adjoint operator in L2( ), namely, the Laplace operator
M
with Neumann boundary condition, ∆. The domain (∆) is defined by
D
(∆) = u H1( ) : D[u,v] = (f,v) for some f L2( ) , (4)
L2(M)
D { ∈ M ∈ M }
where v H1( ) is arbitrary. Analysing condition (4), we see that u
∈ M ∈
(∆) if u H2(Ω ), i = 1,...,I and, on any interface γ Ω Ω ,
i − +
D ∈ ⊂
u = u , [√g ∂ u ] = [√g ∂ u ] . T (5)
− γ + γ − σ − γ + σ + γ
| | | |
where g (s) = det[(g ) (s,0)].
± ± αβ
As, due to the finiteness of , the embedding of H1( ) into L2( ) is
M M M
compact, the spectrum of ∆ is pure discrete,
0 = λ < λ ..., λ ,
1 2 k
≤ → ∞
with the corresponding basis of orthonormal eigenfunctions to be denoted
by ϕ ∞ . Standard considerations, see e.g. [9] or [43] show that ϕ ∞
{ k}k=1 { k}k=1
distinguish points in int, i.e. for p = q int, there is k with ϕ (p) =
k
M 6 ∈ M 6
ϕ (q). Moreover,
k
3 GAUSSIAN BEAMS NEAR INTERFACES 9
Proposition 2.3 Let ϕ ∞ be an orthonormal basis of eigenfunctions
{ k}k=1
of the Laplace operator ∆. Then ϕ ∞ form local coordinates near any
{ k}k=1
p int, i.e. there are k (p),...,k (p) such that (ϕ ,...,ϕ ) form local
∈ M 1 n k1 kn
coordinates near p.
3 Gaussian Beams near interfaces
3.1 Gaussian beams on smooth manifolds
In this section we briefly recall some results on the non-stationary Gaussian
beams on smooth manifolds. Their theory goes back to the pioneering works
[5], [19], [40]. In our exposition we follow mainly section 2.4 of [21]. Non-
stationary Gaussian beams are some (formal) solutions of the wave equation
U ∆U = 0, (6)
tt
−
which are concentrated, at each moment of time t, near a point x(t). The
point x(t) moves with a unit speed along a geodesic on a smooth Rieman-
nian manifold ( , h) with ∆ being the Laplacian corresponding to ( , h).
N N
Introducing a moving frame
y(t) = x x(t),
−
a formal Gaussian beam has a form as a formal series
U (t,y) M exp (iε)−1Θ(t,y) u (t,y)(iε)l. (7)
ε ε l
≍ {− }
l≥0
X
Here M = (πε)−n, 0 < ε 1; Θ and u , l = 0,1,..., are formal series in
ε 4 l
≪
powers of y. They are usually represented as sums of homogeneous polyno-
mials in y with coefficients depending on t,
Θ θ (t,y), u = u (t,y),
m l lm
≍
m≥1 m≥1
X X
θ and u being homogeneous polynomials on y. The polynomials θ and
m lm m
u are chosen so that, considered as formal series with respect to y and (iε),
lm
∂2U ∆U = 0. (8)
t ε − ε
Note that ” ” exactly means that the formal series (7) satisfies formally
≍
equation (8).
Themostimportantpropertiesofthenon-stationaryGaussianbeamsare:
3 GAUSSIAN BEAMS NEAR INTERFACES 10
(a) θ (t,y) = (ξ(t),y(t)) = ξ (t)yj(t), where ξ (t) is the unit covector corre-
1 j j
sponding to the geodesic x(t);
(b) θ (t,y) = H(t)y,y , where H(t) is a symmetric matrix, satisfying
2
h i
Im H(t)y,y C(T) y 2, for T < t < T.
h i ≥ | | −
Remark 3.1 From now on throughout this paper we use the following nota-
tions C (or, C (t)) is a generic constant, C > 0, independent of ε; µ(L) is
L
defined for sufficiently large positive integers L such that µ(L) when
→ ∞
L .
→ ∞
Conditions (a) and (b) imply that U decays exponentially outside an ε1/2 -
neighborhood of x(t). It is important to note that, starting from a formal
GaussianbeamU wecanconstruct afamilyofsolutionstothewaveequation
ε
(6), which ”looks like” U . To this end, we start with a finite series
ε
L
UL = M exp (iε)−1ΘL(t,y) uL(t,y)χ(d2(x,x(t))ε−5/6), (9)
ε ε {− } l
l=0
X
L L
ΘL = Σθ ; uL = Σu ,
m l lm
l=0 l=0
where χ(s) is a smooth cut-off function equal to 1 near s = 0. Then
∂2UL ∆UL C (T)ε−µ(L).
k t ε − ε kCµ(L)(N×[−T,T]) ≤ L
By standard hyperbolic estimates there exists a solution L to (6) such that
Uε
(UL L) C (T)ε−µ(L). (10)
k ε −Uε kCN(N×[−T,T]) ≤ L
Moreover, if we generate a wave inside by a boundary source
N
U = f (t,s). (11)
ε ∂N×[−T,T] ε
|
Let f (t,s), s ∂ , t [ T,T] be given by a formal expansion
ε
∈ N ∈ −
f (t,s) exp (iε)−1Θ(t,s) u (t,s)(iε)l,
ε l
≍ {− }
l≥0
X
b b
where
Θ(t,s) t+ξ sα+ < H((s,t),(s,t)) > + θ (t,s); gαβ(0)ξ ξ < 1,
α m α β
≍ −
m≥2
X
b b b