High-frequency approximation of the interior dirichlet-to-neumann map and applications to the transmission eigenvalues

0.27 MB·

by  Georgi Vodev

HIGH-FREQUENCY APPROXIMATION OF THE INTERIOR
DIRICHLET-TO-NEUMANN MAP AND APPLICATIONS TO THE
TRANSMISSION EIGENVALUES
GEORGI VODEV
7
Abstract. We study the high-frequency behavior of the Dirichlet-to-Neumann map for an
1
arbitrarycompactRiemannianmanifoldwithanon-emptysmoothboundary. Weshowthatfar
0
from the real axis it can be approximated by a simpler operator. We use this fact to get new
2
results concerning the location of the transmission eigenvalues on the complex plane. In some
r
cases we obtain optimal transmission eigenvalue-free regions.
a
M

8
2 1. Introduction and statement of results

] Let (X, ) be a compact Riemannian manifold of dimension d = dimX 2 with a non-
P G ≥
empty smooth boundary ∂X and let ∆ denote the negative Laplace-Beltrami operator on
A X
(X, ). Denote also by ∆ the negative Laplace-Beltrami operator on (∂X, ), which is a
h. G ∂X G0
Riemannian manifold without boundary of dimension d 1, where is the Riemannian metric
0
t − G
a on ∂X induced by the metric . Given a function f Hm+1(∂X), let u solve the equation
m G ∈
∆ +λ2n(x) u =0 in X,
[ X
(1.1)
3 ( (cid:0)u= f (cid:1) on ∂X,
v
8 where λ C, 1 Imλ Reλ and n C∞(X) is a strictly positve function. Then the
∈ ≪ | | ≪ ∈
6 Dirichlet-to-Neumann (DN) map
6
4 (λ;n) : Hm+1(∂X) Hm(∂X)
N →
0
. is defined by
1
(λ;n)f := ∂ u
0 ν ∂X
N |
7 where ν is the unit inner normal to ∂X. Oneof our goals in the present paper is to approximate
1
the operator (λ;n) when n(x) 1 in X by a simpler one of the form p( ∆ ) with a suitable
: N ≡ − ∂X
v complex-valued function p(σ), σ 0. More precisely, the function p is defined as follows
i ≥
X
p(σ) = σ λ2, Rep < 0.
r −
a
Our first result is the following p
Theorem 1.1. Let 0 < ǫ < 1 be arbitrary. Then, for every 0 < δ 1 there are constants
≪
C ,C > 1 such that we have
δ ǫ,δ
(1.2) (λ;1) p( ∆ ) δ λ
kN − − ∂X kL2(∂X)→L2(∂X) ≤ | |
for C Imλ (Reλ)1−ǫ, Reλ C .
δ ǫ,δ
≤ | |≤ ≥
Note that this result has been previously proved in [11] in the case when X is a ball in Rd
and the metric being the Euclidean one. In fact, in this case we have a better approximation
of the operator (λ;1). In the general case when the function n is arbitrary the DN map
N
can be approximated by h ΨDOs, where 0 < h 1 is a semi-classical parameter such that
− ≪
1
2 G.VODEV
Re(hλ)2 = 1. To describe this more precisely let us introduce the class of symbols Sk(∂X),
δ
0 δ < 1/2, as being the set of all functions a(x′,ξ′) C∞(T∗∂X) satisfying the bounds
≤ ∈
∂α∂βa(x′,ξ′) C h−δ(|α|+|β|) ξ′ k−|β|
x′ ξ′ ≤ α,β h i
(cid:12) (cid:12)
for all multi-indices α and (cid:12)(cid:12)β with consta(cid:12)(cid:12)nts Cα,β independent of h. We let OPSδk(∂X) denote
the set of all h ΨDOs, Op (a), with symbol a Sk(∂X), defined as follows
− h ∈ δ
(Oph(a)f)(x′)= (2πh)−d+1 e−hihx′−y′,ξ′ia(x′,ξ′)f(y′)dy′dξ′.
ZT∗∂X
It is well-known that for this class of symbols we have a very nice pseudo-differential calculus
(e.g. see [2]). It was proved in [15] that for Imλ λ 1/2+ǫ, 0 < ǫ 1, the operator h (λ;n)
| | ≥ | | ≪ N
is an h ΨDO of class OPS1 (∂X) with a principal symbol
− 1/2−ǫ
ρ(x′,ξ′) = r (x′,ξ′) (hλ)2n (x′), Reρ <0, n := n ,
0 0 0 ∂X
− |
r0 0 beingthe principalsypmbol of ∆∂X. Note that it is still possibleto construct a semiclas-
≥ −
sical parametrix for the operator h (λ;n) when Imλ λǫ, 0 < ǫ 1, if one supposes that
N | | ≥ | | ≪
the boundary ∂X is strictly concave (see [16]). This construction, however, is much more com-
plex and one has to work with symbols belonging to much worse classes near the glancing region
Σ = (x′,ξ′) T∗∂X : r (x′,ξ′) = 1 , where r = n−1r . On the other hand, it seems that no
{ ∈ ♯ } ♯ 0 0
parametrix construction near Σ is possible in the important region 1 Const Imλ λǫ.
