Identifiability at the boundary for first-order terms
Russell M. Brown
∗†
Department of Mathematics
University of Kentucky
Lexington, Kentucky 40506-0027
USA
Mikko Salo
‡ †
Department of Mathematics and Statistics/RNI
University of Helsinki
P.O. Box 68
00014 University of Helsinki
Finland
Abstract.
Let Ω be a domain in R
n
whose boundary is C
1
if n ≥ 3 or C
1,β
if
n = 2. We consider a magnetic Schr¨odinger operator L
W,q
in Ω and show
how to recover the boundary values of the tangential component of the
vector potential W from the Dirichlet to Neumann map for L
W,q
. We also
consider a steady state heat equation with convection term ∆+2W ·∇ and
recover the boundary values of the convection term W from the Dirichlet
to Neumann map. Our method is constructive and gives a stability result
at the boundary.
1 Introduction
In this note, we consider inverse boundary value problems for the magnetic Schr¨odinger
operator and the Laplacian with a convection term. We recover the boundary values
∗
Research supported, in part, by the U.S. National Science Foundation.
†
Part of this research was carried out while the authors were visiting the University of Washington.
‡
Research supported, in part, by the Academy of Finland and by the Finnish Academy of Science
and Letters, Vilho, Yrj¨o and Kalle V¨ais¨al¨a Foundation.
1
of the coefficients from the Dirichlet to Neumann map. Our method is constructive
and gives a continuous dependence result. Our method is local in the sense that we
only need to know the action of the Dirichlet to Neumann map on functions sup-
ported in an arbitrarily small neighborhood of x
0
in order to recover the value of a
coefficient at x
0
. In three dimensions, we work in domains which are C
1
and require
no smoothness on the coefficients beyond continuity. In two dimensions, we need to
assume that the boundary is C
1,β
for some β > 0 to push our method through.
The identifiability at the boundary is often a first step in recovering the coe fficients
in the interior. We expect that we will be able to use the results of this paper to
establish interior identifiability with less regularity for the coefficient of the operator
and the boundary of the domain.
Before giving more of the background, we give a careful description of the operators
we will study. Throughout this note, Ω will denote a C
1
-domain in R
n
, n ≥ 3 or if
n = 2, a domain which is C
1,β
for some β > 0. Recall that a C
1
-domain (or C
1,β
-
domain) Ω is a bounded connected open set and has a C
1
(or C
1,β
) defining function
ρ : R
n
→ R with Ω = {x : ρ(x) > 0} , ∂Ω = {x : ρ(x) = 0}. In addition, ∇ρ does not
vanish on ∂Ω. Thus, if we fix x
0
∈ ∂Ω, we may normalize ρ so that ∇ρ(x
0
) = −ν(x
0
)
where ν(x) is the normal to ∂Ω which points out of Ω.
We will use L
W,q
to denote a magnetic Schr¨odinger operator with vector potential
W and electric potential q. Thus,
L
W,q
=
n
X
j=1
1
i
∂
∂x
j
+ W
j
!
2
+ q.
The vector potential W :
¯
Ω → C
n
is a continuous function on
¯
Ω and the electric
potential q is in L
∞
(Ω). As we shall see, the electric potential does not enter into
our arguments in any essential way. Our argument allows ve ctor potentials that are
complex, though the main interest is in real-valued potentials.
We will also consider the Laplacian with a convection term
L
W
= ∆ + 2W · ∇
where ∆ is the Laplacian and the convection coefficient W :
¯
Ω → C
n
is a continuous
function.
Next, we define the Dirichlet to Neumann operator associated to the operators
L
W
and L
W,q
. We begin with the magnetic Schr¨odinger operator, L
W,q
. For f in
H
1/2
(∂Ω), we let u denote the solution of the Dirichlet problem,
(
L
W,q
u = 0, in Ω
u = f, on ∂Ω.
(1.1)
Our method requires that we b e able to solve the Dirichlet problem and thus we
assume that zero is not an eigenvalue for the operator L
W,q
with Dirichlet boundary
2
conditions. We define the Dirichlet to Neumann operator acting on f by
Λ
W,q
f =
∂u
∂ν
+ iW · νu.
