Texas Tech University Mathematics Series 1992 ELASTIC WAVES IN ...

Texas Tech University Mathematics Series 1992 ELASTIC WAVES IN A THREE-DIMENSIONAL HALF SPACE: THE LAMB PROBLEM J.R. Schulenberger∗ AN AH Corporation D.S. Gilliam† Department of Mathematics Texas Tech University Lubbock, Texas 79409

Abstract The problem of the title is to find time-harmonic solutions of the elasticity equations in a three dimensional half space with boundary free of normal ∗

12.

Supported in part by the ANAH Corporation, Research Report No.

†

Supported in part by NSF Grant #DMS 8905334, Texas Advanced Research Program Grant #0219-44-5195 and a grant from AFOSR.

1

stresses occasioned by a time-harmonic source located in the half space. This is here accomplished by means of the limiting absorption principle. The asymptotic behavior for large spatial distances of the solutions so obtained is determined, and uniqueness classes containing these solutions are determined. It is further shown in what sense these solutions are approximations for large times to the actual timedependent solution.

1

Introduction

The Lamb problem is to find time-harmonic solutions of the elasticity equations in a medium filling R3+ = {x ∈ R3 : x3 > 0} with boundary {x3 = 0} free of normal stresses and a time-harmonic source located in R3+ (see, e.g., [1, 2]). In the present work we construct such solutions via the principle of limiting absorption, determine their asymptotic behavior as |x| → ∞, define uniqueness classes containing them, and show that these unique solutions are the limits as t → ∞ of the actual time-dependent solution (multiplied by the timeharmonic exponential with argument of sign opposite that of the source) – the principle of limiting amplitude. Our reason for investigating this particular problem is that it is the simplest problem with infinite boundary for the elasticity equations which admits surface waves. The system of elasticity equations admits two positive propagation speeds cs < cp of pressure (P) and shear (S) waves, and with the 2

present boundary condition a P wave, for example, incident on {x3 = 0} gives rise to a reflected P wave as well as an S wave and a surface (or Rayleigh (R)) wave created on {x3 = 0}. A similar statement holds for an incident S wave. This is a more general situation than cases for Maxwell’s equations admitting surface waves [7, 16] and provides additional information as to what can be expected for more general systems as well as providing a reference problem for more general domains. The principle of limiting absorption consists roughly in assuming that the frequency has a small imaginary part and then, with an eventual solution in hand, letting this imaginary part go to zero (i.e., in considering the limit on the spectrum of the resolvent applied to the spatial part of the source). There are three basic problems involved here. The first is to establish the principle of limiting absorption itself, i.e., to show in a rigorous manner that a steady-state solution can actually be constructed in this fashion. The second problem is to find a class of functions in which the solution so constructed is unique (a “radiation condition”). While in problems in exterior domains or with bounded perturbations of the coefficients uniqueness classes are essentially dictated by the asymptotic behavior of the free-space Green functions (see, e.g., [17]), in problems with infinite boundary admitting surface waves, such as the present problem, the asymptotic behavior as |x| → ∞ of these latter must also be 3

considered in defining uniqueness classes (see [16]). Finally, since, strictly speaking, steady-state solutions are physically meaningless (they fail to have finite energy), a third problem is to determine in what sense they are approximations for large times to the actual time-dependent solutions (the principle of limiting amplitude). We proceed to describe the steps taken to resolve these problems in the case of the Lamb problem and to outline the results obtained. In §2 the system of equations for nonstatic solutions, which is second order in both the time and spatial variables, is derived from the elasticity equations as a first-order, symmetric, hyperbolic system, and its relation to the system, first order in time and second order in the spatial variables, is established. In §3 the selfadjoint operator M in L2 (R3+ ; C3 ) engendered by the elasticity operator M (D) in 3 R+ with the condition that {x3 = 0} be free of normal stresses is constructed and given a form suitable for determining the generalized eigenfunctions. These are of three types: 1) ΣR (x, ξ), ξ ∈ R2 , the surface or Rayleigh mode; 2) Ψp (x, η), η ∈ R3 , consisting of an incident and reflected P (pressure) mode plus an SVP mode – an SV (vertical shear) mode created at the boundary by the incident P mode; 3) Ψsh (x, η) + Ψsv (x, η), where the first consists of an incident and reflected SH (horizontal shear) mode, while the second consists of an incident and reflected SV mode plus a PSV mode – a P mode created at the boundary by 4

the incident SV mode. We remark that the PSV mode decays exponentially away from the boundary for η outside the cones C± = {η = |η|s = |η|(s , s3 ), s ∈ S 2 , |s | = n,  s3 = ± (1 − n2 )} where 1 > n = cs /cp is the ratio of the shear and pressure propagation speeds of the medium filling R3+ . The solution of the time-dependent equations is then represented as a superpostion of these modes. Section 5 contains the formulation of the steady-state problem and a representation of the resolvent of M in terms of generalized eigenfunctions. (It may seem somewhat silly to construct the resolvent, then obtain generalized eigenfunctions, and then again represent the resolvent. The point is that the first construction contains the spectral parameter in many untoward places, while in the second it is contained only in denominators.) In §§6-8 the principle of limiting absorption is established, and the asymptotics of the various components of the steady-state solution as |x| → ∞ are determined. In §6 a very simple expression is obtained for the leading term of the asymptotics of the steady-state Rayleigh wave: in 0(|x|−1/2 ) it (multiplied by the time exponential) is an outgoing or incoming cylindrical wave, decaying exponentially in the x3 direction. In §7 an asymptotic expression is obtained for the component of the steady-state solution derived from superposition of the Ψsh (x, η) and Ψsv (x, η) modes: in 0(|x|−1 ) the first is simply an outgoing or incoming SH wave (the reflected wave is manifest only in the coefficient), while the 5

SV part consists of an incoming or outgoing P wave and SV wave, uniformly with respect to direction. In §8 an expression is derived for the component of the steady-state solution derived from superposition of the Ψp (x, η) modes: this consists of a P wave in 0(|x|−1 ) plus an SV wave present in 0(|x|−1 ) only in the cone C˜+ = {|x|ω : ω ∈ S 2 , |ω  | < n}; outside this cone it decays like 0(|x|−1−κ ), κ ∈ (1/2, 1). Thus, although the SV P modes evidence no exceptional behavior on the cones C± above, the SV P steady-state wave exists in 0(|x|−1 ) only in the cone C˜+ , while the P SV wave, for which modes from which it is formed decay exponentially in η-space outside the cones C± , reflects no evidence of this asymptotically. All this is simply a consequence of the location of the critical points of the phases of the P SV and SV P modes. Nevertheless, in view of the usual sort of plane-wavechasing analysis done in many applied works, it should be emphasized that the behavior of the Ψp and Ψs modes from which the steady-state solution is formed gives a totally inverted picture of the asymptotic behavior of the steady-state solution itself. It is an interesting fact that neither the SV P wave nor P SV wave can be observed on the boundary in 0(|x|−1 ): in the first case this happens because the wave vanishes in this order outside the cone C˜+ , while in the second case it happens because the reflection coefficient vanishes on the boundary. In §9 we formulate the asymptotics of the components of the steady-state solution obtained in §§6-8 as 6

Theorem 9.1, define eventual uniqueness classes containing them, and then in Theorem 9.2 prove their uniqueness in these classes. In §10 we establish that the unique steadystate solutions of §9 are indeed approximations to the actual solutions of §4. The appendix contains the proof of a technical result stated in §6 which is needed for the uniqueness theorem. As regards related literature, the Lamb problem in various formulations has been considered in many works (see, e.g., [1, 2] and the bibliographies there). For layered media with infinite interfaces of various shapes the limitingabsorption principle was established for the Helmholtz equation by Eidus [3], but the abstract approach adopted there, while relatively “clean” mathematically, apparently does not afford enough information about the solution so obtained to be able to define uniqueness classes. For Maxwell’s equations with dissipative boundary conditions essentially the same problem as considered here was treated in [16]. Finally, we mention that in the formulation of this problem or analogous problems with infinite boundary it is often stated that the solution should satisfy a “radiation condition” at infinity (see, e.g., [1, 2, 9]). However, to our knowledge the said condition has never been explicitly formulated. For the Helmholtz equations and a restricted class of domains with infinite boundaries or interfaces a Sommerfeld-type radiation condition for the Dirichlet problem is established in [22]. In 7

the present case and in general there is no such radiation condition: the asymptotic properties of the steady-state solutions constructed by limiting absorption are used to define uniqueness classes containing them (which is really what Sommerfeld-type conditions do implicitly in simple cases).

2

The Equations of Elasticity and the Associated Second-Order Systems

In this section we present the equations of elasticity as a first-order, symmetric, hyperbolic system and derive an associated second-order system for nonstatic solutions. This system gives rise to a system in energy space which is first order in time and second order in the spatial variables. The correspondence between these two systems has general character and is established for abstract systems. The correspondence of the resolvents and spectral families of the abstract systems is also established. The results will be applied to the concrete system considered in §3. We write the equations of elasticity with an initial-value or initial-boundary-value problem in mind; source terms will be dealt with in §10. Everywhere below t M denotes the transpose of a matrix M , t¯M denotes the conjugate transpose, and M ∗ denotes the adjoint of a matrix or operator. In terms of the stress tensor u1 = t (σ11 , σ22 , σ33 , σ12 , σ13 , σ23 ) 8

and the time derivative u2 = t (v1 , v2 , v3 ) ≡ t (∂t w1 , ∂t w2 , ∂t w3 ) of the elastic displacement w = t (w1 , w2 , w3 ) with u = t (u1 , u2 ) the equations of elasticity of a homogeneous, isotropic medium occupying a region of three-dimensional space can be written as the first-order hyperbolic system [15] (Dj = −i∂j , j = 1, 2, 3), −i∂t u = E −1 A(D)u,  A(D) =

t

 t

E −1 =  E0−1 =

,

A3×6 03×3

D1 0

  A(D) =  0  0 



A6×3

06×6

0

D2 D3 0

D2 0 0

D1 0

D3 0

E0−16×6

06×3

03×6

I3

D1



  D3   D2 (2.1)

 ,

e3×3 03×3

 ,

03×3 µI3 

c2p λ2 λ

  e =  λ  λ

c2p λ



  λ2  .  2 cp

Here λ, µ are the Lam´e constants in terms of which the 9

propagation speeds of S and P waves, cs and cp , are given by c2s = µ and c2p = λ+2µ respectively; −i∂t u1 = E0−1 A(D)u2 is 3  the time-derivative of Hooke’s law σij = λδij ekk + 2µeij , 1

2eij = ∂i wj + ∂j wi , i, j = 1, 2, 3, and in terms of the displacement w the second equation −i∂t u2 = t A(D)u1 can be written ∂t2 w(x, t) = −t A(D)E0−1 A(D)w(x, t) + it A(D)u10 (x), (2.2) where u10 is the static (time-independent) component of u(x, t). Since in this paper we shall be interested only in nonstatic solutions, we suppose that u10 (x) = 0, so that (2.2) can be written as the 3 × 1 system ∂t2 w = −t A(D)E0−1 A(D)w = Div σ(x, t) (2.3) = [c2p ∂ ⊗ ∂ − c2s rot rot]w(x, t) where the divergence of the stress tensor is [Div σ]j =

3  1

j = 1, 2, 3, and [∂ ⊗ ∂]ij = ∂i ∂j = [grad divij ], i, j = 1, 2, 3, 

0

−∂3 ∂2

  0 rot =  ∂3  −∂2 ∂1 10



  −∂1  .  0

∂k σjk ,

Given w, the stress can be recovered as σ = iE0−1 A(D)w. We shall thus be concerned with the elasticity equations in the form (2.3): ∂t2 w(x, t) + M (D)w(x, t) = 0, M (D) =

t

A(D)E0−1 A(D),

(2.4)

= −c2p ∂ ⊗ ∂ + c2s rot rot. For the symbol M (η) = t A(η)E0−1 A(η), 0 = η = |η|ω ∈ R3 , ω ∈ S 2 , we compute det[M (η) − zI] = (z − c2p |η|2 )(z − c2s |η|2 )2 , z ∈ C, M (η) = c2p |η|2 P (ω) + c2s |η|2 S(ω), I = P (ω) + S(ω), P (ω) = ω ⊗ ω = t ωω = P 2 (ω), S(ω) = −ω ∧ ω∧ = S 2 (ω), = v(ω) ⊗ v(ω) + h(ω) ⊗ h(ω), v(ω) = |ω  |−1 t (ω1 ω3 , ω2 ω3 , −|ω  |2 ), h(ω) = |ω  |−1 t (ω2 , −ω1 , 0), (2.5) 11



−ω3

0

  ω∧ =  ω3  −ω2

0 ω1

ω2



  −ω1  ,  0

(ω∧)2 = ω ∧ ω∧, |ω  |2 = ω12 + ω22 . Here v(ω) ⊗ v(ω) and h(ω) ⊗ h(ω) are, respectively, the SV and SH components of S(ω), i.e., 0 = ω · h(ω) = ω · v(ω) = h(ω) · v(ω) and ω ∧ h(ω) = v(ω), ω ∧ v(ω) = −h(ω). As is known [21], the Fourier transform −n/2 exp(−ixη)f (x)dx Φn f (η) ≡ fˆ(η) = (2π) Rn

is an automorphism of the Schwartz space S(Rn ) of smooth, rapidly decreasing functions which extends by continuity to an automorphism of L2 (Rn ) and by duality to an automorphism of S  (Rn ). The symbol Φ with no index means Φ3 . In terms of the Fourier transform the fundamental solutions √ gp,s of the Helmholtz equation for Im z > 0, z ∈ C\R+ , (∆ + c−2 p,s z)gp,s = −δ, gp,s (x; z) = exp(ikp,s |x|)/4π|x|, kp,s =

√

(2.6) z/cp,s ,

have Fourier transforms gˆp,s (η; z) = (2π)−3/2 c2p,s (c2p,s |η|2 − z)−1 , 12

and for the Fourier transform of I(x; z), [M (D) − zI3 ]I(x; z) = δI3 ,

(2.7)

we have ˆ z) = [M (η) − zI3 ]−1 (2π)3/2 I(η; = (c2p |η|2 − z)−1 P (ω) + (c2s |η|2 − z)−1 S(ω) = z −1 [c2p |η|2 (c2p |η|2 − z)−1 P (ω) +

(2.8)

c2s |η|2 (c2s |η|2 − z)−1 S(ω) − I3 ] gp (η, z) + = (2π)3/2 z −1 [|η|2 P (ω)ˆ gs (η, z) − (2π)−3/2 I3 ] +|η|2 S(ω)ˆ so that, taking the inverse Fourier transform in S  , −I(x; z) = z −1 [∂ ⊗ ∂gp (x; z) − rot rotgs (x; z) + δI3 ]. (2.9) Computed explicitly in S  , −zI(x; z) = −kp2 gp (x; z) + ks2 gs (x; z) + ikp gp (x; z)|x|−1 (I − 3ω ⊗ ω) +iks gs (x; z)|x|−1 (2I + 3ω ∧ ω) +

(2.10)

gp (x; z)|x|−2 (3ω ⊗ ω − I) − gs (x; z)|x|−2 (2I + 3ω ∧ ω) where the last two terms applied to φ ∈ S are to be understood in the sense of singular integrals. (The integrals over S 2 of both I − 3ω ⊗ ω and 2I + 3ω ∧ ω∧ are equal to zero.) 13

Remark 2.1 It is a straightforward computation to verify that I(x; z) can be written in the equivalent form [10] I(x; z) = −zI3 g(x, z) + c2s (−∂ ⊗ ∂)g + c2p rot rot g, g = [z(c2s − c2p )]−1 gp + [z(c2p − c2s )]−1 gs = [z(c2s − c2p )]−1 (gp − gs ). It is interesting to note that the fundamental solution of the second-order elliptic operator M has a singularity at zero of 0(|x|−3 ), while first-order elliptic operators can have a singularity only like 0(|x|−2 ) [18]. Defining the selfadjoint operator M in L2 (R3 ; C3 ) by D(M ) = {f ∈ L2 R3 ; C3 ) : M f ∈ L2 (R3 ; C3 )}, its resolvent is given by (z ∈ C\R+ ) ˆ − z]−1 fˆ(x) ≡ I(z)f (x) = [M − zI]−1 f (x) = Φ∗ [M (2.11)

=

I(x − y; z)f (y)dy.