≪ ≤ | | ≤ | |
Therefore, inthepresentpaperwefollow adifferentapproach whichconsistsof showingthat, for
arbitrary manifold X, the norm of the operator h (λ;n)Op (χ0) is (δ) for every 0 < δ 1
N h δ O ≪
independent of λ, provided Imλ and Reλ are taken big enough (see Proposition 3.3 below).
| |
Here the function χ0 C∞(T∗∂X) is supported in (x′,ξ′) T∗∂X : r (x′,ξ′) 1 2δ2 and
δ ∈ 0 { ∈ | ♯ − | ≤ }
χ0 = 1 in (x′,ξ′) T∗∂X : r (x′,ξ′) 1 δ2 (see Section 3 for the precise definition of χ0).
δ { ∈ | ♯ − | ≤ } δ
Theorem 1.1 is an easy consequence of the following semi-classical version.
Theorem 1.2. Let 0 < ǫ < 1 be arbitrary. Then, for every 0 < δ 1 there are constants
≪
C ,C > 1 such that we have
δ ǫ,δ
(1.3) h (λ;n) Op (ρ(1 χ0)+hb) Cδ
N − h − δ L2(∂X)→Hh1(∂X) ≤
for C Imλ (R(cid:13)eλ)1−ǫ, Reλ C , where C >(cid:13)0 is a constant independent of λ and δ,
δ ≤ | | ≤ (cid:13) ≥ ǫ,δ (cid:13)
and b S0(∂X) is independent of λ and the function n.
∈ 0
Here H1(∂X) denotes the Sobolev space equipped with the semi-classical norm (see Section
h
3 for the precise definition). Thus, to prove (1.3) (resp. (1.2)) it suffices to construct semi-
classical parametrix outside a δ2- neighbourhood of Σ, which turns out to be much easier and
can be done for an arbitrary X. In the elliptic region (x′,ξ′) T∗∂X : r (x′,ξ′) 1 +δ2
♯
{ ∈ ≥ }
we use the same parametrix construction as in [15] with slight modifications. In the hyperbolic
region (x′,ξ′) T∗∂X : r (x′,ξ′) 1 δ2 , however, we need to improve the parametrix
♯
{ ∈ ≤ − }
construction of [15]. We do this in Section 4 for 1 Const Imλ λ 1−ǫ. Then we show
≪ ≤ | | ≤ | |
that the difference between the operator h (λ;n) microlocalized in the hyperbolic region and
N
its parametrix is e−β|Imλ| + λ −M , where β > 0 is some constant and M 1 is
ǫ,M
O O | | ≥
arbitrary. So, we can do it small by taking Imλ and λ big enough.
(cid:0) (cid:1) (cid:0) | (cid:1) | | |
This kind of approximations of the DN map are important for the study of the location
of the complex eigenvalues associated to boundary-value problems with dissipative boundary
conditions (e.g. see [9]). In particular, Theorem 1.2 leads to significant improvements of the
eigenvalue-free regions in [9]. In the present paper we use Theorem 1.2 to study the location
of the interior transmission eigenvalues (see the next section). We improve most of the results
HIGH-FREQUENCY APPROXIMATION OF THE INTERIOR DIRICHLET-TO-NEUMANN MAP 3
in [15] as well as those in [11], [16], and provide a simpler proof. In some cases we get optimal
transmission eigenvalue-free regions (see Theorem 2.1). Note that for the applications in the
anisotropic case it suffices to have an weaker analogue of the estimate (1.3) with the space H1
h
replaced by L2, in which case the operator Op (hb) becomes negligible. In the isotropic case,
h
however, it is essential to have in (1.3) the space H1 and that the function b does not depend
h
on the refraction index n.
Note finally that Theorem 1.2 can be also used to study the location of the resonances for the
exterior transmission problems considered in [1] and [3]. For example, it allows to simplify the
proof of the resonance-free regions in [1] and to extend it to more general boundary conditions.
2. Applications to the transmission eigenvalues
Let Ω Rd, d 2, be a bounded, connected domain with a C∞ smooth boundary Γ = ∂Ω.