In general, Λ
W,q
f is not a function. We give a more careful (and useful) definition
of Λ
W,q
f as an element of H
−1/2
(∂Ω). If g is in H
1
(Ω) (and hence the trace of g on
∂Ω lies H
1/2
(∂Ω)–we will use g to denote both the f unction in Ω and its trace on the
boundary), we have
hΛ
W,q
f, gi =
Z
Ω
∇u(x) · ∇¯g(x) + W (x) · (iu(x)∇¯g(x) − i¯g(x)∇u(x))
+(W (x) · W (x) + q(x))u(x)¯g(x) dx. (1.2)
Since u is a solution of the Dirichlet problem, (1.1), the right-hand side of this ex-
pression depends only on the trace of g on ∂Ω and not on the extension of g into
Ω.
In a similar manner, we define the Dirichlet to Neumann operator for the convec-
tion operator L
W
. For f in the Sobolev space H
1/2
(∂Ω), we let u be the solution of
the Dirichlet problem for L
W
,
(
L
W
u = 0 in Ω
u = f on ∂Ω.
(1.3)
We define Λ
W
f =
∂u
∂ν
where the precise definition of the normal derivative as an
element of H
−1/2
(∂Ω) is given by
hΛ
W
f, gi =
Z
Ω
∇u(x) · ∇¯g(x) − 2¯g(x)W (x) · ∇u(x) dx. (1.4)
Again, because u is a solution of (1.3), the right-hand side of (1.4) depends only on
the boundary values of g and not the particular extension of g into the interior.
In the theorem below and throughout this paper, we will let W
tan
= W − νW · ν
denote the tangential component of the boundary values of W . With these definitions,
we now may state our main results. Our first theorem allows us to recover the
tangential component of the boundary values of W .
Theorem 1.1 Let Ω be a C
1
-domain if n ≥ 3 or C
1,β
-domain for some β > 0 if
n = 2. Let W :
¯
Ω → C
n
be a continuous vector potential and q ∈ L
∞
(Ω), a scalar
potential. Assume that zero is not an eigenvalue for L
W,q
. For each x
0
∈ ∂Ω and
α, a unit tangent vector to ∂Ω at x
0
, we may find a family of functions f
M
with f
M
supported in a ball of radius 1/M about x
0
,
lim
M→∞
kf
M
k
L
2
(∂Ω)
= 1 (1.5)
and so that
lim
M→∞
h(Λ
W,q
− Λ
0,0
)f
M
, f
M
i = α · W (x
0
).
Thus, we may recover the tangential component of W at the boundary from the Dirich-
let to Neumann map.
3
From this theorem, we easily obtain a continuous dependence result. We also will
use kT k to denote the operator norm of T on L
2
(∂Ω).
Corollary 1.2 Let Ω , W
j
and q
j
be as in Theorem 1.1. Then we have
k(W
1
− W
2
)
tan
k
L
∞
(∂Ω)
≤
√
2kΛ
W
1
,q
1
− Λ
W
2
,q
2
k.
Remark. One deficiency of our results, is that we do not have a sharp criterion which
tells us when the operator norm on the right is finite. If Ω and W are smooth, this
norm will be finite.
We have a similar pair of results for the convection operator. As we shall see, the
functions f
M
used in finding the boundary values of the vector potential may also be
used to reconstruct the boundary values of the drift term in the convection operator.
Theorem 1.3 Let Ω be a C
1
-domain if n ≥ 3 or C
1,β
-domain for some β > 0 if
n = 2 and W :
¯
Ω → C
n
a continuous vector-valued function. Assume that 0 is not
an eigenvalue for L
W
in Ω. Then for each x
0
∈ ∂Ω and α which is a unit tangent
vector to ∂Ω at x
0
, we may find a family of functions f
M
with f
M
supported in a ball
about x
0
of radius 1/M,
lim
M→∞
kf
M
k
L
2
(∂Ω)
= 1
and
lim
M→∞
h(Λ
W
− Λ
0
)f
M
, f
M
i = −(iα + ν(x
0
)) · W (x
0
).