The nonnegative, selfadjoint operator M in L2 (R3 ; C3 ) generates a unitary group which solves the Schr¨odinger problem for M , while we are interested in the Cauchy problem for (2.4) with w(x, 0) = w0 (x), wt (x, 0) = v0 (x). The solution of this problem can be expressed in terms of the square root of M . To see this it is convenient to go over to the energy 14

space where the corresponding group directly delivers the solution of the Cauchy problem. The following result is “well known”, but we know of no reference and therefore present the simple proof. Here M ≥ 0 is any nonnegative, selfadjoint operator in a Hilbert space H for which zero is not in its point spectrum σp (M ). Theorem 2.1 Let M ≥ 0 be a selfadjoint operator in a Hilbert space H with domain D(M ) and with 0 ∈ σp (M ). Let HM be the completion of D(M )×H in the norm [|M 1/2 f1 |2 + |f2 |2 ]1/2 . Let   0 I M = −i −M 0 with D(M) = D(M ) × D(M 1/2 ) ⊂ HM . Then M is selfadjoint in HM . Proof. D(M

1/2

Let f, g ∈ D(M).

Since f1 ∈ D(M ), f2 ∈

),

(f, Mg)H = (M 1/2 f1 , −iM 1/2 g2 )H + (f2 , iM g1 )H = (iM f1 , g2 )H + (−iM 1/2 f2 , M 1/2 g1 )H = (Mf, g)HM . Hence, M is symmetric, and so its defect indices are equal. Suppose now that f ∈ HM is orthogonal to the range of

15

M − iI, i.e., for all g ∈ D(M) 0 = (f, (M − iI)g)HM = (M 1/2 f1 , −iM 1/2 g2 − iM 1/2 g1 )H (2.12) + (f2 , iM g1 − ig2 )H . We first take g1 = 0 : 0 = (M 1/2 f1 , M 1/2 g2 ) + (f2 , g2 ) for all g2 ∈ D(M 1/2 ) implies that ∗

M 1/2 f1 ∈ D(M 1/2 = M 1/2 ) and M 1/2 (M 1/2 f1 ) = −f2 . (2.13) We now take g2 = 0: from (2.12), (2.13) (−f2 , g1 ) = (M 1/2 f1 , M 1/2 g1 ) = (f2 , M g1 ) for all g1 ∈ D(M ) implies that f2 ∈ D(M ) and M f2 = −f2 , so f2 = 0, since −1 ∈ σ(M ). Thus, from (2.13) M 1/2 f1 = 0, because 0 ∈ σp (M ): suppose there exists g ∈ D(M 1/2 ) such that M 1/2 g = 0; then for all k ∈ D(M ) 0 = (M 1/2 g, M 1/2 k) = (g, M k) which implies that g ∈ D(M ) and M g = 0, so that g = 0. Thus, |f |HM = 0, so f = 0 in HM . Hence, (M∗ + iI)f = 0 in HM implies that f = 0, and so the defect indices of M are zero. The theorem is proved. Corollary 2.1 If z 2 ∈ ρ(M ), the resolvent set of M , then z ∈ ρ(M), and in terms of the resolvent r(z 2 ) = (M − z 2 I)−1 in H the resolvent R(z) = (M − zI)−1 in HM can be expressed on a dense set in HM , e.g., D(M), as   zr(z 2 ) −ir(z 2 ) R(z) = . (2.14) iz 2 r(z 2 ) + i zr(z 2 ) 16

Proof. Let f ∈ D(M). Then  D(M)  R(z)f =

zr(z 2 )f1 − ir(z 2 )f2 iz 2 r(z 2 )f1 + if1 + zr(z 2 )f2

 .

We must first show that |R(z)f |2HM ≤ c(z)|f |2HM . We suppose that f2 = 0 and note that r(z 2 )M 1/2 ⊂ M 1/2 r(z 2 ) [8, pp. 279, 285]; then |R(z)f |2HM = |M 1/2 zr(z 2 )f1 |2H + |z 2 r(z 2 )f1 + f1 |2H = |z|2 |r(z 2 )M 1/2 f1 |2H + (M r(z 2 )f1 , z 2 r(z 2 )f1 )H +(M r(z 2 )f1 , f1 )H ≤ (|z|2 /|Im z 2 |)|M 1/2 f1 |2H + z 2 |M 1/2 r(z 2 )f1 |2H +(r(z 2 )M f1 , f1 )H ≤ c(z)|M 1/2 f1 |2H . Now let f1 = 0: |R(z)f |2HM = |M 1/2 r(z 2 )f2 |2 + |z|2 |r(z 2 )f2 |2 = (r(z 2 )f2 , M r(z 2 )f2 ) + |z|2 |r(z 2 )f2 |2 = z 2 |r(z 2 )f2 |2 + |z|2 |r(z 2 )f2 |2 + (r(z 2 )f2 , f2 ) ≤ c(z)|f2 |2 . 17

Thus, R(z) is bounded on D(M), and it is straightforward to verify that for f ∈ D(M) (M − zI)R(z)f = f . Hence, (2.14) is an expression for the resolvent on a dense set. Corollary 2.2 Suppose the spectrum of M , σ(M ), is continuous and σ(M ) = [0, ∞). Then σ(M) = R is also continuous, and the spectral family for M in HM , E(dλ), is given in terms of the spectral family e(dλ2 ) of M in H by   e(dλ2 ) −iλ−1 e(dλ2 ) sgn (λ), δ ⊂ R, E(δ) = 2−1 2 2 ) e(dλ ) iλe(dλ δ (2.15) where if δ = (µ1 , µ2 ) ⊂ (−∞, 0), this means that E(µ1 , µ2 ) = 2−1

|µ1 | e(dλ2 ) −iλe(dλ2 )

|µ2 |

iλ−1 e(dλ2 ) e(dλ2 )

 .

Proof. The expression (2.15) is computed directly from (2.14) and Stone’s formula [8] E(µ) = (2πi)−1 lim

µ

→0 −∞

[R(λ + i1) − R(λ − i1)]dλ.

Theorem 2.1 and its corollaries are stated for abstract operators. In the case of the operators M, M considered below which are engendered by the elasticity operator M (D) plus the boundary condition equality (2.15), for example, makes it possible to compute the eigenfunction expansion 18

for M in the energy space directly from the eigenfunction expansion for M in H = L2 (R3+ , C3 ) (see §4). Further for the unitary group U (t)“=”exp(iMt) in HM for f ∈ HM ∞ U (t)f =

exp(iλt)E(dλ)f,

(2.16)

−∞

so that for f ∈ D(M) from (2.15) u(·, t) = [U (t)f ]1 ∞ √ √ = [cos(t ν)f1 + ν −1/2 sin(t ν)f2 ]e(dν) 0

= cos(M 1/2 t)f1 + M −1/2 sin(M 1/2 t)f2 , which is the usual expression for the solution of the Cauchy problem utt + M u = 0, u(0) = f1 , ut (0) = f2 .

(2.17)

(cf. [20, p. 15]). We remark that in the case of a boundary value problem where the operator E −1 A(D) plus a boundary condition engenders a selfadjoint operator its resolvent and spectral family (on the complement of the null space) can also be expressed in terms of those for M . We shall have no need for them in the present work. 19

3

The Elasticity Operator in R3+ with Boundary Free of Normal Stresses

The operator M (D) = t A(D)E0−1 A(D) of (2.4) and the boundary operator (n = (0, 0, 1))   µD3 0 µD1     B(D) ≡ t A(n)E0−1 A(D) =  0 µD3 µD2    2 λD1 λD2 cp D3

(3.1)

give rise to a selfadjoint operator in H = L2 (R3+ , C3 ). We recall from §2 that the components of the stress tensor are recovered from the displacement w by σ = iE0−1 A(D)w. From (2.1) with D → n B(D)w(x , 0) = 0 then says that σi3 (x , 0) = 0, i = 1, 2, 3, i.e., the normal components of the stress are zero on the boundary. Remark. In general, for any vector n = (n1 , n2 , n3 ) and displacement vector w (sum on i) t

A(n)E0−1 A(D)w = t (ni σi1 , ni σi2 , ni σi3 ).

(3.2)

◦

Definition 3.1 We denote by D(M ) the set of compactly ¯ 3 such that B(D)f (x , 0) = supported, smooth functions f in R + 0. We define M as the graph closure in H = L2 (R3+ , C3 ) of ◦

M (D) on D(M ) and denote its domain by D(M ). Theorem 3.1 The operator M is a nonnegative, selfadjoint operator in H. 20

◦

It is clear by integration by parts on D(M ) that M is symmetric and nonnegative. It is possible to show that its defect indices are zero by introducing the necessary Sobolev machinery as in [8, 9], but, rather than do this, we show that the equation M u + iIu = f has a solution for any f ∈ H. This demonstrates that the operator M has zero defect indices, and the operator H(z), z ∈ C\R+ , delivering such solutions is its resolvent. We proceed to construct this operator explicitly. Let φ ∈ S = S(R3 , C3 ), and let I and Iˆ be the distributions of (2.7), (2.8). We write R3  η = (ξ, ρ), ξ ∈ R2 , ρ ∈ R, and denote by χh the characteristic function of {η = (ξ, ρ) : |ξ| < h, |ρ| < h}. Omitting the z-dependence of I, Iˆ to simplify notation, we then have ∗ˆ ∗ −3/2 ˆ exp(iηx)φ(x)dxdη ˆ I(φ) = Φ I(φ) = I(Φ φ) = (2π) I(η) =

lim Ih (φ),

h→∞

3/2

Ih (x) = (2π)

ˆ exp(iηx)χh (η)I(η)dη.

For φ ∈ D = C0∞ (R3 , C3 ) by the continuity of differentiation and convolution in the dual space D BI ∗ φ(x) =

lim BIh ∗ φ(x),

h→∞

−3/2

BIh (x − y) = (2π)

ˆ exp[iη(x − y)]χh (η)B(η)I(η)dη, 21

(3.3)

−1/2

Φ2 BIh (ξ, 0, y) = (2π)

h ˆ χh (ξ) exp(−iξy ) exp(−iy3 ρ)B(η)I(η)dρ. 

−h

The idea is now to evaluate the last integral explicitly in order to construct a function satisfying the boundary conditions. From (2.8) for ξ = 0 P (ω) = |η|−2 η ⊗ η and S(ω) = |η|−2 (−η ∧ η∧) are analytic functions of ρ and so ˆ z). We extend ρ to τ ∈ C, so that P (ξ, τ ) = then also is I(η; (|ξ|2 + τ 2 )−1 (ξ, τ ) ⊗ (ξ, τ ), S(ξ, τ ) = (|ξ|2 + τ 2 )−1 [−(ξ, τ ) ∧ 2 2 1/2 by Imτp,s > 0 for (ξ, τ )∧] and define τp,s = c−1 p,s (z − cps |ξ| ) z ∈ C\R+ , noting that c2p,s |ξ|2 + c2p,s τ 2 − z = c2p,s (τ − τp,s ) (τ + τp,s ). With y3 > 0 we now choose h in (3.3) so large that the semicircle of radius h in C− encloses −τp,s . Then by the residue theorem Φ2 BIh (ξ, 0, y; z) = = i(2π)−1 χh (ξ) exp(−iy  ξ)[exp(iy3 τp )(2c2p τp )−1 B(ξ, −τp ) P (ξ, −τp ) + exp(iy3 τ2 )(2c2s τs )−1 B(ξ, −τs )S(ξ, −τs )] + oh (1), π oh (1) ≤ const

exp(−y3 h sin θ)dθ, y3 > 0. 0

Hence, Φ2 BI ∗ φ(ξ, 0) =

lim Φ2 BIh φ(ξ.0)

h→∞

22

=

lim

Φ2 BIh (ξ, 0, y; z)φ(y)dy

h→∞

=

Φ2 BI(ξ, 0, y; z)φ(y)dy,

Φ2 BI(ξ, 0, y; z) = = i(2π)−1 exp(−iy  ξ)[(2c2p τp )−1 exp(iy3 τp )B(ξ, −τp ) P (ξ, −τp ) + (2c2s τs )−1 exp(iy3 τs )B(ξ, −τs )S(ξ, −τs )], P (ξ, −τp ) = c2p z −1 (ξ, −τp ) ⊗ (ξ, −τp ),

(3.4)

S(ξ, −τs ) = c2s z −1 [−(ξ, −τs ) ∧ (ξ, −τs )]. We define P± =

t

(ξ, ±τp ),

V± = |ξ|−1 t (±ξ1 τs , ±ξ2 τs , −|ξ|2 )

(3.5)

√ √ −1 H = |ξ|−1 t (c−1 s ξ2 z, −cs ξ1 z, 0). Then t t c−2 s zS(ξ, −τs ) = V− V− + H H, t c−2 p zP (ξ, −τp ) = P− P− ,

Φ2 BI(ξ, 0, y; z) = αp B(ξ, −τp )P− t P− + αs B(ξ, −τs )[V− t V− + H t H], 23

(3.6)

αp,s = i(2π)−1 exp(−iy  ξ)(2zτp,s )−1 exp(iy3 τp,s ). We now construct an operator H(z) = I(z) − R(z) where I(z) is given in (2.11) and for f ∈ D(R3+ ) = C0∞ (R3+ ; C3 ) R(z)f (x) = R(x, y; z)f (y)dy. (3.7) R(z) is to be constructed so that (M − zI)R(z)f = 0 and BI(z)f (x , 0) = BR(z)f (x , 0). We seek R(x, y; z) in the form Φ2 R(ξ, x3 , y; z) = αp [rpp exp(iτp x3 )P+ t P− + rsp exp(iτs x3 )V+ t P− ] + αs [rss exp(iτs x3 )V+ t V− + rps exp(iτp x3 )P+ t V− ] + αs rhh exp(iτs x3 )H t H. Setting B(±τp,s ) = B(ξ, ±τp,s ), we see from (3.6), (3.7) that Φ2 BI(ξ, 0, y; z) = Φ2 BR(ξ, 0, y; z) if and only if B(−τp )P− = rpp B(τp )P+ + rsp B(τs )V+ , B(−τs )V− = rps B(τp )P+ + rss B(τs )V+ , B(−τs )H = rhh B(τs )H. Solving this system of equations, we obtain ˜ rpp = rss = ∆/∆, 24

(3.8)

∆ = (τs2 + |ξ|2 )2 + 4|ξ|2 τp τs , ˜ = (τ 2 + |ξ|2 )2 − 4|ξ|2 τp τs , ∆ s

(3.9)

rsp = −4|ξ|τp (τs2 − |ξ|2 )/∆, rps = 4|ξ|τs (τs2 − |ξ|2 )/∆. We observe that 0 = [M (D)−zI]R(x, y; z) = Φ∗2 [M (ξ, D3 )−zI]Φ2 R(ξ, x3 , y; z), (3.10) since 0 = [M (ξ, τp )−zI]P+ = [M (ξ, τs )−zI]V+ = [M (ξ, τs )−zI]H. Further, for f ∈ D(R3+ ) H(z)f (x) = H(x, y; z)f (y)dy = I(z)f (x) − R(z)f (x) (3.11) = I(x, y; z)f (y)dy − R(x, y; z)f (y)dy satisfies the boundary conditions by construction. This can easily be checked directly from (3.6), (3.8), (3.9). ¯ + H(z) is Theorem 3.2 For any f ∈ H and z ∈ C\R bounded in H, H(z)f ∈ D(M ), and (M − zI)H(z)f = f . In particular, M is selfadjoint, and H(z) is its resolvent. 25

Proof. For f ∈ H |I(z)f | ≤ |Im z|−1 |f |. From (3.6), (3.7), (3.8) Φ2 R(z)f (ξ, x3 ) = = i(2zτp )−1 [rpp exp(iτp x3 )P+ t P− + rsp exp(iτs x3 )V+ t P− ]hp + i(2zτs )−1 [rss exp(iτs x3 )V+ t V− + rps exp(iτp x3 )P+ t V− ]hs − i(2zτs )−1 exp(iτs x3 )H t Hhs , hp,s (ξ, z) = exp(iτp,s y3 )Φ2 f (ξ, y3 )dy3 2 −1 |hp,s (ξ, z)| ≤ [2Im τp,s (ξ, z)] |Φ2 f (ξ, y3 )|2 dy3 . For large |ξ| from (3.5), (3.9) rpp P+ t P− , rss V+ t V− , etc. are at most 0(|ξ|2 ), and hence ∞ |Φ2 R(z)f (ξ, x3 )|2 dx3 ≤ c(z)(1+|ξ|)[|hp (ξ, z)|2 +|hs (ξ, z)|2 ]. 0

Therefore,



|R(z)f | = |Φ2 R(z)f | 2

2

≤ c(z)

(1 + |ξ|)[|hp (ξ, z)|2 + |hs (ξ, z)|2 ]dξ R2

∞ |Φ2 f (ξ, y3 )|2 dy3 dξ = c(z)|f |2 .