⊂ ≥
A complex number λ C, Reλ 0, will be said to be a transmission eigenvalue if the following
∈ ≥
problem has a non-trivial solution:
c (x) +λ2n (x) u = 0 in Ω,
1 1 1
∇ ∇
(2.1) c (x) +λ2n (x) u = 0 in Ω,
 (cid:0) 2 2 (cid:1) 2
∇ ∇

 u =u , c ∂ u = c ∂ u on Γ,
(cid:0)1 2 1 ν 1 2(cid:1) ν 2
where ν denotes the Euclidean unit inner normal to Γ, c ,n C∞(Ω), j = 1,2 are strictly
j j
∈
positive real-valued functions. We will consider two cases:
(2.2) c (x) c (x) 1 in Ω, n (x) = n (x) on Γ, (isotropic case)
1 2 1 2
≡ ≡ 6
(2.3) (c (x) c (x))(c (x)n (x) c (x)n (x)) = 0 on Γ. (anisotropic case)
1 2 1 1 2 2
− − 6
In Section 6 we will prove the following
Theorem 2.1. Assume either the condition (2.2) or the condition
(2.4) (c (x) c (x))(c (x)n (x) c (x)n (x)) < 0 on Γ.
1 2 1 1 2 2
− −
Then there exists a constant C > 0 such that there are no transmission eigenvalues in the region
(2.5) λ C: Reλ > 1, Imλ C .
{ ∈ | |≥ }
Remark. It is proven in [15] that under the condition (2.2) (as well as the condition (2.6)
below) there exists a constant C > 0 such that there are no transmission eigenvalues in the
region
e
λ C: 0 Reλ 1, Imλ C .
∈ ≤ ≤ | | ≥
n o
This is no longer true under the condition (2.4) in which case there exist infinitely many trans-
e
mission eigenvalues very close to the imaginary axis.
Note that the eigenvalue-free region (2.5) is optimal and cannot be improved in general.
Indeed, it follows from the analysis in [7] (see Section 4) that in the isotropic case when the
domain Ω is a ball and the refraction indices n and n constant, there may exist infinitely
1 2
many transmission eigenvalues whose imaginary parts are bounded from below by a positive
constant. Note also that the above result has been previously proved in [11] in the case when
the domain Ω is a ball and the coefficients constant. In the isotropic case the eigenvalue-free
region (2.5) has been also obtained in [14] when the dimension is one. In the general case of
arbitrary domains transmission eigenvalue-free regions have been previously proved in [5], [6]
4 G.VODEV
and [12] (isotropic case), [15] and [16] (both cases). For example, it has been proved in [15] that,
under the conditions (2.2) and (2.4), there are no transmission eigenvalues in
λ C : Reλ > 1, Imλ Cε(Reλ)21+ε , Cε > 0,
∈ | | ≥
n o
for every 0 < ε 1. This eigenvalue-free region has been improved in [16] under an additional
≪
strict concavity condition on the boundary Γ to the following one
λ C :Reλ > 1, Imλ C (Reλ)ε , C > 0,
ε ε
{ ∈ | | ≥ }
forevery0 < ε 1. Whenthefunctionintheleft-handsideof(2.3)isstrictlypositive,parabolic
≪
eigenvalue-free regions have been proved in [15] for arbitrary domains, which however are worse
than the eigenvalue-free regions we have under the conditions (2.2) and (2.4). In Section 7 we
will prove the following
Theorem 2.2. Assume the conditions
(2.6) (c (x) c (x))(c (x)n (x) c (x)n (x)) > 0 on Γ
1 2 1 1 2 2
− −
and
n (x) n (x)
1 2
(2.7) = on Γ.
c (x) 6 c (x)
1 2
Then there exists a constant C > 0 such that there are no transmission eigenvalues in the region
(2.8) λ C :Reλ > 1, Imλ Clog(Reλ+1) .
{ ∈ | | ≥ }
Note that in the case when (2.6) is fulfilled but (2.7) is not, the method developed in the
present paper does not work and it is not clear if improvements are possible compared with the
results in [15]. To our best knowledge, no results exist in the degenerate case when the function
in the left-hand side of (2.3) vanishes without being identically zero.
It has been proved in [10] that the counting function N(r) = # λ trans.eig. : λ r ,
{ − | | ≤ }
r > 1, satisfies the asymptotics
N(r)= (τ +τ )rd+ (rd−κ+ε), 0< ε 1,
1 2 ε
O ∀ ≪
where 0 < κ 1 is such that there are no transmission eigenvalues in the region
≤
λ C : Reλ >1, Imλ C(Reλ)1−κ , C > 0,
∈ | |≥
n o
and
ω n (x) d/2
d j
τ = dx,
j (2π)d c (x)
ZΩ(cid:18) j (cid:19)
ω being the volume of the unit ball in Rd. Using this we obtain from the above theorems the
d
following
Corollary 2.3. Under the conditions of Theorems 2.1 and 2.2, the counting function of the
transmission eigenvalues satisfies the asymptotics
(2.9) N(r) = (τ +τ )rd+ (rd−1+ε), 0< ε 1.
1 2 ε
O ∀ ≪
This result has been previously proved in [16] under an additional strict concavity condition
on the boundary Γ. In the present paper we remove this additional condition to conclude that
in fact the asymptotics (2.9) holds true for an arbitrary domain. We also expect that (2.9) holds
with ε = 0, but this remains an interesting open problem. In the isotropic case asymptotics for
the counting function N(r) with remainder o(rd) have been previously obtained in [4], [8], [13].
