Inverse Hyperbolic Problems with Time-Dependent Coefficients

Abstract

We consider the inverse problem for the second order self-adjoint hyperbolic equation in a bounded domain in R n with lower order terms depending analytically on the time variable. We prove that, assuming the BLR condition, the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism and a gauge transformation. As a by-product we prove a similar result for the nonself-adjoint hyperbolic operator with time-independent coefficients.

Full text

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arXiv:math/0508161v2 [math.AP] 2 Mar 2006
Inverse hyperbolic problems with
time-dependent coefficients.
G.Eskin,Department of Mathematics, UCLA,
Los Angeles, CA 90095-1555, USA. E-mail: eskin@math.ucla.edu
February 8, 2008
Abstract
We consider the inverse problem for the second order self-adjoint
hyperbolic equation in a bounded domain in Rnwith lower order terms
depending analytically on the time variable. We prove that, assuming
the BLR condition, the time-dependent Dirichlet-to-Neumann oper-
ator prescribed on a part of the boundary uniquely determines the
coefficients of the hyperbolic equation up to a diffeomorphism and a
gauge transformation. As a by-product we prove a similar result for
the nonself-adjoint hyperbolic operator with time-independent coeffi-
cients.
1Introduction.
Let Ω ⊂ Rnbe a bounded domain with a smooth boundary ∂Ω and let Γ0
be an open subset of ∂Ω. Consider a hyperbolic equation in Ω × (0,T0) of
the form:
Lu
def
=
?
−i∂
∂t+ A0(x,t)
?2
u(x,t)(1.1)
−
n
?
j,k=1
1
?g(x)
?
−i∂
∂xj
+ Aj(x,t)
??
g(x)gjk(x)
?
−i
∂
∂xk
+ Ak(x,t)
?
u
−V (x,t)u = 0,
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where ?gjk(x)?−1is the metric tensor in Ω, g(x) = det?gjk?−1, Aj(x,t), 0 ≤
j ≤ n,and V (x,t) are smooth in x ∈ Ω and real analytic in t, t ∈ [0,T0]. We
assume that
(1.2)
u(x,0) = ut(x,0) = 0 inΩ,
and
(1.3)
u??∂Ω×(0,T0)= f(x′,t),
where
(1.4)supp f(x′,t) ⊂ Γ0× (0,T0].
Let Λf be the D-to-N (Dirichlet-to-Neumann) operator on Γ0× (0,T0) :
(1.5)
Λf =
n
?
j,k=1
gjk(x)
?∂u
∂xj
+ iAj(x,t)u
?
νk
?
n
?
p,r=1
gpr(x)νpνr
?−1
2??Γ0×(0,T0),
where ν = (ν1,...,νn) is the unit exterior normal to ∂Ω with respect to the
Euclidian metric. We shall study the inverse problem of the determination
of the coefficients of (1.1) knowing the D-to-N operator on Γ0× (0,T0) for
all smooth f with supports in Γ0× (0,T0].
Let
(1.6)
y = y(x)
be a diffeomorphism of Ω onto some domain Ω0such that ∂Ω0⊃ Γ0and
(1.7)
y(x) = x
onΓ0.
The equation (1.1) will have the following form in y-coordinates:
L0v
def
=
?
??
−i∂
∂t+ A(0)
0(y,t)
?2
−i∂
v(y,t)(1.8)
−
n
?
j,k=1
1
?g0(y)
?
−i∂
∂yj
+ A(0)
j(y,t)
g0(y)gjk
0(y)
?
∂yk
+ A(0)
k(y,t)
?
v(y,t)
−V(0)(y)v(y,t) = 0,
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where v(y(x),t) = u(x,t),
(1.9)?gjk
0(y(x))? =
?Dy
Dx
?
?gjk(x)?
?Dy
Dx
?T
,
(1.10)
g0(y) = det?gjk
0(y)?−1,
Dy
Dxis the Jacobi matrix of (1.6), and
(1.11)
A(0)
0(y(x),t) = A0(x,t),V(0)(y(x),t) = V (x,t),
(1.12)
Ak(x,t) =
n
?
j=1
A(0)
j(y(x),t)∂yj(x)
∂xk
,
1 ≤ k ≤ n.
Denote by G0(Ω0× [0,T0]) the group of C∞(Ω0× [0,T0]) complex valued
functions c(y,t) such that c(y,t) ?= 0 and
(1.13)
c(y,t) = 1on Γ0× [0,T0].
We shall call G0(Ω0× [0,T0]) the gauge group. We say that the poten-
tials A(0)
gauge equivalent if there exists c(y,t) ∈ G0(Ω0× [0,T0]) such that
0(y,t) − ic−1(y,t)∂c
0(y,t),...,A(0)
n (y,t),V(0)(y,t) and A(1)
0(y,t),...,A(1)
n (y,t),V(1)(y,t) are
(1.14)
A(1)
0(y,t) = A(0)
∂t,
V(1)(y,t) = V(0)(y,t),
(1.15)
A(1)
j(y,t) = A(0)
j(y,t) − ic−1(y,t)∂c
∂yj,
1 ≤ j ≤ n.
Note that if
(1.16)
v(1)(y,t) = c−1(y,t)v(y,t),
then v(1)satisfies equation of the form (1.8) with potentials {A(0)
placed by the gauge equivalent {A(1)