≤ c(z) R2 0

Thus, H(z) is bounded. For f ∈ D(R3+ ) it is easy to see from (2.11), (3.7), (3.8) that H(z)f (x) is smooth and B(D)H(z)f (x , 0) = 26

0. Let φ be a smooth, scalar-valued function on (0, ∞) such that φ(t) = 1 for 0 < t < 1 and φ(t) = 0 for t > 2; let φN (x) = φ(|x| − N ). Then φN H(z)f ∈ D(M ) and converges to H(z)f in H. Furthermore, the derivatives of φN are supported in {N − 1 ≤ |x| ≤ N + 1} and are uniformly bounded with respect to N ; hence, M φN H(z)f = φN M H(z)f + o(1), so that M φN H(z)f → M H(z)f . Thus, H(z)f 1D(M ), and (M − zI)H(z)f = f . For any f ∈ H we approximate f by functions in D(R+ 3 ) and use the fact that H(z) is bounded and M is closed (cf., e.g., [6]). Because of the way z occurs in R(x, y; z), the present form of H(x, y; z) is not convenient for establishing the limiting-absorption principle. For this an expression for H(z) in terms of generalized eigenfunctions will be obtained in the next section. In order to be able to compute them we multiply H(x, y; z) by the characteristic function χ+ = χ+ (y), y ∈ R3+ , and obtain an expression for the Fourier transform Φ∗y χ+ H(x, η; z). For fixed x ∈ R3+ let λx ∈ D(R3+ ), supp λx ⊂ Nδ (x) ⊂ R3+ , λx (y) = 1 for y ∈ Nδ/2 (x). For each x ∈ R3+ and ψ ∈ S(R3 ) χ+ (·)H(x, ·; z)(ψ) = H(x, ·; z)(λx ψ)+χ+ (·)H(x, ·; z)[(1−λx )ψ] Now λx ψ ∈ D(R3+ ), so λx χ(·)H(x, ·; z) = λx H(x, ·; z) ∈ S  (R3 ); [1−λx (y)] H(x, y; z) is a smooth function, so χ+ (·)H(x, ·; z) ∈ S  (R3 ) for each x ∈ R3+ . We note that this depends on x being an interior point of R3+ : if x3 = 0+ , then χ+ (·)λx (·)H(x, ·; z) 27

does not exist in general: e.g., in (2.10) the singular integrals won’t exists, in general, if the integration in I ∗ψ(x) goes only over the hemisphere S+2 . Thus, χ+ (·)H(x, ·; z) is a family of elements of S  (R3 ) depending continuously on x ∈ R3+ , but the limit as x3 → 0 of the family does not exist in general. (This illustrates the fact that no meaning can be assigned to the product of two distributions whose singular supports intersect.) For y ∈ R3− , x ∈ R3+ χ− (·)I(x, ·; z) is a smooth function and so in S  (R3 ). Thus, χ+ (·)I(x, .; z) = I(x, ·; z) − χ− (·)I(x, ·; z) is also in S  (R3 ). We wish to compute Φ∗y χ+ I(x, ·; z). As above in (3.3), using the fact that I(x, ·; z) is the limit in S  of Ih (x, ·; z), we compute Φ∗y I(x, ξ, y3 ; z) = (2π)−1/2 exp(ix ξ)

∞ ˆ exp[i(x3 −y3 )ρ]I(η)dρ.

−∞

For x3 ∈ R+ , y3 ∈ R− (x3 − y3 ) > 0, so this can be evaluated by the residue theorem in C+ . Taking the one-dimensional Fourier transform on y3 , we obtain Φ∗y χ− I(x, η; z) = (2π)−3/2 exp(ix ξ)[(c2p |η|2 − z)−1 (2τp )−1 exp(iτp x3 ) (ρ + τp )P (ξ, τp ) + (c2s |η|2 − z)−1 (2τs )−1 exp(iτs x3 ) (ρ + τs )S(ξ, τs )].

(3.12) 28

Using the fact that τ¯p,s (ξ, z¯) = −τp,s (ξ, z), from (3.5), (3.6), (3.8), (3.9) it is straightforward to verify that exp[−iξ(x + y  )]t [Φ2 R(ξ, y3 , x; z¯)] = Φ2 R(ξ, x3 , y; z). ¯

Hence, Φ∗y R(x, ξ, y3 ; z) = t [Φy R(ξ, y3 , x; z¯)] = exp[i(x +y  )]Φ2 R(ξ, x3 , y; z), ¯

so that from (3.8) we compute −(2π)3/2 exp(ix ξ)Φ∗y χ+ R(x, η; z) = (c2p |η|2 − z)−1 (2τp )−1 (ρ − τp )[Rpp exp(iτp x3 ) + Rsp exp(iτs x3 )] + (c2s |η|2 − z)−1 (2τs )−1 (ρ − τs )[Rss exp(iτs x3 ) + Rps exp(iτp x3 )], Rpp = c2p z −1 rpp P+ t P− , Rsp = c2p z −1 rsp V+ t P− , Rss = c2s z −1 [rss V+ t V− − H t H], Rps = c2s z −1 rps P+ t V− , where rpp , V+ , etc. are given in (3.5), (3.9). Thus, for x ∈ R3+ , χ+ = χ+ (y3 ) in S  (R3 ) there exists Φ∗y χ+ H(x, η; z) = Φ∗y χ+ I(x, η; z) − Φ∗y χ+ R(x, η; z) which by (2.11), (3.12), (3.13) is given by Ψ(x, η; z) ≡ Φ∗y χ+ H(x, η; z) = (c2p |η|2 − z)−1 Ψp (x, η; z) + (c2s |η|2 − z)−1 Ψs (x, η; z), 29

(3.13)

Ψp (x, η; z) = (2π)−3/2 exp(ix ξ){exp(iρx3 )P (ω) − (2τp )−1 (ρ + τp ) exp(iτp x3 )P (ξ, τp ) + (2τp )−1 (ρ − τp )[exp(iτp x3 )Rpp + exp(iτs x3 )Rsp ]},

(3.14)

Ψs (x, η; z) = (2π)−3/2 exp(ix ξ){exp(iρx3 )S(ω) − (2τs )−1 (ρ + τs ) exp(iτs x3 )S(ξ, τs ) + (2τs )−1 (ρ − τs )[exp(iτs x3 )Rss + exp(iτp x3 )Rps ]}. We note that [M (D)−zI]Ψp,s (x, η; z) = (c2p,s |η|2 −z)(2π)−3/2 e(ixη) Q(ω), Q = P, S. (3.15)

4

Solution of the Time-Dependent Problem in Terms of Generalized Eigenfunctions

In this section we obtain a representation of the solution of problem (2.17) in terms of an expansion in generalized eigenfunctions. This also affords a representation of the resolvent of M which is used in §§6-8 to establish the principle of limiting absorption.

30

With Ψ(x, η; z) of (3.14) for f ∈ D(R3+ ) we define ¯ t¯ Ψ(x, η; z)f (x)dx = t [Φ∗y χ+ H(x, η; z)]f (x)dx. Ψ(z)f (η) = (4.1) Lemma 4.1 Ψ(z)f (η) = Φχ+ H(¯ z )f (η). Proof. For ψ ∈ L2 (R3 ) let ψ+ = χ+ ψ, and let {ψn } ⊂ D(R3 ) be such that χ+ ψn ∈ D(R3+ ), χ+ ψn → ψ+ in L2 (R3+ ), and ψn → ψ in L2 (R3 ). Then denoting by ·, · the inner product in L2 (R3 ), z )f  = ψ, χ+ H(¯ z )f  = (ψ+ , H(¯ z )f ) = (H(z)ψ+ , f ) = Φψ, Φχ+ H(¯ lim(H(·, y; z)χ+ (y)ψn (y), f ) = limχ+ (y)H(·, y; z)ψn , f  = limΦ∗y χ+ H(·, η; z)Φψn , f  = limΦψn , t [Φ∗y χ+ H(·, η; z)]f  = ¯

Φψ, t [Φ∗y χ+ H(·, η; z)f ]. ¯

Now let f, g ∈ D(R3+ ), and let e(·) be the spectral measure of M in H. Setting e(N ) = e(0, N ), by Stone’s formula [21], the resolvent identity, (4.1), Lemma 4.1, and the fact that λ → Ψ(λ ± iκ)h, h = f, g, is a continuous, L2 -valued function of λ, κ > 0, we have 31

(f, e(N )g) = = lim (2πi)−1

N (f, [H(λ + iκ) − H(λ − iκ)]g)dλ

κ→0

0

= lim π

−1

N κ(f, H(λ ± iκ)H(λ ∓ iκ)g)dλ

κ→0

0

= lim π −1

N κ(H(λ ∓ iκ)f, H(λ ∓ iκ)g)dλ

κ→0

0

= lim π −1

n κΦχ+ H(λ ∓ iκ)f, Φχ+ H(λ ∓ iκ)gdλ(4.2)

κ→0

0

= lim π

−1

N κΨ(λ ± iκ)f, Ψ(λ ± iκ)gdλ

κ→0

0

= lim π

−1



N 0

= lim π −1

R3



N

R3

¯

κt [Ψ(λ ± iκ)f (η)]Ψ(λ ± iκ)g(η)dλ.

dη

κ→0

t¯

[Ψ(λ ± iκ)f (η)]Ψ(λ ± iκ)g(η)dη

κdλ

κ→0

0

We now pass to the limit under the last integral and evaluate the result. The justification and technique for doing this are essentially the same as in [5, 6, 14] and are not reproduced 32

here. To state the result we must compute the generalized eigenfunctions of M from (3.9), (3.13), and (3.14). Details of such a computation can be found in the aforecited references. Denoting the characteristic functions of the hemispheres S±2 by χ± (q3 ), from (3.14) we have with the notation of (2.5), η = (ξ, ρ) = |η|q, q ∈ S 2 , q˜ = (q  , −q3 ), η˜ = (ξ, −ρ) 2 2 Ψ± s (x, η) ≡ Ψs (x, η; cs |η| ± i0) ± = Ψ± sv (x, η) + Ψsh (x, η), −3/2 [exp(ixη) + exp(ix˜ η )]h(q) ⊗ h(q) Ψ± sh (x, η) = χ∓ (q3 )(2π) −3/2 ± [exp(ixη)v(q) − exp(ix˜ η )rss (q)v(˜ q) Ψ± sv (x, η) = χ∓ (q3 )(2π) ± − exp(ixηs± )rps (q)θ± (q)] ⊗ v(q), ± ˜ ± (q)/∆± (q), (q) = ∆ rss s s

(4.3)

± (q) = ±4n|q3 | |q  |(q32 − |q  |2 )/∆± rps s (q), 2  2  2 ± ∆± s (q) = (q3 − |q | ) ± 4n|q | |q3 |θ3 , 2  2  2 ± ˜± ∆ s (q) = (q3 − |q | ) ∓ 4n|q | |q3 |θ3 ,

θ± (q) =

t

(θ1 , θ2 , θ3± ) ≡ t (n−1 q1 , n−1 q2 , θ3± ),

 θ3± (q) = ±χ(0,n) (|q  |) (1 − n−2 |q  |2 ) +  iχ(n,1) (|q  |) (n−2 |q  |2 − 1), 33

ηs± = |η|(q  , nθ3± ), n = cs /cp . From (3.15) and by direct verification 2 2 ± M (D)Ψ± sv (x, η) = cs |η| Ψsv (x, η), 2 2 ± M (D)Ψ± sh (x, η) = cs |η| Ψsh (x, η),

(4.4)

±   B(D)Ψ± sv (x , 0, η) = B(D)Ψsh (x , 0, η) = 0,

so Ψ± s (x, η) are generalized eigenfunctions of M correspond± ing to shear modes, and Ψ± sv (x.η) and Ψsh (x.η) are, respectively, the vertical and horizontal components. The second component corresponds to the reflected S mode, while the third component of Ψ± sv (x, η), ± (q)θ± (q), exp(ixηs± )rps

is the reflected pressure mode due to the incident shear mode. We note that the phase ηs± has a purely imaginary third √ component for |q  | > n = cs /cp < 1/ 2. Further, from (3.14), retaining the notation of (4.3), 2 2 Ψ± p (x, η) ≡ Ψp (x, η; cp |η| ± i0)

η )rpp (q)˜ q− = χ∓ (q3 )(2π)−3/2 [exp(ixη)q − exp(ix˜ ± (q)v ± (φ)] ⊗ q, − exp(ixηp± )rsp

˜ rpp (q) = ∆(q)/∆(q), 34

± rsp (q) = ∓rsp (q), rsp (q) = 4|q3 | |q  |(φ23 − |φ |2 )

φ = nq  , φ3 =



(1 − |φ |2 ),

φ3 ) ηp± = n−1 |η|φ± ≡ n−1 |η|(φ , ±φ3 ) = |η|(q  , ±n−1 (4.5) ∆p (q) = (φ23 − |φ |2 ) + 4n|φ |2 φ3 |q3 |, ˜ p (q) = (φ2 − |φ |2 ) − 4n|φ |2 φ3 |q3 |, ∆ 3 v ± (φ) =

t

(±φ3 φ1 , ±φ3 φ1 , −|φ |2 ).