HIGH-FREQUENCY APPROXIMATION OF THE INTERIOR DIRICHLET-TO-NEUMANN MAP 5
3. A priori estimates in the glancing region
Let λ C, Reλ > 1, 1 < Imλ θ Reλ, where 0 < θ < 1 is a fixed constant, and set
0 0
∈ | | ≤
h = µ−1, where
2
Imλ
µ = Reλ 1 Reλ λ .
s − Reλ ∼ ∼ | |
(cid:18) (cid:19)
Clearly, we have Re(hλ)2 = 1 and
λ2 = µ2(1+izh), z = 2µ−1ImλReλ 2Imλ.
∼
Given an integer m 0, denote by Hm(X) the Sobolev space equipped with the semi-classical
≥ h
norm
kvkHhm(X) = h|α|k∂xαvkL2(X).
|α|≤m
X
We define similarly the Sobolev space Hm(∂X). It is well-known that
h
kvkHhm(∂X) ∼ kOph(hξ′im)vkL2(∂X) ∼ kvkL2(∂X) +kOph((1−η)|ξ′|m)vkL2(∂X)
for any function η C∞(T∗∂X) independent of h. Hereafter, ξ′ = (1+ ξ′ 2)1/2.
∈ 0 h i | |
Given functions V L2(X) and f L2(∂X), we let the function u solve the equation
∈ ∈
∆ +λ2n(x) u= λV in X,
X
(3.1)
( u(cid:0) = f (cid:1) on ∂X,
and set g = h∂ u . We will first prove the following
ν ∂X
|
Lemma 3.1. There is a constant C > 0 such that the following estimate holds
(3.2) u C Imλ −1 V +C Imλ −1/2 f 1/2 g 1/2 .
k kHh1(X) ≤ | | k kL2(X) | | k kL2(∂X)k kL2(∂X)
Proof. By Green’s formula we have
Im(λ2) n1/2u 2 = Im λV,u +Im ∂ u ,f
k kL2(X) h iL2(X) h ν |∂X iL2(∂X)
which implies
(3.3) Imλ u 2 . V u + f g .
| |k kL2(X) k kL2(X)k kL2(X) k kL2(∂X)k kL2(∂X)
On the other hand, we have
u 2 Re(λ2) n1/2u 2 = Re λV,u Re ∂ u ,f
k∇X kL2(X) − k kL2(X) − h iL2(X)− h ν |∂X iL2(∂X)
which yields
(3.4) h u 2 . u 2 + (h2) V 2 + (h) f g .
k ∇X kL2(X) k kL2(X) O k kL2(X) O k kL2(∂X)k kL2(∂X)
Since h. Imλ −1, the estimate (3.2) follows from (3.3) and (3.4). ✷
| |
We now equip X with the Riemannian metric n . We will write the operator n−1∆ in
X
G
the normal coordinates (x ,x′) with respect to the metric n near the boundary ∂X, where
1
G
0 < x 1 denotes the distance to the boundary and x′ are coordinates on ∂X. Set Γ(x ) =
1 1
≪
x X : dist(x,∂X) = x , Γ(0) = ∂X. Then Γ(x ) is a Riemannian manifold without
1 1
{ ∈ }
boundaryofdimensiond 1withaRiemannianmetricinducedbythemetricn ,whichdepends
− G
smoothly in x . It is well-known that the operator n−1∆ writes as follows
1 X
n−1∆ = ∂2 +Q(x )+R
X x1 1
6 G.VODEV
where Q(x ) = ∆ is the negative Laplace-Beltrami operator on Γ(x ) and R is a first-
1 Γ(x1) 1
order differential operator. Clearly, Q(x ) is a second-order differential operator with smooth
1
(n)
coefficients and Q(0) = ∆ is the negative Laplace-Beltrami operator on ∂X equipped with
∂X
the Riemannian metric induced by the metric n .
G
Let χ C∞(R), 0 χ(t) 1, χ(t) = 1 for t 1, χ(t) = 0 for t 2. Given a parameter
∈ 0 ≤ ≤ | | ≤ | | ≥
0 < δ 1 independent of λ and an integer k 0, set φ (x ) = χ(2−kx /δ ). Given integers
1 k 1 1 1
≪ ≥
0 s s we define the norm u by
≤ 1 ≤ 2 k ks1,s2,k
s1 s2−ℓ1 ∞
u 2 = u 2 + (h∂ )ℓ1(φ u)(x , ) 2 dx .
k ks1,s2,k k kHhs1(X) ℓX1=0 ℓX2=0 Z0 k x1 k 1 · kHhℓ2(∂X) 1
Clearly, we have
kukHhs1(X) ≤ kuks1,s2,k . kukHhs2(X).