As a consequence, we may recover the boundary values of W .
Remark. If W is real-valued, then the maximum principle implies that 0 is not an
eigenvalue for L
W
, thus our hypothesis is satisfied in the case that is most interesting.
Corollary 1.4 If Ω, W
1
and W
2
as in the theorem, then we have
kW
1
− W
2
k
L
∞
(∂Ω)
≤
√
3kΛ
W
1
− Λ
W
2
k.
Ours are not the first results on identifiability at the boundary for first-order
terms. Nakamura, Sun and Uhlmann [9] consider the magnetic Schr¨odinger operator
in smooth domains in R
n
, n ≥ 3, and with C
∞
-coefficients. Their methods should
also give boundary identifiability when n = 2. They go on to study the problem
of recovery in the interior. The problem of identifiability of a vector potential was
first studied by Sun [11]. Other authors who have considered this problem include
Eskin and Ralston [5] and Tolmasky [13]. In his Ph.D. thesis [10], Salo gives recovery
of a continuous vector p otential in domains that are C
1,1
. We relax the regularity
hypothesis for the boundary determination to just C
1
.
Interior identifiability of a convection term has been considered by several authors
including Cheng, Nakamura and Somersalo [3] and Salo [10] in dimensions three and
4
higher and Cheng and Yamamoto [4], Tong, Cheng and Yamamoto [14] and Tamasan
[12] in two dimensions. Tong, Cheng, and Yamamoto work with L
p
coefficients and do
not need identifiability at the boundary as a first step towards interior identifiability.
Cheng, Nakamura and Somersalo prove a result on identifiability at the boundary
in dimensions three and higher. Their methods likely extend to two dimensions.
Isakov [7] proves a theorem on identifiability at the boundary for nonlinear equations
that includes the operators we consider, however he requires additional smoothness
assumptions on the domain and the coefficients. Salo [10] proves the coeffic ient W
is uniquely determined by the Dirichlet to Neumann map in dimensions three and
higher. Salo’s argument at the boundary uses the singular solutions of Alessandrini
[1]. It would be interesting to adapt Alessandrini’s techniques to give a reconstruction
of W at the boundary, rather than just showing uniqueness.
Our proof for recovering W at the boundary will follow the argument used by Salo
for recovering the the vector potential for a magnetic Schr¨odinger operator [10]. Salo’s
method is based on work of Brown [2]. Our proof requires estimates for the L
2
(Ω)
norm of one of the functions appearing in the expressions (1.2) and (1.4) defining the
Dirichlet to Neumann maps. By contrast, the recovery of a conductivity in the work
of Brown [2] only requires the standard energy estimate for the gradient of a solution.
The new estimate is obtained using methods developed by Je rison and Kenig [8] in
their study of harmonic functions in Lipschitz domains.
In our proofs, we will use c and C to denote constants that depend only on the
domain Ω.
2 Construction of approximate solutions
In this section, we write down the boundary functions f
M
that we will use to recover
W . We will prese nt these functions as the boundary values of an approximate solution
to the Laplacian. The remainder of this section is devoted to proving estimates for
solutions of boundary value problems with this data.
We will fix a point x ∈ ∂Ω and a unit vector α which is tangent to ∂Ω. By a
change of coordinates we may assume that x is the origin and that the normal to ∂Ω
at the origin is −e
n
. We recall that ρ is a defining function for Ω and we normalize
ρ so that |∇ρ(0)| = 1.
Next, we let ω be a modulus of continuity for ∇ρ. Thus ω : [0, ∞) → [0, ∞) is
a strictly increasing continuous function with ω (0) = 0. We let η ∈ C
∞
0
(R
n
) be a
function which is supported in the ball of radius 1/2 centered at 0 and which satisfies
Z
R
n−1
η(x
0
, 0)
2
dx
0
= 1. (2.1)
We put η
M
(x) = η(M(x
0
, ρ(x)) where we are using x
0
= (x
1
, . . . , x
n−1
) to denote the
first n − 1 coordinates. For M large, we will have η
M
supported in a ball of radius
5