As above, 2 2 ± M (D)Ψ± p (x, η) = cp |η| Ψp (x, η),  B(D)Ψ± p (x , 0, η) = 0,

so Ψ± p (x, η) are generalized eigenfunctions corresponding to a P mode. Here q ∈ S 2 is the direction of the incident P mode, q˜ ∈ S 2 is that of the reflected P mode, and φ is that of the reflected SP mode – the S mode due to the incident P mode. There is a further contribution to the limit (4.2) due to the pole of the reflection and transmission coefficients (3.9) entering in (3.14), i.e., due to the real zero of ∆(ξ; z). This zero gives rise to the Rayleigh or surface mode. For each R2  ξ = 0 ∆(ξ; z) has a simple zero [4, 14] R2 (ξ) = R02 c2s |ξ|2 , R0 ∈ (0, 1), 35

(4.6) (2 − R02 )2 = 4(1 − R02 )1/2 (1 − n2 R02 )1/2 with 2 ∂z ∆(ξ; z)|z=R = 2c−2 s |ξ|

(4.7)

α(R0 )α(R0 ) = (1−R02 )−1/2 (1−n2 R02 )[1+n2 −2n2 R02 −(2−R02 )/4] > 0. (Note that R2 (ξ) lies to the left of the least branch point c2s |ξ|2 .) Passing to the limit z = λ + iκ → R2 in (4.2) and then integrating on ρ [4, 5, 14], we see that the Rayleigh mode has the form Σ(x, ξ) = (2π)−1 λ(R0 )1/2 |ξ|2 exp(ix ξ)γ(ξ, x3 ),

γ(ξ, x3 ) = (1 − n2 R02 )−1/2 (R02 − 2)π(β) exp[−x3 |ξ| (1 − n2 R02 )]

−2iσ(β) exp[−x3 |ξ| (1 − R02 )], π(β) =

t

(β1 , β2 , i (1 − n2 R02 )),

σ(β) =

t

2 (iβ1 (1 − R0 ), iβ2 (1 − R02 ), −1),

βi = ξi /|ξ|, i = 1, 2, λ(R0 ) = 2−1 τ (R0 )α(R0 )−2 R0−4 , τ (R0 ) = (1 − R02 )−1/2 [(1 − R02 )(2 − n2 R02 ) + 36

(4.8)

(1 − n2 R02 )(2 − R02 ) − (2 − R02 )3 /2] Using (4.6), (4.7), it can be verified directly (but perhaps not so easily [4]) that τ (R0 )2 = α(R0 )R04 (1 − n2 R02 ).

(4.9)

This may then be used in λ(R0 ) in place of the lengthy expression above. It follows by direct verification that M (D)Σ(x, ξ) = R2 (ξ)Σ(x, ξ) = c2s R02 |ξ|2 Σ(x, ξ) (4.10) B(D)Σ(x , 0, ξ) = 0. Thus, Σ(x, ξ) is a generalized eigenfunction of M corresponding to surface modes. Multiplied by exp(−iνt), they propagate as cylindrical modes along the boundary {x3 = 0} while decaying exponentially in the normal direction. Lemma 4.2 Let f ∈ D(R3+ ). Then t¯ ± ± Ψp,s (x, η)f (x)dx, Ψp,s f (η) = Σf (ξ) =

(4.11) t¯

Σ(x, ξ)f (x)dx

extend to bounded mappings from H to L2 (R3 , C3 ) and L2 (R2 , C3 ) respectively. Their adjoints are ∗ Ψp,s g(x) = Ψ± p,s (x, η)g(η)dη, 37

Σ∗ h(x) =

(4.12)

Σ(x, ξ)h(ξ)dξ.

The proof proceeds by direct verification using (4.3), (4.5), (4.8) (cf., e.g., [3, 4]). Remark 4.1 In the proof of Theorem 4.1 below we shall need to know the following facts which follow directly from (4.3), (4.5), (4.8), (4.11) and (4.12): Σ∗ h(x) = λ(R0 )Φ∗2 [| · |1/2 γ(·, x3 )h(·)](x ), ± q Φf (˜ η ) − rsp (q)v ± (φ)Φf (ηp± )], Ψ± p f (η) = χ∓ (q3 )q ⊗ [qΦf (η) − rpp (q)˜

η )] + Ψ± s f (η) = χ∓ (q3 ){h(q) ⊗ h(q)[Φf (η) + Φf (˜ ± (q)v(˜ q )Φf (˜ η) v(q) ⊗ [v(q)Φf (η) − rss ± (q)θ−± (q)Φf (ηs± )]}, −rps

Φf (ηp± ) Φf (ηs± )

−1/2



= (2π)

−1/2

= (2π)



exp(∓in−1 |η|φ3 x3 )Φ2 f (ξ, x3 )dx3 , exp(in|η|θ¯3± x3 )Φ2 f (ξ, x3 )dx3 .

Denoting now by χp,N  χs,N  and χR,N the characteristic functions of the respective sets {η : c2p |η|2 ∈ (0, N )}, {η : c2s |η|2 ∈ (0, N )}, and {ξ : R2 (ξ) = c2s |ξ|2 R02 ∈ (0, N )}, in

38

terms of (4.11), (4.12) the expression (4.2) can now be writ− ten (Ψp,s denotes either Ψ+ p,s or Ψp,s ) (f, e(N )g) = (Ψp f, χp,N Ψp g)+(Ψs f, χs,N Ψs g)+(Σf, χR,N Σg). (4.13) Theorem 4.1 The spectral measure e(N ) = e(0, N ) of M is given by e(N ) = Ψ∗p χp,N Ψp + Ψ∗s χs,N Ψs + Σ∗ χR,N Σ

(4.14)

which for N → ∞ gives the Parseval identity for M : I = Ψ∗p Ψp + Ψ∗s Ψs + Σ∗ Σ.

(4.15)

For f ∈ D(M ) Ψp,s M f (η) = c2p,s |η|2 Ψp,s f (η), (4.16) ΣM f (ξ) = R(ξ)2 Σf (ξ), and M f = Ψ∗p c2p | · |2 Ψp f + Ψ∗s c2s | · |2 Ψs f + Σ∗ R(·)2 Σf.

(4.17)

The operators Ψ∗p Ψp +Ψ∗s Ψs and Σ∗ Σ are mutually orthogonal orthoprojectors in H. Proof. Equality (4.14) is just (4.13). To prove (4.16), ◦ let f ∈ D(M ) and recall that B(D) = t A(n)E −1 A(D), n = 39

(0, 0, 1) (see (3.1). All three relations(4.16) are verified in the same way simply by integrating by parts and recalling that f and the generalized eigenfunctions satisfy the boundary conditions. Consider, for example, Ψp M (D)f (η) = Ψp t A(D)E −1 A(D)f (η) t¯ Ψ(x, η) t A(D)E −1 A(D)f (x)dx = = R3+

=

[A(D)Ψp (x, η)]E −1 A(D)f (x)dx +

t¯

R3+

+

it Ψp (x , 0, η)B(D)f (x , 0)dx ¯

R2

=

t¯

[M (D)Ψp (x, η)]f (x)dx +

R3+



it [B(D)Ψp (x , 0, η)]f (x , 0)dx = c2p |η|2 Ψp f (η). ¯

R2

The result for any f ∈ D(M ) now follows in an obvious way (see, e.g., [5], Lemma 4.13). Equality (4.17) follows formally from (4.15) and rigorously from (4.14) and the ensuing functional calculus [14]. From (4.8), (4.10), (4.12) M (D)Σ∗ h(ξ) = λ(R0 )1/2 Φ∗2 [| · |1/2 γ(·, x3 )R2 (·)h(·)](x ). 40

From Remark 4.1 Ψp,s M (D)Σ∗ h(η) = R(ξ)2 Ψp,s Σ∗ h(η). From (4.16) Ψp,s M (D)Σ∗ h(η) = c2p,s |η|2 Ψp,s Σ∗ h(η). Subtraction gives [c2p,s |η|2 − R(ξ)2 ]Ψp,s Σ∗± h(η) = 0 which implies the orthogonality relation. The fact that the operators are projections follows from this and (4.15) (it can be verified by direct computation that Σ∗ Σ is a projection [4]). Remark 4.2 It was stated in [14] that the operators Ψ∗p Ψp and Ψ∗s Ψs are mutually orthogonal by appeal to a proof of a similar result in [13]. That proof was subsequently found to be incorrect, and a valid proof for the case of Maxwell’s equations was given in [5]. However, that proof does not work in the present situation. We very much doubt if such an assertion is true (see the form of the asymptotics of the steady-state waves in §§7-8). Be this as it may, it has no bearing on the present work. We now recall from §2 the energy space HM associated with M . 41

Theorem 4.2 In terms of the mappings ˜ ±p,s : HM → L2 (R3 ; C3 ). Ψ (4.18) ˜ ± : HM → L2 (R2 ; C3 ) Σ defined by  ˜ ±p,s f = 2−1/2 [cp,s | · |I3 , ∓iI3 ] Ψ  ˜ ± f = 2−1/2 [R(·)I3 , ∓iI3 ] Σ

Ψp,s f1

 ,

Ψp,s f2 Σf1



(4.19)

Σf2

with adjoints  ˜ ∗ g = 2−1/2 Ψ ±p,s

±iΨ∗p,s g 

˜ ∗ h = 2−1/2 Σ ±

Ψ∗p,s (cp,s | · |−1 )g

∗

−1

Σ R(·) h ∗

±iΣ h

 , g ∈ L2 (R3 , C3 ), (4.20)

 , h ∈ L2 (R2 , C3 ),

the solution of the initial boundary value problem ∂t2 f1 (x, t) + M f1 (x, t) = 0, x ∈ R3+ , t ∈ R, (4.21) f1 (x, 0) = f10 (x), ∂t f1 (x, 0) = f20 (x), Bf1 (x , 0, t) = 0 42

is given by  f1 (x, t) = [U (t)f 0 ]1 (x), f 0 =

f10 f20

 ,

where U (t)“=”exp(iMt) is the unitary group in the energy space HM generated by the selfadjoint operator M of Theorem 2.1 and has the representation  ˜ ∗ exp(ijcp | · |t)Ψ ˜ jp f 0 + Ψ ˜ js exp(ijcs | · |t)Ψ ˜ js f 0 {Ψ U (t)f 0 = jp j=±1

˜ ∗j exp(ijR(·)t)Σ ˜ j f 0 }. + Σ

(4.22)

˜ ±p,s , Σ ˜ ± are obtained on the basis of The mappings Ψ ˜ ±p,s denotes four mappings each of (2.13), (4.14); here Ψ ± which contains one of the two mappings Ψ± p , Ψs . The representation of U (t) follows from (2.13), (2.14), and (4.14).

5

The Steady-State Problem and the Resolvent in Terms of Generalized Eigenfunctions

The Lamb problem for time-harmonic solutions is that of finding solutions of the problem ∂t2 u(x, t) + M (D)u(x, t) = f (x) exp(−iνt) B(D)u(x , 0, t) = 0,

ν ∈ R+ , f ∈ D(R3+ ), x ∈(5.1) R3+ . 43

If v(x) is a solution of the steady-state problem [M (D) − ν 2 I]v(x) = f (x), B(D)v(x , 0) = 0, ν > 0,

(5.2)

then u(x, t) = exp(iνt)v(x) will be a time-harmonic solution. To proceed we set z = (ν ± i1)2 , 1 ∈ (0, 10 ] and write the resolvent for M in terms of generalized eigenfunctions in the form v(x; z) ≡ H(z)f (x) = Ψ∗p (c2p | · |2 − z)−1 Ψp f (x) + Ψ∗s (c2s | · |2 − z)−1 Ψs f (x) + Σ∗ (R2 (·) − z)−1 Σf (x)

(5.3)

≡ vp (x; z) + vs (x; z) + vR (x; z). We must now demonstrate that each of these terms has a limit as 1 → 0 – the principle of limit absorption. Because of (5.2) with ν 2 replaced by z = (ν ± i1)2 , the resulting limit will automatically be a solution of (5.2).

6

The Steady-State Rayleigh Wave

In this section we derive the surface-wave component of the steady-state solution and find its asymptotics as |x| → ∞. To simplify notation we consider only Dβ vR for |β| = 0, but 44

it is obvious that an expression of exactly the same form holds for |β| > 0. With the notation of (4.8), (5.3) vR (x; z) = Σ∗ (R2 (·) − z)−1 Σf (x) −1 dξ[R(ξ)2 − z]−1 exp(ix ξ)|ξ|γ(ξ, x3 ) × = λ(R0 )(2π) R2

∞

t¯

γ(ξ, y3 )Φ2 f (ξ, y3 )dy3 ,

(6.1)

0

where z = (ν ±i1)2 , ν ∈ (ν0 −δ, ν0 +δ), ν0 −4δ > 0, 1 ∈ (0, 10 ]. Let ψ ∈ C0∞ (R), supp ψ ⊂ {|cs R0 r − ν0 | < 4δ} ⊂ (0, ∞), ψ(r) = 1 for r ∈ {|cs R0 r − ν0 | < 3δ}, ψ(r) = 0 for r ∈ {|cs R0 r − ν0 | > 4δ}. We define χ ∈ C0∞ (R2 \{0}) by χ(ξ) = χ(rq) = ψ(r), q ∈ S 1 . Then vR (x; z) = I(x; z) + J(x; z),

(6.2)

where, as in [16], formulas (3.5)-(3.8), J(x; z) = −1 = λ(R0 )(2π) dξ(R2 − z)−1 exp(ix ξ)[1 − χ(ξ)]|ξ|γ(ξ, x3 ) ×

t¯

γ(ξ, y3 )Φ2 f (ξ, y3 )dy3

= 0(|x|−2 ), 45

(6.3)

I(x; z) = −2





= (2π) λ(R0 )

dξ R2

exp(ix ξ)(R2 − z)−1 χ(ξ)|ξ|γ(ξ, x3 ) ×

R3+

γ(ξ, y3 ) exp(−iy  ξ)f (y)dy −2 = (2π) λ(R0 ) D(x, y; z)f (y)dy, t¯

(6.4)

R3+

∞ ¯ 2 2 2 −1 2 (cs R0 r − z) ψ(r)r dr γ(ξ, x3 )t γ(ξ, y3 ) × D(x, y; z) = 0

S1

exp[i(x − y  )ξ]dφ ξ = (rcosφ, rsinφ) ≡ rq, q1S 1 . Setting |ξ| = r, from (4.8) we now compute γ(ξ, x3 )t γ(x, y3 ) = (R02 − 2)2 (1 − n2 R02 )−1 π(q)t π(q) ¯

¯

exp[−(x3 + y3 )r (1 − n2 R02 )] +

(6.5)

2i(R02 − 2)(1 − n2 R02 )−1/2 π(q)t σ(q) ¯

(1 −

exp[−x3

n2 R02 )

− y3 r (1 − R02 )] −

−2i(R02 − 2)(1 − n2 R02 )−1/2 σ(q)t π(q) ¯

exp[−x3 r 46

(1 −

R02 )

− y3 r

(1 − n2 R02 )] +

4σ(q) σ(q) exp[−(x3 + y3 )r (1 − R02 )] t¯

We define φ(r) = r3/2 ψ(r)/(cs R0 r + ν ± i0) and set φ1 (r) = φ(r) exp[−(x3 + y3 )r φ2 (r) = φ(r) exp[−x3 r

(6.6)

(1 − n2 R02 )]

(1 − n2 R02 ) − y3 r (1 − R02 )]

2 φ (r) = φ(r) exp[−x3 r (1 − R0 ) − y3 r (1 − n2 R01(6.7) )] 3

4

φ (r) = φ(r) exp[−(x3 + y3 )r

(1 − R02 )].