Throughout this paper η C∞(T∗∂X), 0 η 1, η = 1 in ξ′ A, η = 0 in ξ′ A+1, will
∈ 0 ≤ ≤ | | ≤ | | ≥
be a function independent of λ, where A > 1 is a parameter we may take as large as we want.
We will now prove the following
Lemma 3.2. Let u solve the equation (3.1) with V Hs−1(X) and f H2s(∂X) for some
∈ ∈
integer s 1. Then the following estimate holds
≥
1/2 1/2
(3.5) u . u + V + Op (1 η)f g .
k k1,s+1,k k kHh1(X) k k0,s−1,k+s−1 k h − kHh2s(∂X)k kL2(∂X)
Proof. Note that
u . u + u
k k1,s+1,k k kHh1(X) k s,kkHh1(X)
where the function u = Op ((1 η)ξ′ s)(φ u) satisfies the equation
s,k h − | | k
h2∂2 +h2Q(x )+1+ihz u = U
x1 1 s,k s,k
with (cid:0) (cid:1)
U = h2Q(x ),Op ((1 η)ξ′ s) (φ u)+Op ((1 η)ξ′ s) h2∂2 ,φ φ u
s,k 1 h − | | k h − | | x1 k k+1
(cid:2) h2Op ((1 η)ξ′ s)φ Rφ(cid:3) u+h2λOp ((1 η)ξ′(cid:2)s)(φ V). (cid:3)
− h − | | k k+1 h − | | k
We also have
f := u = Op ((1 η)ξ′ s)f,
s s,k|x1=0 h − | |
g := h∂ u = Op ((1 η)ξ′ s)g ,
s x1 s,k|x1=0 h − | | ♭
where g := h∂ u . Integrating by parts the above equation and taking the real part, we
♭ x1 |x1=0
get
h∂ u 2 (h2Q(x )+1)u ,u
k x1 s,kkL2(X) − 1 s,k s,k L2(X)
Us,k,us,k L(cid:10)2(X) +h fs,gs L2(∂X)(cid:11)
≤ h i h i
. u ( V + u )
k(cid:12)(cid:12) s,kkHh1(X) k k(cid:12)(cid:12)0,s−1,(cid:12)(cid:12)k k k1,s,k+1(cid:12)(cid:12)
(3.6) + Op ((1 η)ξ′ s)∗Op ((1 η)ξ′ s)f g
h − | | h − | | L2(∂X)k ♭kL2(∂X)
The principal symbol(cid:13)r of the operator Q(x ) satisfies r(cid:13)(x,ξ′) C′ ξ′ 2, C′ > 0, on suppφ ,
(cid:13) − 1 (cid:13) ≥ | | k
provided δ is taken small enough. Therefore, we can arrange by taking the parameter A big
1
enough thatr 1 C ξ′ onsupp(1 η)φ , whereC > 0is someconstant. Hence, by Ga¨rding’s
k
− ≥ h i −
inequality we have
(3.7) (h2Q(x )+1)u ,u C Op ( ξ′ )u 2
− 1 s,k s,k L2(X) ≥ k h h i s,kkL2(X)
(cid:10) (cid:11)
HIGH-FREQUENCY APPROXIMATION OF THE INTERIOR DIRICHLET-TO-NEUMANN MAP 7
with possibly a new constant C > 0. Since the norms of g and g are equivalent, by (3.6) and
♭
(3.7) we get
u . V + u + u
k s,kkHh1(X) k k0,s−1,k k kHh1(X) k s−1,k+1kHh1(X)
1/2 1/2
(3.8) + Op (1 η)f g .
k h − kH2s(∂X)k kL2(∂X)
h
We may now apply the same argument to u . Thus, repeating this argument a finite
s−1,k+1
number of times we can eliminate the term involving u in the RHS of (3.8) and obtain
s−1,k+1
the estimate (3.5). ✷
Let the functions χ C∞(R), 0 χ (t) 1, j = 1,2,3, be such that χ +χ +χ 1,
j j 1 2 3
∈ ≤ ≤ ≡
χ = χ, χ (t) = 1 for t 2, χ (t) = 0 for t 1, χ (t) = 0 for t 1, χ (t) = 1 for t 2.
2 1 1 3 3
≤ − ≥ − ≤ ≥
Given a parameter 0 < δ 1 independent of λ, set
≪
χ−(x′,ξ′)= χ ((r (x′,ξ′) 1)/δ2),
δ 1 ♯ −
χ0(x′,ξ′) =χ ((r (x′,ξ′) 1)/δ2),
δ 2 ♯ −
χ+(x′,ξ′)= χ ((r (x′,ξ′) 1)/δ2),
δ 3 ♯ −
where r = n−1r is the principal symbol of the operator ∆(n). Since (r 1)kχ0 = (δ2k),
♯ 0 0 − ∂X ♯ − δ O
we have
(3.9) (h2∆(n) +1)kOp (χ0)= (δ2k) :L2(∂X) L2(∂X)
∂X h δ O →
for every integer k 0. Clearly, we also have
≥
Op (χ0) = (1) : L2(∂X) Hm(∂X), m 0,
h δ O → h ∀ ≥
uniformly in δ. Using (3.9) we will prove the following
Proposition 3.3. Let u solve (3.1) with f 0 and V Hs(X) for some integer s 0. Then
≡ ∈ ≥
the function g = h∂ u satisfies the estimate
ν ∂X
|
(3.10) g Hs(∂X) C′ Imλ −1/2 V 0,s,s
k k h ≤ | | k k
with a constant C′ > 0 independent of λ.