transformation (1.16) the gauge transformation.
j,V(0)} re-
j,V(1)}, 0 ≤ j ≤ n. We shall call the
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Denote
(1.17)
T∗= sup
x∈Ω
d(x,Γ0),
where d(x,Γ0) is the distance in Ω with respect to metric tensor ?gjk?−1from
x ∈ Ω to Γ0.
We shall assume that the BLR condition (see [BLR]) is satisfied for t =
T∗∗. This means roughly speaking that any bicharacteristics of L in T∗
[0,T∗∗]) intersects ((Γ0× [0,T∗∗]) × Rn+1\ {0}.
Note that since BLR condition is determined by the geometry of Ω and
Γ0 and the second order terms of L it holds when [0,T∗∗] is replaced by
[t0,T∗∗+ t0], ∀t0> 0.
We shall prove the following theorem:
0(Ω ×
Theorem 1.1. Let L and L0be two operators of the form (1.1) and (1.8) in
Ω and Ω0respectively, with coefficients analytic in t ∈ [0,T0] and let L and L0
be formally self-adjoint, i.e coefficients Aj,V and A(0)
0 ≤ j ≤ n. Suppose that the BLR condition for L is satisfied when t = T∗∗,
and the D-to-N operators Λ and Λ(0)coresponding to L and L(0)respectively
are equal on Γ0× (0,T0) for all f satisfying (1.4) where Γ0 ⊂ ∂Ω ∩ ∂Ω0.
Suppose T0> 2T∗+ T∗∗. Then there exists a diffeomorphism y = y(x) of Ω
onto Ω0, y(x) = x on Γ0, such that (1.9) holds. Moreover, there exists a
gauge transformation c0(x,t) ∈ G0(Ω × [0,T0]) such that
(1.18)
c0◦ y−1◦ L0= L
in Ω × (0,T0).
Remark 1.1 Let L∗be formally adjoint operator to L. To prove Theorem
1.1 we need to know in the addition to D-to-N operator Λ the D-to-N operator
Λ∗corresponding to L∗. In the case when L∗= L we have, obviously, that
Λ∗= Λ. When A0= 0 and the coefficients of L are independent of t one
can show ( see, for example, [KL1], [E1] ) that we can recover Λ∗from Λ.
Therefore the proof and the result of Theorem 1.1 hold for the case when
A0= 0 and Aj(x), 1 ≤ j ≤ n, V (x) are complex-valued and independent of
t, and this gives a new proof of the corresponding result in [KL1].
The first inverse problem with boundary data on a part of the boundary
was considered in [I]. The most general results were obtained by the BC-
methods in the case of self-adjoint hyperbolic operators with time-independent
j,V(0)are real-valued,
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coefficients ( see [B1], [B2], [KKL], [KK] ). The nonself-adjoint case with
time-independent coefficient was considered in [B3], [KL1], [KL2], [KL3].
The inverse problems for the wave equations with time-dependent potentials
were considered in [St], [RS] (see also [I]).
The present paper is a generalization of the paper [E1]. We shall widely
use the notations, results and the proofs from [E1]. Note that the case of
equations with Yang-Mills potentials was considered in [E2].
In §2 we recover the coefficients of L (modulo a diffeomorphism and a
gauge transformation) locally near Γ0(Theorem 2.1). Following [E1] in §3
we prove the global result. In §4 we prove some lemmas used in §2.
2The local result.
Let Γ be an open subset of Γ0and U0⊂ Rnbe a neighborhood of Γ. Let
(x′,xn) be coordinates in U0such that the equation of Γ is xn= 0 and x′=
(x1,...,xn−1) are coordinates on ∂Ω∩U0. Introduce semi-geodesic coordinates
y = (y1,...,yn) for L in U0(c.f. [E1]):
(2.1)
y = ϕ(x),
where ϕ(x′,0) = x′,
(2.2)ˆ gnn(y) = 1, ˆ gnj= ˆ gjn= 0, 1 ≤ j ≤ n − 1,
?ˆ gjk(y)?−1is the metric tensor in the semi-geodesic coordinates. The equa-
tion (1.1) has the following form in the semi-geodeic coordinates
ˆLˆ u
def
=
?
−i∂
∂t+ˆA0(y,t)
?2
?
(2.3)
−
n
?
j,k=1
1
?ˆ g(y)
?
−i∂
∂yj
+ˆAj(y,t)
??
−ˆV (y,t)ˆ u(y,t) = 0,
ˆ g(y)ˆ gjk(y)−i∂
∂yk
+ˆAk(y,t)
?
ˆ u(y,t)
where, as in §1, ˆ u(ϕ(x),t) = u(x,t), ˆ g(y) = det?ˆ gjk(y)?−1,
(2.4)
ˆA0(ϕ(x),t) = A0(x,t),
ˆV (ϕ(x),t) = V (x,t),
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