Then D(x, y; z) of (6.4) can be written D(x, y; z) = ∞ =

r1/2 [cs R0 − (ν ± i1)]−1 dr{[(R02 − 2)2 /(1 − n2 R02 )]

0

1

φ (r)

exp[i(x − y  )rq]π(q)t π(q)dφ + ¯

(6.8)

S1

[2i(2 −

R02 )/

(1 −

n2 R02 )]φ2 (r)

exp[i(x − y  )rq]π t σdφ + ¯

S1

[2i(2 −

R02 )/

(1 − n2 R02 )]φ (r) 3

S1

47

exp[i(x − y  )rq]σ t πdφ + ¯

4

4φ (r)

exp[i(x − y  )rq]σ(q)t σ(q)dφ}. ¯

S1

We now need the principle of stationary phase [16, 19]: for large |x|, uniformly with respect to r in bounded intervals of R+ and y  in compact sets, for a smooth scalar function ψ and a smooth matrix-valued function R with x = |x |α, α = (cos φx , sin φx ) exp(irx q)ψ(r, y  , q)R(q)dφ = (2π)1/2 |rx |−1/2 × S1



{exp(ijr|x | − ijπ/4)ψ(r, y, jα)R(jα)} + q(rx ),

j=±1

|Dβ q(x )| = 0(|x |−3/2 ), |β| ≥ 0.

(6.9)

From (6.7), (6.8), (6.9), and a remainder estimate as in [11, 16] we have D(x, y; z) = = (2π)1/2 |x |−1/2



exp(ijπ/4){[R02 − 2)2 /(1 − n2 R02 )]

j=±1

π(jα)I1± (tj , ν, 1) σ(jα)I2± (tj , ν, 1)

t¯

π(jα)

t¯

π(jα)

+

[2i(R02

− 2)/ (1 − n2 R02 )]

−

[2i(R02

− 2/ (1 − n2 R02 )] (6.10)

σ(jα)t π(jα)I3± (tj , ν, 1) + 4σ(jα)t σ(jα)I4± (tj , ν, 1)} + R(x), ¯

¯

48

Ik± (tj , ν, 1) =

∞

β k (r) exp(irtj /cs R0 )[r − (ν ± i1)]−1 dr,

0

β k (r) = φk (r/cs R0 )(cs R0 )−1 , k = 1, 2, 3, 4, |R(x)| ≤ const (1 + x3 ) exp(−ax3 )|x |−1−κ , (0, 1/2)  κ arbitrarily close to 1/2, a = a(ν0 , δ) > 0, tj = j(|x | − αy  ). Using the Fourier transform in the usual way [16], we write (χ± are the characteristic functions of R± ) Ik± (tj , ν, 1) = ±i(2π)1/2

∞ χ± (tj /cs R0 − τ ) exp[i(ν ± i1) ×

−∞

(tj /cs R0 − τ )]Φ∗1 β k (τ )dτ (6.11) from which it follows that Ik± (tj , ν, 1) are bounded and continuous for x3 , y3 ∈ R3+ , ν ∈ [ν0 − δ, ν0 + δ], 1 ∈ [0, 10 ] which with (6.3), (6.4), (6.10) establishes the principle of limiting absorption for vR . Computing in the usual manner [16], we obtain for |x | large ¯ 2  −1/2 γ± (x) t γ± (y)f (y)dy+R(x) vR (x; (ν±i0) ) = c± (R0 , ν)|x |

49

γ± (x) = exp(±iν|x



|/cs R0 ){(R02

exp[−x3 ν

− 2) (1 − n2 R02 )π(±α)

(1 − n2 R02 )/cs R0 ] −

2iσ(±α) exp[−x3 ν

(6.12)

(1 − R02 )/cs R0 ]},

c± (R0 , ν) = ±i(2π)−1/2 exp(±iπ/4)ν 1/2 (cs R0 )−5/2 λ(R0 ), κ = |x |α, α = (cos φx , sin φx ), π(α) =

t

σ(α) =

t

(α1 , α2 , i (1 − n2 R02 ),

(iα1

(1 −

R02 ), iα2

(1 − R02 ), −1), 

|R(x)| ≤ const (1 + x3 ) exp(−ax3 )|x |1−κ , κ ∈ (0, 1/2) arbitrarily close to 1/2, a = a(ν) > 0. We check that [M (D) − ν 2 I]γ± (x) = 0, (6.13) B(D)γ± (x , 0) = 0, so that, in particular, [M (D) − ν 2 I]vR ∈ L2 . We thus see that in order |x |−1/2 vR (x; ν + i0) exp(−iνt) is an outgoing cylindrical wave, while vR (x; ν − i0) exp(−iνt) is an incoming cylindrical wave. They both decay exponentially away from the boundary {x3 = 0}. The leading term 50

satisfies the homogeneous equation and the boundary condition to 0(|x |−3/2 ). Finally, in formulating uniqueness classes in §8, it is essential to observe the following fact: for all α ∈ S 1 independent of x ∞ ±

γ± (x)A(α, 0)E0−1 A(D)γ± (x)dx3 = const > 0. (6.14)

t¯

0

The verification of this is simply a computation, but it requires some organization in order to make it amenable. We therefore present the proof in the appendix.

7

The Steady-State SH and PSV Waves

The component vS (x; (ν ± i0)2 ) of the steady-state solution consists of two parts. The first is an SH wave due to the incident and reflected SH modes. The second is formed by an SV wave and a P wave due to incident and reflected in SV modes and reflected P modes; this part we call the PSV wave. In this section we derive these components of the solution and determine their asymptotic behavior as |x| → ∞. To simplify notation the arguments are presented only for Dβ vS with |β| = 0. It is obvious that entirely similar expressions hold for |β| > 0. Below we shall need the principle of stationary phase in 51

the following two forms. Suppose R(q) is a smooth, matrixvalued function on the unit sphere S 2  q, x = |x|ω ∈ R3+ , y ∈ R3+ , r > 0, and ψ(r, y, q) is a smooth function. Then for large |x|, uniformly with respect to r in bounded intervals of R+ and y in compact sets,  exp(irxq)ψ(r, y, q)R(q)dSq = 2π {exp(ijr|x| − ijπ/2) s2

j=±1

ψ(r, y, jω)R(jω)|rx|−1 }

(7.1)

+q(rx), Dα q(x) = 0(|x|−2 ) and smooth (see [18]). Suppose for each ω ∈ S 2 F (ω, q) has an isolated, nondegenerate critical point qc (ω) ∈ S 2 , [qc (ω)]3 = 0, qc (ω) smooth. Let φ(q) ∈ C0∞ (S 2 ), qc ∈ supp φ, φ(qc ) = 1, supp φ ∩ {q3 = 0} = ∅, and let R(q) be a smooth matrix-valued function. Then I(r, x, y) = φ(q) exp[ir|x|F (ω, q) − iryq]R(q)dSq (7.2) S2

φ(q) exp[ir|x|F (ω, q) − iryq]R(q)|n(q)|dq1 dq2

= R2

= (2π)|F  (ω, qc )|−1/2 exp[iπsgnF  (ω, qc )/4]R(qc )|n(qc )| exp[ir|x|F (ω, qc ) − irqc ]|rx|−1 + 0(|rx|−2 ). 52

Since H(z)f (x) in (5.3) has the two representations with and Ψ− p,s , we may add them together to obtain 2H(z)f (x).

Ψ+ p,s

It is convenient to do this because of the characteristic functions of the half spaces in Ψ± p,s . Thus, from (4.3), (5.3) ∗ 2 2 −1 + 2vS (x; z) = (Ψ+ s ) (cs | · | − z) Ψs f (x) + ∗ 2 2 −1 − (Ψ− s ) (cs | · | − z) Ψs f (x) ∗ 2 2 −1 + = (Ψ+ sh ) (cs | · | − z) Ψsh f (x) + ∗ 2 2 −1 − (Ψ− sh ) (cs | · | − z) Ψsh f (x) + ∗ 2 2 −1 + (Ψ+ sv ) (cs | · | − z) Ψsv f (x) +

(7.3)

∗ 2 2 −1 − (Ψ− sv ) (cs | · | − z) Ψsv f (x)

≡ 2vsh (x; z) + 2vsv (x; z), where we have used the fact that h(q) is orthogonal to v(q) (see (2.6), (4.4)). Now let ν ∈ [ν0 − δ], ν0 + δ, ν0 − 4δ > 0,ψ ∈ C0∞ (R), supp ψ ⊂ {r : |cs r − ν0 | ≤ 4d} ⊂ (0, ∞), ψ(r) = 1 for |cs r − ν0 | ≤ 3δ, χ ∈ C0∞ (R3 \{0}), χ(η) = χ(rq) ≡ ψ(r), q ∈ η ) = Φg(η), S 2 , z = (ν ± i1)2 , 1 ∈ (0, 10 ]. Setting Φf (η) + Φf (˜ from (4.3), (5.3) after straightforward changes of the variable of integration we have −3/2 vsh (x; z) = (2π) (c2s |η|2 − z)−1 exp(ixη)h(q) ⊗ h(q)Φg(η)dη 53

≡ Ih (x; z) + Jh (x; z), −3/2 Jh (x; z) = (2π) (c2s |η|2 − z)−1 [1 − χ(η)] exp(ixη) × h(q) ⊗ h(q)Φg(η)dη

(7.4)

= 0(|x|−3+µ ), µ > 0, −3/2 (c2s |η|2 − z)−1 χ(η) exp[i(x − y)] × Ih (x; z) = (2π) R3+ R3

h(q) ⊗ h(q)g(y)dηdy −3 Dh (x, y; z)g(y)dy, = (2π) R3+

∞ 2 2 −1 2 Dh (x, y; z) = (cs r −z) ψ(r)r dr exp[ir(x−y)q]h(q)⊗h(q)dS, 0

S2

where the estimate for Jh (x; z) follows from the fact that it can be viewed as a singular integral operator [12, 16]. The principle of stationary phase (7.1) applied to the integral over S 2 for x = |x|ω, ω ∈ S 2 , in the expression for Dh gives [16, 19] exp[ir(x − y)q]h(q) ⊗ h(q)dS = (2π)|rx|−1 h(ω) ⊗ h(ω) × (7.5) S2

 j=±1

54

exp(irtj − ijπ/2) + q(rx),

q(x) = 0(|x|−2 ), tj = j|x| − jωy. Applying (7.5) to Dh , we obtain Dh (x, y; z) = (2π)|x|−1 h(ω) ⊗ h(ω)



exp(−ijπ/2) ×

j=±1

∞ [r − (ν ± i1)]−1 φ(r) exp(itj r/cs )dr 0

+0(|x|−κ−1 ), κ1(1/2, 1) arbitrarily close −1 to one, φ(r) = rc−2 s ψ(r/cs )(r + ν ± i1) .

(The remainder estimate is obtained as in [11, 16].) Using now the Fourier transform in the usual way [16], we find that Dh (x, y; ν ± i1) are bounded and continuous for x, y ∈ R3+ , ν1[ν0 − δ, ν0 + δ], 1 ∈ [0, 10 ], and for large |x| −1 Dh (x, y; ν ± i0) = 2π 2 c−2 2 |x| h(ω) ⊗ h(ω) exp[iν(±|x| ∓ yω)/cs ] +

0(|x|−1−κ ).

(7.6)

Recalling now the definition of g above and noting that y ), y˜ = Φf (˜ η ) = Φf˜(η) where supp f˜ ⊂ R3− , f˜(y) = f (˜  (y , −y3 ), from (7.4), (7.6) we have finally vsh (x; (ν ± i0)2 ) = gs± (x; ν)h(ω) ⊗ h(ω)Fh (±ω; ν) + 0(|x|−1−κ ), gs± (x; ν) = (4π|x|)−1 exp(±iν|x|/cs ), 55

Fh (ω; ν) =

c−2 s

exp(−iνωy/cs )[f (y) + f˜(y)]dy, (7.7)

x = |x|ω, ω ∈ S 2 , κ ∈ (1/2, 1) arbitrarily close to one, ν ∈ R+ . Thus, in R3+ , as expected because of the lack of coupling, in order |x|−1 vsh (x; (ν ± i0)2 ) exp(−iνt) are outgoing (+) and incoming (−) hemispherical SH waves. (Note that gs± are the Green functions for S waves; see §2.) We proceed to vsv (x; z). In the notation of (4.3) let ± πs± (x, η) = exp(ixη)v(q) − exp(ix˜ η )rss (q)v(˜ q) ± (q)θ± (q). − exp(ixηs± )rps

(7.8)

Then from (4.3), (5.3), (7.3) ∗ 2 2 −1 + vsv (x; z) = 2−1 [(Ψ+ sv ) (cs | · | − z) Ψsv f (x) + ∗ 2 2 −1 − (Ψ− sv ) (cs | · | − z) Ψsv f (x)] 1 2 (x; z) + vsv (x; z), ≡ vsv 1 (x; z) vsv

≡

1 vsv (x; z; ψ)

−1

(7.9) −3/2



= 2 (2π)

(c2s r2 − z)−1 ψ(r)×

R+ S 2

+ {χ− (q3 )πs+ (x, η) ⊗ [v(q)Φf (η) −rss (q)v(˜ q )Φf (˜ η) + (q)θ¯+ (q)Φf (¯ ηs+ )] −rps

56

− +χ+ (q)πs− (x, η) ⊗ [v(q)Φf (η) −rss (q)v(˜ q )Φf (˜ η) − (q)θ¯− (q)Φf (¯ ηs− )]}dη, −rps 2 1 (x; z) ≡ vsv (x; z; 1 − ψ). vsv

Replacing the interval |cs r − ν0 | ≤ 2δ by a circle of radius 2δ < 10 in the lower (z = (ν + i1)2 ) or upper (z = (ν − i1)2 ) half plane and extending ψ to this disk by 1, it follows that 1 (x; (ν ± i0)2 ) exists and is continuous in (x, ν), ν ∈ [ν0 − vsv 1 (x; (ν ± i1)2 ) exists and is continuous on δ, ν0 + δ]. Hence, vsv R3+ × [ν0 − δ, ν0 + δ] × [0, 10 ]. From the estimates below it 1 2 follows that vsv is also bounded. Since the integrand of vsv contains no singularity at cs r = ν ±i0, it is also bounded and continuous. In summary, vsv (x; z) is bounded and continuous on R3+ × [ν0 − δ, ν0 + δ] × [0, 10 ]. The same obviously applies to Dβ vsv (x; z) i.e., the principle of limiting absorption holds. To study the asymptotics of vsv (x; (ν ± i0)2 ), we write (7.9) as vsv (x; z) = k s (x; z) − Cs (x; z) − ms (x; z), where k s (x; z) = −1

−3/2

= 2 (2π)



(c2s |η|2 − z)−1 [χ− (q3 )πs+ (x, η) +

χ+ (q3 )πs− (x, η)] ⊗ v(q)Φf (η)dη 57

(7.10)

−1

−3/2



= 2 (2π)

η) (c2s |η|2 − z)−1 {exp(ixη)v(q) − exp(ix˜

+ − [χ− (q3 )rss (q) + χ+ (q3 )rss (q)]v(˜ q ) − [χ− (q3 ) exp(ixηs+ ) + − (q)θ+ (q) + χ+ (q3 ) exp(ixηs− )rps (q)θ− (q)]} × rps

v(q)Φf (η)dη, ≡ k1s (x; z) + k2s (x; z) + k3s (x; z), C(x; z) = −1

−3/2



= 2 (2π)

+ (q) + (c2s |η 2 − z)−1 [χ− (q3 )π + (x, η)rss

− (q)] ⊗ v(˜ q )Φf (˜ η )dη (7.11) χ− (q3 )rss −1 −3/2 + (q) + (c2s |η|2 − z)−1 {exp(ixη)[χ− (q3 )rss = 2 (2π) − + (q)]v(q) − exp(ix˜ η )[χ− (q3 )rss (q)2 + χ+ (q3 ) χ+ (q3 )rss − + + (q)2 ]v(˜ q ) − [χ− (q3 ) exp(ixηs+ )rps (q)rss (q)θ+ (q) + rss − − (q)rss (q)θ− (q)]} ⊗ v(˜ q )Φf (˜ n)dη, χ+ (q3 ) exp(ixηs− )rps