Let u solve (3.1) with f replaced by Op (χ0)f and V Hs+2(X) for some integer s 0.
h δ ∈ ≥
Then the function g = h∂ u satisfies the estimate
ν ∂X
|
(3.11) g Hs(∂X) C δ+ Imλ −1/4 f L2(∂X) +C δ1/2 + Imλ −1/8 V 0,s+2,s+2
k k h ≤ | | k k | | k k
(cid:16) (cid:17) (cid:16) (cid:17)
for 1< Imλ δ2Reλ, Reλ C 1, with a constant C > 0 independent of λ and δ.
δ
| | ≤ ≥ ≫
Proof. Set w = φ (x )u. We will first show that the estimates (3.10) and (3.11) with s 1
0 1
≥
follow from (3.10) and (3.11) with s = 0, respectively. This follows from the estimate
(3.12) kgkHhs(∂X) . kgkL2(∂X)+kh∂x1vs|x1=0kL2(∂X)
where the function v = Op ((1 η)ξ′ s)w satisfies the equation (3.1) with V replaced by
s h − | |
V = nOp ((1 η)ξ′ s)φ n−1V +λ−1n n−1∆ ,Op ((1 η)ξ′ s)φ u.
s h − | | 0 X h − | | 0
We can write the commutator as (cid:2) (cid:3)
∂2 +R,φ (x ) Op ((1 η)ξ′ s)φ (x )+φ Q(x )+R,Op ((1 η)ξ′ s) φ (x ).
x1 0 1 h − | | 1 1 0 1 h − | | 1 1
There(cid:2)fore, if f 0, in(cid:3)view of Lemmas 3.1 and 3.2,(cid:2)the function Vs satisfies the b(cid:3)ound
≡
(3.13) V . V + u . u + V . V .
k sk0,0,0 k k0,s,0 k k1,s+1,1 k kHh1(X) k k0,s,s k k0,s,s
8 G.VODEV
Clearly, the assertion concerning (3.10) follows from (3.12) and (3.13). The estimate (3.11) can
be treated similarly. Indeed, in view of Lemma 3.2, the function V satisfies the bound
s
V . V + u
s 0,2,2 0,s+2,0 1,s+3,1
k k k k k k
(3.14) . u + V + Op (1 η)Op (χ0)f 1/2 g 1/2 .
k kHh1(X) k k0,s+2,s+2 k h − h δ kHh2s+4(∂X)k kL2(∂X)
Taking the parameter A big enough we can arrange that suppχ0 supp(1 η) = . Hence
δ ∩ − ∅
(3.15) Op (1 η)Op (χ0) = (h∞): L2(∂X) Hm(∂X), m 0.
h − h δ O → h ∀ ≥
By (3.14) and (3.15) together with Lemma 3.1 we conclude
V . u + V + (h∞) f 1/2 g 1/2
k sk0,2,2 k kHh1(X) k k0,s+2,s+2 O k kL2(∂X)k kL2(∂X)
. V + Imλ −1/2 +h∞ f 1/2 g 1/2 .
k k0,s+2,s+2 O | | k kL2(∂X)k kL2(∂X)
We now apply (3.11) with s = 0 to the(cid:16)function v and(cid:17)note that
s
v = Op ((1 η)ξ′ s)Op (χ0)f = (h∞)f.
s|x1=0 h − | | h δ O
Hence
h∂ v (h∞) f + δ1/2 + Imλ −1/8 V
k x1 s|x1=0kL2(∂X) ≤ O k kL2(∂X) O | | k sk0,2,2
(cid:16) (cid:17)
(3.16) δ1/2 + Imλ −1/8 V + Imλ −1/2 +h∞ f 1/2 g 1/2 .
≤ O | | k k0,s+2,s+2 O | | k kL2(∂X)k kL2(∂X)
(cid:16) (cid:17) (cid:16) (cid:17)
Therefore, the assertion concerning (3.11) follows from (3.12) and (3.16).