≡ C1 (x; z) + C2 (x; z) + C2 (x; z), m(x; z) = −1

−3/2

= 2 (2π)



+ (q)θ+ (q) (c2s |η|2 − z)−1 {χ− (q3 )π + (x, η) ⊗ rps

58

− Φf (ηs+ ) + χ+ (q3 )π − (x, η) ⊗ rps (q)θ− (q)Φf (ηs− )}dη −1

−3/2

= 2 (2π)



+ (q)v(q) ⊗ (c2s |η|2 − z)−1 {exp(ixη)[χ− (q3 )rps

− θ+ (q)Φf (ηs+ ) + χ+ (q3 )rps (q)v(q) ⊗ θ− (q)φf (ηs− )] − + + (q)rps (q)v(˜ q ) ⊗ θ+ (q)Φf (ηs+ ) + exp(ix˜ η )[χ− (q3 )rss − − (q)rps (q)v(˜ q ) ⊗ θ− (q)Φf (ηs− )] − χ+ (q3 )rss + (q))2 θ+ (q) ⊗ θ+ (q)Φf (ηs+ ) + [χ− (q3 ) exp(ixηs+ )(rps − (q))2 θ− (q) ⊗ θ− (q)Φf (ηs− )]}dη χ+ (q3 ) exp(ixηs− )(rps

≡ m1 (x; z) + m2 (x; z) + m3 (x; z). We first consider k s in all detail, treating k1s , k2s , k3s successively. The arguments for Cs and ms then follow one of the patterns established, and so we state only the final result. Introducing the function χ as above, we write s s (x; z) + k12 (x; z), k1s (x; z) = k11 s −1 −3/2 k12 (x; z) = 2 (2π) (c2s |η|2 − z)−1 [1 − χ(η)]

exp(ixη)v(q) ⊗ v(q)Φf (η)dη = 0(|x|−3+µ ), µ > 0,

59

s k11 (x; z)

−1

−3/2

−1

−3



= 2 (2π)

(c2s |η|2 − z)−1 χ(η)v(q) ⊗ v(q)Φf (η)dη



≡ 2 (2π)

K11 (x, y; z)f (y)dy,

(7.12)

∞ 2 2 −1 2 (cs r − z) ψ(r)r dr K11 (x, y; z) =

S2

0

exp[ir(x − y)]v(q) ⊗ v(q)dSq . Applying (7.1) to the integral over S 2 and then using the Fourier transform in the usual way [16], for x = |x|ω, |x| large K11 (x, y; (ν ± i0)2 ) = 2π 2 c2s |x|−1 v(ω) ⊗ v(ω) exp[iν(±|x| ∓ yω)/cs ] + 0(|x|−1−κ ), κ ∈ (1/2, 1) (7.13) arbitrarily close to one. From (7.10), (7.12), (7.13) k1s (x; (ν ± i0)2 ) = gs± (x; ν)v(ω) ⊗ v(ω)K1s (±ω; v) + 0(|x|−1−κ ), s −1 −2 (7.14) K1 (ω; ν) = 2 cs exp(−iνωy/cs )f (y)dy, κ ∈ (1/2, 1) arbitrarily close to one. We now proceed to k2s . With the function χ above s s (x; z) + k22 (x; z), −k2s (x; z) = k21

60

s k21 (x; z)

−1

−3/2



= 2 (2π)

+ η )χ(η)[χ− (q3 )rss (q) (c2s |η|2 − z)−1 exp(ix˜

− +χ+ (q3 )rss (q)]v(˜ q ) ⊗ v(q)Φf (η)dη,

s (x; z) = 2−1 (2π)−3/2 k22



(7.15) exp(ix˜ η )(c2s |η|2 − z)−1 [1 − χ(η)]

+ − [χ− (q3 )rss (q) + χ+ (q3 )rss (q)]v(˜ q ) ⊗ v(q)Φf (η)dη ≡ exp(ix˜ η )[g + (η; z) + g − (η; z)]dη

in an obvious notation; we have g + (ξ, 0; z) = g − (ξ, 0; z), ˜ − /∆− for q3 = 0 (see (4.3)). ˜ + /∆+ = ∆ since ∆ For x = |x|ω ∈ R3+ , ω ˜ = (ω  , −ω3 ) let L = ω ˜ · ∇η ; ˜ ]. With then for a scalar function g L∗ g(η) = −∇η [g(η)t ω 3 η = (ξ, ρ) ∈ R to simplify notation we drop the vector indices (writing, for example, g in place of gj ) and compute s k22 (x; z) =    −1 = (i|x|) [L exp(ix˜ η )]g + (η; z)dη+  3 R−



[L exp(ix˜ η )]g − (η; z)dη

R3+

61

    

= ω3 (i|x|)−1

 

exp(ix ξ)[g + (ξ, 0− ; z) − g − (ξ, 0+ , ; z)]dξ



R2

(i|x|)−1

  

exp(ix˜ η )L∗ g + (η; z) +

 3

R−

= −|x|−2

    3

R3+

  

+

   ∗ − exp(ix˜ η )L g (η; z)  

[L exp(ix˜ η )]L∗ g + (η; z)dη+

(7.16)

R−



[L exp(ix˜ η )]L∗ g − (η; z)dη

R3+

= ω3 |x|−2

  

    

exp(ix ξ)[L∗ g + (ξ, 0; x) − L∗ g − (ξ, 0; z)]dξ

R2

−

|x|−2

    3

exp(ix˜ η )(L∗ )2 g + (η; z)dη+

R−



exp(ix˜ η )(L∗ )2 g − (x; z)dη

R3+

    

.

Here the integrand in the integral over R2 is at worst 0(|ξ|−1 ) near 0, while that of the integrals over R3± are at worst 62

  

0(|η|−2 ) near 0. These integrals are thus bounded by const(f ), and k22 (x; z) is 0(|x|−2 ). From (7.15) s k21 (x; z)

−1

−3/2



= 2 (2π)

K21 (x, y; z)f (y)dy, R3

∞ (c2s r2

K21 (x, y; z) =

−1



− z) ψ(r)r dr

0

(7.17) exp(ir|x|˜ ω q − iyrq)

2

S2

+ − (q) + χ+ (q3 )rss (q)]v(˜ q ) ⊗ v(q)dSq , [χ− (q3 )rss

ω ˜ = (ω  , −ω3 ), ω3 > 0. Now let φ+ and φ− in C0∞ (S 2 ) be equal to one in small neighborhoods of the critical points ω ˜ and −˜ ω respectively, whereby suppφ± ∩ {q3 = 0} = ∅, and set φ˜ = 1 − φ+ − φ− . Identifying ± and ±1, we have + − ˜ 21 (x, y; z), (x, y; z) + K21 (x, y; z) + K K21 (x, y; z) = K21

j (x, y; z) K21

∞ = (c2s r2 − z)−1 I j (r, x, y)dr, 0

j φj (q) exp(ir|x|˜ ω q − iryq)rss (q)v(˜ q ) ⊗ v(q)dS

j

I (r, x, y) = S2

(7.18)

63

∞ ˜ 21 (x, y; z) = ˜ x, y)dr, (c2s r2 − z)−1 ψ(r)r2 I(r, K 0 ˜ x, y) = I(r, S2

+ ˜ exp(ir|x|˜ φ(q) ω q − iryq)[χ− (q3 )rss (q) +

− χ+ (q3 )rss (q)]v(˜ q ) ⊗ v(q)dS.

˜ y, j = ±1, and applying (7.1), we have Setting tj = j|x| − ij ω j (j ω ˜ ) exp(irtj )v(ω)⊗v(˜ ω )|rx|−1 +0(|rx|−2 ), I j (r, x, y) = −ij2πrss

so that from (7.16) j (x, y; z) K21

=

j −ij2πrss (j ω ˜ )v(ω)

−1

⊗ v(˜ ω )|x|

∞ [r − (ν ± i1)]−1 λ(r) 0

exp(irtj /cs )dr,

(7.19)

−2 λ(r) = rc−2 s ψ(r/cs )(r + ν ± i1) .

From this it follows in the usual way [16] that j j (x, y; (ν + ij0)2 ) = 2π 2 rss (ω)v(ω) ⊗ v(˜ ω )|x|−1 K21

(7.20)

exp[iν(j|x| − j ω ˜ y)/cs ] + 0(|x|−1−κ ), j (x, y; (ν − ij0)2 ) = 0(|x|−2 ), j = ±1, κ ∈ (1/2, 1). K21

By localizing in a neighborhood of {q3 = 0} and introducing ˜ x, y) = polar coordinates, it can be shown that in (7.18) I(r, 64

˜ 21 (x, y; z) = 0(|x|−1−κ ), κ ∈ (1/2, 1), 0(|rx|−2 ), whence K uniformly with respect to y in compact sets, ν ∈ [ν0 − δ, ν0 + δ], and 1 ∈ (0, 10 ] (see the arguments following (8.11) below). From (7.15), (7.16), (7.17), (7.20) we finally obtain for x = |x|ω, |x| large, k2s (x; (ν ± i0)2 ) = ± = −gs± (x; ν)rss (ω)v(ω) ⊗ v(˜ ω )K2s (±˜ ω ; ν) + 0(|x|−1−κ ),

K2s = K1s of (6.14), κ ∈ (1/2, 1).

(7.21)

We now consider the term k3s of (7.12): s s −k3s (x; z) = k31 (x; z) + k32 (x; z), s + (c2s |η|2 − z)−1 χ(η)[χ− (q3 ) exp(ixηs+ )rps (q)θ+ (q) + k31 (x; z) = − (q)θ− (q)] ⊗ v(q)Φf (η)dη, χ+ (q3 ) exp(ixηs− )rps

s k32 (x; z)

=

(7.22)

+ (c2s |η|2 − z)−1 [1 − χ(η)][χ− (q3 ) exp(ixηs+ )rps (q)θ+ (q)

− (q)θ− (q)] ⊗ v(q)Φf (η)dη +χ+ (q3 ) exp(ixηs− )rps

≡

exp(ixηs+ )g + (η; z)dη

R3−

+

exp(ixηs− )g − (η; z)dη,

R3+

0 = g + (ξ, 0− ; z) = g − (ξ, 0+ ; z), ± where the last line follows from rps (q) = 0 for q3 = 0∓ (see (4.3)).

65

s We proceed to estimate k32 (x; z) of (7.22). From (4.3) for  2 q = (q , q3 ) ∈ S     ω  η  ± [η 2 − (n−2 − 1)|η  |2 ], |q  | < n,  . ω · ηs =  ω  η  + iω [(n−2 − 1)|η  |2 − η 2 ], |q  | > n  3 3

In the integral (7.22) we make the change of variable η  = n ˜ η˜ , ˜ = (n−2 − 1)−1/2 , dη = n ˜ 2 d˜ η , and we then retain η3 = η˜3 , n the previous notation with η in place of η˜. Then     C± (ω, η) = n ˜ ω  η  ± ω3 [η32 − |η  |2 ], |η  | < |η3 |,  ω·ηs = .   C(ω, η) = n ˜ ω  η  + iω3 [|η  |2 − η32 ], |η  | > |η3 |  (7.23) Integration by parts is permissible in the interior (|η  | < |η3 |) and exterior (|η  | > |η3 |) of the cones C± = {±η3 = √ |η  |} = {η± (s, t) ∈ R3 : η± (s, t) = (t/ 2)(cos s, sin s, ±1), √ s ∈ (0, 2π], t > 0} with inner unit normals ν ± = (1/ 2)(− cos s, − sin s, ±1) √ and element of surface area dS = (1/ 2)tdsdt. We denote the interior of the cones C± by [C±< ] the exterior by [C±> ], and define for a scalar function g ¯ η) · ∇, L> = |∇C(ω, η)|−2 ∇C(ω, −2 L± < = |∇C± (ω, η)| ∇C± (ω, η) · ∇,

¯ η)g/|∇C(ω, η)|2 ], L∗> g = −∇ · [∇C(ω, ∗ 2 (L± < ) g = −∇ · [∇C± (ω, η)g/|∇C± (ω, η)| ].

66

(7.24)

2 ¯ Since ν ± ·∇C/|∇C| and ν ± ·∇C± /|∇C± |2 are zero on the cones ¯ η  , 0) = 0 (see (7.27), (7.28) below), dropping C± and ∂3 C(ω,

the vector indices on g, etc., we have by integration by parts in (7.22) s i|x|k32 (x; z) = [L> exp(ixηs+ )]g + (η; z)dη + > [C− ]

+ + [L+ < exp(ixηs )]g (η; z)dη + < [C− ]



[L> exp(ixηs− )]g − (η; z)dη

(7.25)

> [C+ ]



− − [L− < exp(ixηs )]g (η; z)dη

< [C+ ]



exp(ixηs+ )L∗> g + (η; z)dη +

> [C− ]

=

∗ + exp(ixηs+ )(L+ < ) g (η; z)dη +

< [C− ]



exp(ixηs− )L∗> g − (η; z)dη +

> [C+ ]

67



∗ − exp(ixηs− )(L− < ) g (η; z)dη.

< [C+ ]

A further integration by parts introduces a singularity on the cones C± from either side of them. However, Providence has designed that the singularities occur with null coefficients. Some care is needed to reveal this design. To this end we introduce new coordinates based on the cones C± : for 0 < t < ∞, s ∈ (0, 2π], u ∈ (0, t) in [C±< ] : η = η(s, t, u) = η± (s, t) + uν ± √ = (1/ 2)((t − u) cos s, (t − u) sin s, ±t ± u), √ |∂(η1 , η2 , η3 )/∂(s, t, u)| = (t − u)/ 2, η32 − |η  |2 = 2ut, |η|2 = t2 + u2

(7.26)

in [C±> ] : η = η(s, t, u) = η± (s, t) − uν ± √ = (1/ 2)((t + u) cos s, (t + u) sin s, ±t ∓ u), √ |∂(η1 , η2 , η3 )/∂(x, t, u)| = (t + u)/ 2, |η  |2 − η32 = 2ut, |η|2 = t2 + u2 . From (7.23), (7.24) with η = η(s, t, u) in the corresponding regions we compute in [C±> ] :

¯ η) = (2ut)−1/2 t (˜ ∇C(ω, nω1 (2ut)1/2 − iω3 η1 , 68

n ˜ ω2 (2ut)1/2 − iω3 η2 , iω3 η3 ) =n ˜ t (ω1 , ω2 , 0) + (2ut)−1/2 iω3 t (−η1 , −η2 , η3 ) ¯ η)M (ω, η), ¯ η)/|∇C(ω, η)|2 = 2ut∇C(ω, ∇C(ω, M (ω, η) = [2˜ n|ω  |2 ut + ω32 (t2 + u2 )]−1 , ¯ η) = −iω3 /(2ut)1/2 + iω3 (t2 + u2 )/(2ut)3/2 , ∇C(ω, ¯ η)/|∇C(ω, η)|2 = ia1 (ω, η)(2ut)−1/2 , ∇C(w, 0 < a1 (ω, η) = ω3 (t − u)2 M (ω, η), lim a1 (ω, η) = ω3−1 ,

u→0

∇C¯ · ∇|∇C|−2 = ia2 (ω, η)(2ut)−1/2 + a3 (u, t),

(7.27)

a2 (ω, η) = 4ω3 ut[n2 |ω  |2 (t2 + u2 ) + 2ω32 ut]M (ω, η) − 2ω3 (t2 + u2 )M (ω, η), lim a2 (ω, η) = −2ω3−1 ,

u→0

n[M − 2ut(˜ n|ω  |2 + ω32 )M 2 ]ω  η  , a3 (ω, η) = 2˜ 2 ¯ ) + ∇C¯ · ∇|∇C|−2 ] − (∇C¯ · ∇g± )/|∇C|2 L∗> g± = −g± [(∇C/|∇C|

= −g± [ia(ω, η)(2ut)−1/2 + a3 (ω, η)] − 2utM (ω, η)∇C¯ · ∇g± ,

69

a(ω, η) = a1 (ω, η) + a2 (ω, η), lim a(ω, η) = −ω3−1 . u→0

in [C±< ] :

∇C± (ω, η) = n ˜ t (ω1 , ω2 , 0) ± (2ut)−1/2 ω3 t (−η1 , −η2 , η3 ), ∇C± (ω, η)/|∇C± (ω, η)|2 = 2ut∇C± (ω, η)M± (ω, η), n(2ut)1/2 ω  ∓ ω3 η  |2 + ω32 η32 ]−1 , M± (ω, η) = [|˜ ∆C± (ω, η) = ∓ω3 (2ut)−1/2 ∓ ω3 (t2 + u2 )(2ut)−3/2 , −1/2 , ∆C± (ω, η)/|∇C± (ω, η)|2 = ∓b± 1 (ω, η)(2ut) 2 0 < b± 1 (ω, η) = ω3 (t + u) M± (ω, η), −1 lim b± 1 (ω, η) = ω3 ,

u→0

−1/2 ∇C± · ∇|∇C± |−2 = ±b± ± b± 2 (ω, η)(2ut) 3 (ω, η), ± 1 b± 2 , b3 ∈ C , −1 lim b± 2 = 2ω3 ,

u→0

2 2 2 b± 2 (ω, η) = 2ω3 {(t + u )M± (ω, η) ∓ 2utM± (ω, η)

[(˜ n(2ut)1/2 ω  ∓ ω3 η  )η  + ω3 η32 ]}, nM± (ω, η)[2utω3 (˜ n(2ut)1/2 |ω  |2 ∓ ω3 ω  η  ) b± 3 (ω, η) = 2˜ M± (ω, η) ∓ ω  η  ], 70

(7.28)

∗ 2 −2 (L± < ) g± = −g± [∆C± /|∇C± | + ∇C± · ∇|∇C± | ] −

(∇C± · ∇g± )/|∇C± |2 = −g± [±b± (ω, η)(2ut)−1/2 + b± 3 (ω, η)] − 2utM± (ω, η)∇C± · ∇g± , b± (ω, η)

± 1 = b± 2 (ω, η) − b1 (ω, η) ∈ C ,

lim b± (ω, η) = ω3−1 .

u→0

In (7.23) we now replace the cones C± by C±< (u0 ) = C± + u0 ν ± and C±> (u0 ) = C± − u0 ν ± and denote their interiors by [C±< (u0 )] and their exteriors by [C±> (u0 )], u0 > 0 small. Omitting the explicit dependence of η on s, t, u, we have s (x; z) = −|x|2 k32

   = lim

u0 →0   > [C− (u0 )]



L> exp(ixηs+ )L∗> g + (η; z)dη+

+ + ∗ + (L+ < ) exp(ixηs )(L< ) g (η; z)dη +

< [C− (u0 )]



L> exp(ixηs− )L∗> g − (η; z)dη +

> [C+ (u0 )]

71



− − ∗ − (L− < ) exp(ixηs )(L< ) g (η; z)dη

< [C+ (u0 )]

   =

lim

u0 →0  > C− (u0 )



    

−(ν − · ∇C/|∇C|2 ) exp(ixηs+ )L∗> g + (η;(7.29) z)dS

∗ + (ν − · ∇C− /|∇C− |2 ) exp(ixηs+ )(L+ < ) g (η; z)dS

< C− (u0 )



−(ν + · ∇C/|∇C|2 ) exp(ixηs− )L∗< g − (η; z)dS

> C+ (u0 )



∗ − (ν + · ∇C+ /|∇C+ |2 ) exp(ixηs− )(L− < ) g (η; z)dS

< C+ (u0 )

   + lim

u0 →0   > [C− (u0 )]



exp(ixηs+ )(L∗> )2 g + (η; z)dη+

∗ 2 + exp(ixηs+ )[(L+ < ) ] g (η; z)dη +

< [C− (u0 )]



exp(ixηs− )(L∗> )2 g − (η; z)dη +

> [C+ (u0 )]

72

    



∗ 2 − exp(ixηs− )[(L− < ) ] g (η; z)dη

< [C+ (u0 )]

    

.

The first limit exists and in modulus is ≤ c(f ), since, e.g., ν − · ∇C/|∇C|2 is proportional to (2ut)1/2 , while L∗> g + is proportional to (2ut)−1/2 (see (7.27)); there is thus no singularity on the cone C−> , and the surface integral is ≤ c(f ) < ∞ in modulus. Since the left side of (7.29) does not depend on u and the first limit is ≤ c(f ) < ∞, it follows that the second limit is = F (x), |F (x)| < ∞. Our task is to show that in fact |F (x)| ≤ c(f ) < ∞. This is true, since the singularity of the term (L∗ )2 g ∝ (2ut)−1 has a coefficient which vanishes at u = 0, as we now proceed to demonstrate. We consider, e.g., [C+> (u0 )] and [C+< (u0 )]; the other two regions can be treated in the same way. From (7.27), (7.28) for η ∈ [C+> (u0 )], [C+< (u0 )], respectively, [L∗> ]2 g − = = −L∗> g − [ia(ω, η)(2ut)−1/2 + a3 (ω, η)] − 2utM (ω, η)∇C− · ∇L∗> g − = g − [ia(ω, η)(2ut)−1/2 + a3 (ω, η)]2 + 2utM (ω, η)∇C¯ · ∇g − [ia(ω, η)(2ut)−1/2 + a3 (ω, η)] + 2utM (ω, η)∇C¯ · ∇ {[ia(ω, η)(2ut)−1/2 + a3 (ω, η)]g − + 2utM (ω, η)∇C¯ · ∇g − } (7.30) = −g − (η; z)a(ω, η)2 (2ut)−1 + 2iuta(ω, η)M (ω, η) 73

g − ∇C¯ · ∇(2ut)−1/2 + (2ut)1/2 h− (ω, η) + k − (ω, η) = −g − (ω, η)a(ω, η)[a(ω, η) + ω3 M (ω, η)(t2 + u2 )](2ut)−1 − n ˜ g − (η; z)a(ω, η)M (ω, η)ω  η  (2ut)−1/2 + h− (ω, η)(2ut)1/2 + k − (ω, η), h− , k − 1C 1 ∩ L1 ; ∗ 2 [(L− < ) ] g− = ∗ − − −1/2 = −(L− + b− < ) g [−b (ω, η)(2ut) 3 (ω, η)] − ∗ − 2utM− (ω, η)∇C− · ∇(L− 0. For |q  | < n we have  ∂F± /∂qi = ωi ∓ n−1 ω3 qi / (1 − n−2 |q  |2 ), i = 1, 2,  critical points of F± : qs± = (±nω  , ∓ (1 − n2 |ω  |2 )), F± (ω, qs± ) = ±n|ω  |2 ± nω3



 Hess F± (qs± ) = ∓n−1 ω3−2

(1 − |ω  |2 ) = ±n,

(1 − ω22 ) ω1 ω2

75

ω1 ω2 (1 − ω12 )

 , (7.33)



| det HessF± (qs± )| = n−1 ω3−1 ,

exp[iπsgn Hess F± (qs± )/4] = i, θ± (qs± ) = t (±ω), ± 2  2 2  2 ∆± s (qs ) = (1 − 2n |ω | ) + 4n |ω | ω3

˜ ± (q ± ) = (1 − 2n2 |ω  |2 ) − 4n2 |ω  |2 ω3 ∆ s s



(1 − n2 |ω  |2 ),

 (1 − n2 |ω  |2 ),

± ± ± ˜± rss (qs± ) = ∆ s (qs )/∆s (qs ),

 ± ± ± (qs ) = ±4n2 |ω  |(1 − 2n2 |ω  |2 ) (1 − n2 |ω  |2 )/∆± rps s (qs ), v(qs± ) = −|ω  |−1 t (ω 



(1 − n2 |ω  |2 ), |ω  |2 ).

Now let φ± ∈ C0∞ (S 2 ) with φ± = 1 in neighborhoods of qs± and suppφ± ∩ {|q  | = n} = ∅ (suppφ± depends on |ω  | < 1). We set φ˜ = 1 − φ+ − φ− ; then in an obvious notation we have from (7.22) s s s s (x; z) = k31 (x; z)+ + k31 (x; z)− + k˜31 (x; z) k31

(7.34)

It is straightforward to show that s (x; z) = 0ω (|x|−2 ). k˜31

We proceed to the interesting cases: with j = ± j s j −1 −3 (x, y; z)f (y)dy, k31 (x; z) = 2 (2π) K31 76

(7.35)

j K31 (x, y; z)

∞ j = (c2s r2 − z)−1 ψ(r)r2 I31 (r, x, y)dr,

(7.36)

0

j (r, x, y) I31

j φj (q) exp[ir|x|Fj (ω, q) − iryq]rps (q)θj (q) ⊗ v(q)dS.

= S2

From (7.2), (7.33), (7.36) j j (r, x, y) = i2πnω3 rps (qsj )(1 − n2 |ω  |2 )−1/2 exp(irtj )(jω) ⊗ v(qsj ) I31

|rx|−1 + 0ω (|rx|−2 ), tj = jn|x| − qsj y, so that j j (x, y; (ν ± i1)2 ) = ij2πnω3 rps (qsj )(1 − n2 |ω  |2 )−1/2 ω ⊗ v(qsj )|x|−1 K31

I ± (tj , ν, 1) + 0ω (|x|−1−κ ), κ ∈ (1/2, 1), (7.37) −1 where with φ(r) = c−2 s rψ(r/cs )(r + ν ± i1)

∞ I (tj , ν, 1) = [r − (ν ± i1)]−1 φ(r) exp(irtj /cs )dr. ±

0

In the usual way we now have t j /cs

exp[i(ν + i1)(tj /cs − τ )]Φ∗1 φ(τ )dτ,

I + (tj ; ν, 1) = i(2π)1/2 −∞

77

I − (tj ; ν, 1) = −i(2π)1/2

∞

exp[i(ν − i1)(tj /cs − τ )]Φ∗1 φ(τ )dτ,

tj /cs −1 exp[iν(±n|x| − qs± y)/cs ] + 0(|x|−1 ), I ± (t±1 , ν, 0) = ±i2πc−2 s ν(2ν)

I ± (t∓1 , ν, 0) = 0(|x|−1 ). Hence, ± −1 ± ± 2  2 (x, y; (ν ± i0)2 ) = −(2π)2 c−2 K31 s 2 nω3 rps (qs )(1 − n |ω | )

exp[iν(±n|x| − qs± y)/cs ]|x|−1 ω ⊗ v(qs± ) + 0ω (|x|−1−κ ),

(7.38)

∓ (x, y; (ν ± i0)2 ) = 0ω (|x|−1−κ ), κ ∈ (1/2, 1). K31

Thus, from (7.36), (7.38) k31 (x; (ν ± i0)2 )± = ± ± 2  2 −1/2 = −4−1 (2π)−1 c−2 s nω3 rps (qs )(1 − n |ω | ) −1

exp(±iν|x|/cp )|x| ω ⊗

v(qs± )

f (y)dy



exp[(−iνqs± y)/cs ] × (7.39)

± ± (qs )ω ⊗ v(qs± )K3s (ω, ν) + 0ω (|x|−1−κ ), = −gp± (x; ν)rps

78

K3s (ω, ν) = = 2−1 (cp cs )−1 (1 − n2 |ω  |2 ) exp(−iνqs± y/cs )f (y)dy, gp± (x; ν) = (4π|x|)−1 exp(±iν|x|/cp ), κ ∈ (1/2, 1), s (x; (ν ∓ i0)2 )± = 0ω (|x|−1−κ ). k31

We thus have k31 (x; (ν ± i0)2 ) = = −gp± (x; ν)rps (qs± )ω ⊗ v(qs± )K3s (ω; ν) + 0ω (|x|−1−κ ). (7.40) From (7.32), (7.40) k3 (x; (ν ± i0)2 ) = = gp± (x; ν)rps (qs± )ω ⊗ v(qs± )K3s (ω; ν) + 0ω (|x|−1−κ ). (7.41) Finally, from (7.12), (7.14), (7.21), (7.41) k s (x; (ν ± i0)2 ) = = gs± (x; ν)v(ω) ⊗ v(ω)K12 (±ω; ν) − ± (ω)v(ω) ⊗ v(˜ ω )K2s (±˜ ω ; ν) + gs± (x; ν)rss

(7.42)

gp± (x; ν)rps (qs± )ω ⊗ v(qs± )K3s (ω; ν) + 0ω (|x|−1−κ ), κ ∈ (1/2, 1). In the same manner from (7.12) we have 79

Cs (x; (ν ± i0)2 ) = ± = gs± (x; ν)rss (ω)v(ω) ⊗ v(˜ ω )Ls1 (±˜ ω ; ν) − ± (ω)2 v(ω) ⊗ v(ω)Ls2 (±ω; ν) + gs± (x; ν)rss ± ± ± ± (qs )rss (qs )ω ⊗ v(˜ qs± )Ls3 (ω; ν) gp± (x; ν)rps

+ 0(|x|−1−κ ), Ls1 (ω; ν)

=

Ls2 (ω; ν) −1

=

(7.43) 2−1 c−2 s

−1

exp(iνωy/cs )f (y)dy,  2

= 2 (cp cs ) ω3 (1 − n |ω | )

Ls3 (ω; ν)

2



exp(iνqs± y/cs )f (y)dy.

Also, m(x; (ν ± i0)2 ) = ∓ = gs± (x; ν)rps (ω)v(ω) ⊗ θ∓ (±ω)M1s (ω; ν) − ± ± (ω)rss (ω)v(ω) ⊗ θ± (±ω)M2s (ω; ν) + gs± (x; ν)rps ± ± 2 (qs ) ω ⊗ ωM3 (ω; ν) + 0ω (|x|−1−κ ),(7.44) gp± (x; ν)rps

M1s (ω; ν) M2s (ω; ν)

=

2−1 c−2 s

=

2−1 c−2 s



exp[−iνy(±ω  , nθ3∓ (ω  )/cs ]f (y)dy, exp[−iνy(±ω  , nθ3± (ω  )/cs ]f (y)dy,

M3s (ω; ν) = 2−1 (cp cs )−1 ω3 (1 − n2 |ω  |2 )−1/2 × 80

exp[∓iνωy/cp ]f (y)dy. Finally, from (7.10), (7.12), (7.42), (7.43), (7.44) ± ± vsv (x; (ν ± i0)2 ) = gs± (x; ν)v(ω)Fsv (ω; ν) + gp± (x; ν) t ωFps (ω; ν)

+ 0(|x|−1−κ ),

(7.45)

± ± and Fps can be read off from the aforewhere the values of Fsv mentioned formulas. Observe that we have written 0(|x|−1−κ ) in place of 0ω (|x|−1−κ ), since the latter is equal to a sum of bounded functions of ω. The original apparent anistropy of

the estimate occurred only because of our crude methods of analysis. In the end it disappears.