We now turn to the proof of (3.10) and (3.11) with s = 0. In view of Lemma 3.1, the function
U := h(n−1∆ +λ2)w = h[n−1∆ ,φ (x )]u+hλn−1φ V
X X 0 1 0
satisfies the bound
U . u + V
k kL2(X) k kHh1(X) k kL2(X)
(3.17) . V + Imλ −1/2 f 1/2 g 1/2 .
k kL2(X) O | | k kL2(∂X)k kL2(∂X)
(cid:16) (cid:17)
Observe now that the derivative of the function
E(x ) = h∂ w 2+ h2Q(x )+1 w,w ,
1 k x1 k 1
and , being the norm and the scalar pr(cid:10)o(cid:0)duct in L2(∂X(cid:1) ), sa(cid:11)tisfies
k·k h· ·i
E′(x ) = 2Re h2∂2 +h2Q(x )+1 w,∂ w + h2Q′(x )w,w
1 x1 1 x1 1
= 2Re ((cid:10)U(cid:0) izw hRw),h∂ (cid:1)w + h2(cid:11)Q′(x(cid:10) )w,w . (cid:11)
h − − x1 i 1
If we put g := h∂ u , we have
♭ x1 |x1=0 (cid:10) (cid:11)
∞
g 2+ h2∆(n) +1 Op (χ0)f,Op (χ0)f = E(0) = E′(x )dx
k ♭k ∂X h δ h δ − 1 1
D(cid:16) (cid:17) E Z0
. U + z w + hRw h∂ w + w 2
k kL2(X) | |k kL2(X) k kL2(X) k x1 kL2(X) k kHh1(X)
(cid:0) (cid:1)
(3.18) (z ) h∂ w w + Imλ −1 F2
≤ O | | k x1 kL2(X)k kL2(X) O | |
where we have used Lemma 3.1 together with (3) and we h(cid:0)ave put (cid:1)
F = f 1/2 g 1/2 + V .
L2(X)
k k k k k k
HIGH-FREQUENCY APPROXIMATION OF THE INTERIOR DIRICHLET-TO-NEUMANN MAP 9
Clearly, (3.10) with s = 0 follows from (3) applied with f 0 and Lemma 3.1. To prove (3.11)
≡
with s = 0, observe that (3.9) and (3) lead to
(3.19) g (δ) f + Imλ −1/2 F + (Imλ 1/2) h∂ w 1/2 w 1/2 .
k k ≤ O k k O | | O | | k x1 kL2(X)k kL2(X)
(cid:16) (cid:17)
We need now to bound the norm h∂ w in the RHS of (3.19) better than what the
k x1 kL2(X)
estimate (3.2) gives. To this end, observe that integrating by parts yields
h∂ w 2 h2Q(x )+1 w,w
k x1 kL2(X)− 1 L2(X)
= hRe (U hR(cid:10)(cid:0)w),w (cid:1)hRe(cid:11)f,g
− h − iL2(X) − h ♭i
(3.20) (h) w 2 + (h) U 2 + (h) f g (h)F2.
≤ O k kHh1(X) O k kL2(X) O k kk k ≤ O
By (3.19) and (3) together with Lemma 3.1 we get
g (δ) f + (Imλ 1/2) w 1/4 w 3/4
k k ≤ O k k O | | k 1kL2(X)k kL2(X)
+ (h1/4 Imλ 1/2)F1/2 w 1/2 + Imλ −1/2 F
O | | k kL2(X) O | |
(cid:16) (cid:17)
(3.21) (δ) f + (Imλ 1/8) w 1/4 F3/4+ Imλ −1/2 +h1/4 Imλ 1/4 F
≤ O k k O | | k 1kL2(X) O | | | |
(cid:16) (cid:17)
where we have put w := h2Q(x )+1 w. We need now the following
1 1
Lemma 3.4. The functio(cid:0)n w satisfies(cid:1)the estimate
1
Imλ 1/2 w δ2+ Imλ −1+h∞ f 1/2 g 1/2
1 L2(X)
| | k k ≤ O | | k k k k
(cid:0) (cid:1)
(3.22) + h1/2 f + Imλ −1/2 V .
0,2,2
O k k O | | k k
(cid:16) (cid:17) (cid:16) (cid:17)
Let us see that this lemma implies the estimate (3.11) with s = 0. Set
F = f 1/2 g 1/2 + V F.
0,2,2
k k k k k k ≥
By (3) and (3.4),
e
g (δ) f + δ1/2 + Imλ −1/8+h∞ F
k k ≤ O k k O | |
(cid:16) (cid:17)
+ h1/8 ( f +F)+ Imλ −1/2+h1/4 Imλ 1e/4 F
O k k O | | | |
(cid:16) (cid:17) (cid:16) (cid:17)
(3.23) δ+h1/8 f + δ1/2 + Imλ −1/8 +h1/8+h1/4 Imλ 1/4 F.
≤ O k k O | | | |
(cid:16) (cid:17) (cid:16) (cid:17)
Since by assumption h1/4 Imλ 1/4 = δ1/2 , one can easily see that (3.11) with s = 0 follows
e
from (3). | | O ✷
(cid:0) (cid:1)
Proof of Lemma 3.4. Observe that the function w satisfies the equation
1
h2∂2 +h2Q(x )+1+ihz w = hU
x1 1 1 1
where (cid:0) (cid:1)
U := h2Q(x )+1 (U hRw)+2h3Q′(x )∂ w+h3Q′′(x )w.
1 1 − 1 x1 1
We also have
(cid:0) (cid:1)
f := w = (h2Q(0)+1)Op (χ0)f,
1 1|x1=0 h δ
g := h∂ w = (h2Q(0)+1)g +h2Q′(0)Op (χ0)f.
1 x1 1|x1=0 ♭ h δ
10 G.VODEV
Integrating by parts the above equation and taking the imaginary part, we get
z w 2 U ,w + f ,g
| |k 1kL2(X) ≤ h 1 1iL2(X) |h 1 1i|
U w +(cid:12) (1) (h2Q(0)(cid:12)+1)2Op (χ0)f g
≤ k 1kL2(X)k 1kL2(X) (cid:12)O (cid:12) h δ k k
+O(h) Oph(χ0δ)f Hh2(∂X)(cid:13)(cid:13)(h2Q(0)+1)Oph(χ0δ)f (cid:13)(cid:13)
U (cid:13) w (cid:13) + (cid:13)(δ4) f g + (h) f 2(cid:13)
≤ k 1k(cid:13)L2(X)k 1kL(cid:13)2(X) O(cid:13) k kk k O k k(cid:13)
where we have used (3.9). Hence
(3.24) z w 2 z −1 U 2 + (δ4) f g + (h) f 2.
| |k 1kL2(X) ≤ O | | k 1kL2(X) O k kk k O k k
Recall that the function U is of th(cid:0)e form(cid:1) (2h∂ +a(x))φ (x )u+hλn−1φ V, where a is some
x1 1 1 0
smooth function. Hence the function U satisfies the estimate
1
U . u + V
1 L2(X) 1,3,1 0,2,0
k k k k k k
(3.25) . u + V + (h∞) f 1/2 g 1/2
k kHh1(X) k k0,2,2 O k kL2(∂X)k kL2(∂X)
where we have used Lemma 3.2 together with (3.15). By (3.24) and (3),
z w 2 z −1 u 2 + z −1 V 2
| |k 1kL2(X) ≤ O | | k kHh1(X) O | | k k0,2,2
(cid:0) (cid:1) (cid:0) (cid:1)
(3.26) + (δ4 +h∞) f g + (h) f 2.
O k kk k O k k
Clearly, (3.4) follows from (3) and Lemma 3.1. ✷
4. Parametrix construction in the hyperbolic region
Let λ be as in Theorems 1.1 and 1.2, and let h, z, δ, r , n , r , χ and χ− beas in the previous
0 0 ♯ δ
sections. Set θ = Im(hλ)2 = hz = (hǫ), θ h, and
O | |≫
ρ(x′,ξ′) = r (x′,ξ′) (1+iθ)n (x′), Reρ < 0.
0 0
−
It is easy to see that ρχ− S0(∂Xp). In this section we will prove the following
δ ∈ 0
Proposition 4.1. There are constants C,C > 0 depending on δ but independent of λ such that
1
(4.1) h (λ;n)Op (χ−) Op (ρχ−) C h+e−C|Imλ| .
N h δ − h δ L2(∂X)→Hh1(∂X) ≤ 1
Proof. To(cid:13)prove (4.1) we will build a par(cid:13)ametrix near the bou(cid:16)ndary of the s(cid:17)olution to the
(cid:13) (cid:13)
equation (1.1)withf replacedbyOp (χ−)f. Letx = (x ,x′),x > 0,bethenormalcoordinates
h δ 1 1
with respect to the metric , which of course are different from those introduced in the previous
G
section. In these coordinates the operator ∆ writes as follows
X
∆ = ∂2 +Q+R
X x1
where Q 0 is a second-order differential operator with respect to the variables x′ and R is a
≤ e e
first-order differential operator with respect to the variables x, both with coefficients depending
smoothely on x. Let (x0,ξ0) suppχ− and let T∗∂X be a small open neighbourehood
∈ δ U ⊂
of (x0,ξ0) contained in r 1 δ2/2 . Take a function ψ C∞( ). We will construct a
{ ♯ ≤ − } ∈ 0 U
parametrix u− of the solution of (1.1) with u− = Op (ψ)f in the form u− = φ(x ) −f,
ψ ψ|x1=0 h ψ 1 K
where φ(x ) = χ(x /δ ), 0 < δ 1 being a parameter independent of λ to be fixed later on
1 1 1 1
≪
depending oen δ, and e e
( −f)(x) = (2πh)−d+1 ehi(hy′,ξ′i+ϕ(x,ξ′,θ))a(x,ξ′,λ)f(y′)dξ′dy′.
K
Z Z