8

The Steady-State SVP Wave

The component of the solution vp (x; (ν ± i0)2 ) consists of a P wave due to the incident and reflected P modes and an SV wave due to S modes created at the boundary {x3 = 0}. We call this the SV P wave. In this section we establish the existence and asymptotics of SV P . Let ν ∈ (ν0 − δ, ν0 + δ), ν0 − 4δ > 0, ψ ∈ C0∞ (R), supp ψ ⊂ {r : |cp r − ν0 | ≤ 4δ} ⊂ (0, ∞), ψ(r) = 1 for |cp r − ν0 | ≤ 3δ, χ ∈ C0∞ (R3 \{0}), χ(η) = χ(rq) ≡ ψ(r), q ∈ S 2 , z = (ν ± i1)2 , 1 ∈ (0, 10 ]. In the notation of (4.3) let ± η )rpp (q)t q˜−exp(ixηp± )rsp (q)v ± (φ). πp± (x, ν) = exp(ixη)t q+exp(ix˜ (8.1)

81

Then from (4.3), (4.11), (5.3) ∗ 2 2 −1 + vp (x; z) = 2−1 [(Ψ+ p ) (cp | · | − z) Ψp f (x) + ∗ 2 2 −1 − (Ψ− p ) (cp | · | − z) Ψp f (x)]

(8.2)

≡ vp1 (x; z) + vp2 (x; z), vp1 (x; z)

≡

vp1 (x; z; ψ)

−1

−3/2



= 2 (2π)

(c2p r2 − z)−1 ψ(r){χ− (q3 )

R+ S 2 + πp+ (x; η) ⊗ [qΦf (η) − rpp (q)˜ q Φf (˜ η ) − rsp (q)v + (φ)Φf (ηp+ )]

q Φf (˜ η) − +χ− (q3 )πp− (x, η) ⊗ [qΦf (η) − rpp (q)˜ − rsp (q)v − (φ)Φf (ηp− )]}r2 dSdr,

vp2 (x; z) ≡ vp1 (x; z; 1 − ψ). Replacing the interval |cp r − ν0 | ≤ 2δ by a circle of radius 2δ < 10 in the lower (z = (ν + i1)2 ) or upper (z = (ν − i1)2 ) half plane and extending ψ to this half disk by one, it follows that vp1 (x; (ν ± i0)2 ) exists and is continuous in (x, ν), ν ∈ [ν0 − δ, ν0 + δ]. Hence vp1 (x; (ν ± i1)2 ) exists and is bounded and continuous on R3+ × [ν0 − δ, ν0 + δ] × [0, 10 ]. Since the integrand of vp2 (x; z) contains no singularity at cp r = ν ± i0, the same applies to it. In summary, vp (x; z) exist and is bounded and continuous on R3+ × [ν0 − δ, ν0 + δ] × [0, 10 ]. The same obviously applies to Dβ vp (x; z), i.e., the principle 82

of limiting absorption holds. To study the asymptotics of vp (x; (ν ±i0)2 ) we write (8.2) as (cf (7.9)) vp (x; z) = k p (x; z) − Cp (x; z) − mP (x; z), k p (x; z) = −1

−3/2



= 2 (2π)

(c2p |η|2 − z)−1 [χ− (q3 )πp+ (x, η) + χ+ (q3 )

πp− (x, η)] ⊗ qΦf (η)dη −1

−3/2



= 2 (2π)

η )rpp (q)˜ q (c2p |η|2 − z)−1 {exp(ixη)q − exp(ix˜

± (q)[χ− (q3 ) exp(ixηp+ )v + (φ) + χ+ (q3 ) exp(ixηp− )v − (φ)]} ⊗ −rsp

qΦf (η)dη ≡ k1p (x; z) + k2p (x; z) + k3p (x; z), Cp (x; z) = −1

−3/2

= 2 (2π)



(c2p |η|2 − z)−1 [χ− (q3 )π + (x, η) + χ+ (q3 )

(8.3)

q Φf (˜ η )dη π − (x, η)] ⊗ rpp (q)˜ −1 −3/2 η )rpp (q)˜ q (c2p |η|2 − z)−1 {exp(ixη)q − exp(ix˜ = 2 (2π) ± −rsp (q)[χ− (q3 ) exp(ixηp+ )v + (φ) + χ+ (q3 ) exp(ixηp− )v − (φ)]} ⊗

83

rpp (q)˜ q Φf (˜ η )dη ≡ Cp1 (x; z) + Cp2 (x; z) + Cp3 (x; z), mp (x; z) = −1

−3/2



= 2 (2π)

(c2p |η|2 − z)−1 {χ− (q3 )π + (x, η) ⊗ v + (φ)Φf (ηp+ ) +

χ+ (q3 )π − (x, η) ⊗ v − (φ)Φf (ηp− )}rsp (q)dη −1

−3/2

= 2 (2π)



± (qq ⊗ [χ− (q3 ) (c2p |η|2 − z)−1 {exp(ixη)rsp

± η )rsp (q) v + (φ)Φf (ηp+ ) + χ+ (q3 )v − (φ)Φf (ηp− )] − exp(ix˜

q ⊗ [χ− (q3 )v + (φ)Φf (ηp+ ) + χ+ (q3 )v − (φ)Φf (ηp− )] − rpp (q)˜ rsp (q)2 [χ− (q3 ) exp(ixηp+ )v + (φ) ⊗ v + (φ)Φf (ηp+ ) + χ+ (q3 ) exp(ixηp− )v − (φ) ⊗ v − (φ)Φf (ηp− )]}dη ≡ mp1 (x; z) + mP2 (x; z) + mp3 (x; η). As in §7, we treat k p in all detail; the computations for Cp and mp then follow one of the patterns established. From (8.3) with the functions ψ and χ introduced above we have p p (x; z) + k12 (x; z), k1p (x; z) = k11 p −1 −3/2 k12 (x; z) = 2 (2π) (c2p |η|2 − z)−1 [1 − χ(η)]

84

exp(ixη)q ⊗ qΦf (η)dη = 0(|x|−3+µ ), µ > 0, p −1 −3/2 K11 (x, y; z)f (y)dy, k11 (x; z) = 2 (2π) ∞ K11 (x, y; z) =

(8.4)

(c2p r2 − z)−1 ψ(r)r2 exp[ir(x − y)]q ⊗ qdSdr,

0 S2

so that exactly as in (7.14) k1p (x; (ν ± i0)2 ) = gp± (x; v)ω ⊗ ωK1p (±ω; ν) + 0(|x|−1−κ ), (8.5) K1p (ω; ν)

2−1 c−2 p

=

exp(−iνωy/cp )f (y)dy, κ ∈ (1/2, 1).

In the same way k2p (x; (ν ± i0)2 ) = −gp± (x; ν)rpp (ω)ω ⊗ ω ˜ K2p (±˜ ω ; ν) + 0(|x|−1−κ ), K2p = K1p , κ ∈ (1/2, 1).

(8.6)

Finally, p p −k3p (x; z) = k31 (x; z) + k32 (x; z), p (x; z) k31

≡

p k31 (x; z; ψ)

∞ (c2p r2

K31 (x, y; z) =

−1

−3



= 2 (2π)

−1

K31 (x, y; z)f (y)dy,



− z) ψ(r)r dr 2

S2

0

85

(8.7) + {rsp (q) ×

[χ− (q3 ) exp(irxn−1 φ+ − iryq)v + (φ)] − (q)[χ+ (q3 ) exp(irxn−1 φ− − iryq)v − (φ)]} ⊗ qdSq , +rsp p p (x; z) = k31 (x; z; 1 − ψ). k32

To find the critical points of the phase in K31 , we write  (8.8) G± (ω, q) = n−1 ωφ± = ω  q  ± ω3 (1 − n2 |q  |2 ),  where G± corresponds to ∓ (1 − |q  |2 ). Comparing (7.33) (F± with n → n−1 is G± ), we thus have critical points only for |ω  | < n, and these are (see also (4.5))  ∂G± /∂qi = ωi ∓ nω3 qi / (1 − n2 |q  |2 ),  qp± = (±n−1 ω  , ∓ (1 − n−2 |ω  |2 )), G± (ω, qp± ) = ±n−1 ,

| det Hess G± (ω, qp± )| = nω3−1 ,

(8.9)

exp(iπsgn Hess G± (ω, qp± )/4) = i, v ± (φ(qp± )) = v(ω),   ∆p (qp± ) = (1 − 2|ω  |2 ) + 4n|ω  | (1 − |ω  |2 ) (1 − n−2 |ω  |2 ),   ˜ p (q ± ) = (1 − 2|ω  |2 ) − 4n|ω  | (1 − |ω  |2 ) (1 − n−2 |ω  |2 ), ∆ p  ± ± (qp ) = ∓4n−1 |ω  |(1 − 2|ω  |2 ) (1 − n−2 |ω  |2 )/∆p (qp± ) rsp 86

(→ 0 as |ω  | → n), ˜ p (q ± )/∆p (q ± ), rpp (qp± ) = ∆ p p |n(qp± )| = (1 − n−2 |ω  |2 )−1/2 . Let λ± ∈ C0∞ (S 2 ), λ± = 1 in a neighborhood of qp± , supp ˜ = 1 − λ+ − λ− (supp λ± depends on λ± ∩ {q3 = 0} = ∅, λ |ω  | < n). Then from (7.2), (8.7) for |ω  | < n + − ˜ 31 (x, y; z), (x, y; z) + K31 (x, y; z) + K K31 (x, y; z) = K31 j (c2p r2 − z)−1 ψ(r)r2 I j (x, y, r)dr, K31 (x, y; z) =

j

I (x, y; z) = S2

j λj (q)rsp (q) exp[ir|x|Gj (ω, q) − irqy]v j (q) ⊗ qdSq

(8.10) j (qpj )(1 − n−2 |ω  |2 )−1/2 exp(irtj )v(ω) ⊗ qpj = i2πω3 n−1 rsp

|rx|−1 + 0ω (|rx|−2 ), tj = jn−1 |x| − yqpj . From this we have j (x, y; (ν ± i1)2 ) = i2πn−1 ω3 rsp (qpj )(1 − n−2 |ω  |2 )v(ω) ⊗ qpj |x|−1 K31

∞ [r − (ν ± i1)]−1 φ(r) exp(irtj /cp )dr 0 −1 φ(r) = c−2 p ψ(r/cp )(r + ν ± i1) ,

87

so that (j = ±) j j K31 (x, y; ν + ji0) = −j(2π)2 (cp cs )−1 ω3 rsp (qpj )(1 − n−2 |ω  |2 )−1/2

v(ω) ⊗ qpj |x|−1 exp[iν(jn−1 |x| − qpj y)/cp ] + 0ω (|x|−1−κ ),

(8.11)

uniformly with respect to y in compact sets. ˜ 31 (x, y; z) it suffices to estimate an integral To estimate K of the form

λ(q)rsp (q) exp(ixηp+ − irqy)v + (φ) ⊗ qdSq +

I(r, x, y) = {q3 0}

where λ is smooth with support in a neighborhood of {q3 = 0} which does not intersect the supports of λ± ; it is assumed to be equal to one in a smaller neighborhood. Let h± (q) ≡ h± (r, y, q) = [λ(q)rsp (q) exp(−irqy)v ± (φ) ⊗ q]ij , i, j = 1, 2, 3, h± (q  , 0∓ ) = 0 (see (3.5)). In polar coordinates 0 < θ < π, 0 < φ < 2π with G± (ω, q) of (8.8) exp(ixηp± ) = exp(ir|x|G± (ω, q)), 88

G± (ω, q) = ω1 sin θ cos φ + ω2 sin θ sin φ ± ω3 n−1



(1 − n2 sin θ),

∇q G± (ω, q) = 0 on supp λ\{q3 = 0}, lim ∇q G± (ω, q) = 0 as |q  | → 1 for |ω  | < n, L± = |∇q G± |−2 ∇q G± · ∇q = = |∇q G± |−2 [(∂θ G± )∂θ + (sin θ)−2 (∂φ G± )∂φ ], L∗± h± = −(1/ sin θ){∂θ [sin θh± |∇q G± |−2 (∂θ G± )]−∂φ [(∂φ G± )|∇q G± |−2 h± ]}, dS = sin θdθdφ. We thus have ir|x|I(x, y, r) = = h+ (q)L+ exp[ir|x|G+ (ω, q)]dS + 2 S−

h− (q)L− exp[ir|x|G− (ω, q)dS 2 S+

=

exp[ir|x|G+ (ω, q)]L∗+ h+ (q)]dS +

2 S−



exp[ir|x|G− (w, q)]L∗− h− (q)dS

2 S+

(8.12) 89

−r2 |x|2 I(x, y, r) = L∗+ h+ (q).L+ exp[ir|x|G+ (ω, q)]dS + =

(8.13)

2 S−



L∗− h− (q) · L− exp[ir|x|G− (ω, q)]dS

2 S+



|∇G+ |−2 [L∗+ h+ (q)](∂θ G+ ) exp(ir|x|G+ )|θ=π/2 dφ +

=

|∇G− |−2 [L∗− h− (q)](∂θ G− ) exp(ir|x|G− )|θ=π/2 dφ +

    

exp[ir|x|G+ (ω, q)](L∗+ )2 h+ (q)dS+

{q3 0}

    

.

Now L∗± h± (q) is bounded at θ = π/2, and ∂θ G± (ω, q) = 0 there. The term in braces is bounded by a constant c = c(ω) < ∞, |ω  | < n, uniformly with respect to r, y in compact sets. Thus from (8.7), (8.10), (8.11), (8.12) for |ω  | < n uniformly with respect to y in compact sets ˜ 31 (x, y; (ν ± i0)2 ) = 0ω (|x|−1−κ ), K

(8.14)

K31 (x, y; (ν ± i0)2 ) = ∓(2π)2 (cp cs )−1 ω3 rsp (qp± )v(ω) ⊗ qp± |x|−1 90

exp[iν(±n−1 |x| − qp± y)/cp ] + 0ω (|x|−2−κ ). For |ω  | > n the phases in the integrals of (8.7) have no ˜ 31 critical points, and K31 (x, y; z) can be estimated just as K above. Thus, finally from (8.7), (8.14) p (x; (ν ± i0)2 ) = ∓χ(0,n) (|ω  |)gs± (x; ν)rsp (qp± )v(ω) ⊗ qp± K3p (ω; ν) k31

+0ω (|x|−1−κ ), K3p (ω, ν) = 2−1 (cp cs )−1 ω3 (1−n−2 |ω  |2 )−1/2



(8.15) exp(−iνqp± y/cp )f (y)dy.

In the now familiar way p (x; z) = 0ω (|x|−2 ), ω3 = n, k32

(8.16)

uniformly with respect to ν ∈ [ν0 −δ, ν0 +δ], 1 ∈ (0, 10 ], where the condition ω3 = n enters due to our crude methods; it goes away in the end. From (8.7), (8.15), (8.16) we have for ω3 = n k3p (x; (ν ± i0)2 ) = ±χ(0,n) (|ω  |)gs± (x; ν)rsp (qp± )v(ω) ⊗ qp± K3p (ω, ν) +0ω (|x|1−κ ).

(8.17)

Thus, from (8.3), (8.5), (8.6), (8.17) ω K2p (±˜ ω ; ν)] k p (x; (ν ± i0)2 ) = gp± (x; ν)ω ⊗ [ωK1p (±ω; ν) − rpp (ω)˜ ± χ(0,n) (|ω  |)gs± (x; ν)rsp (qp± )v(ω) ⊗ qp± K3p (ω; ν) + 0ω (|x|−1−κ ), κ ∈ (1/2, 1), ω = n. 91

(8.18)

We note that from (8.9) rsp → 0 as |ω  | → n, so that the asymptotics are continuous in this sense. In a similar manner from (8.3) Cp (x; (ν ± i0)2 ) = 2 ω rpp (ω)Lp1 (±˜ ω ; ν) − ωrpp (ω)Lp2 (±ω; ν)] ± = gp± (x; ν)ω ⊗ [˜

qp± rsp (qp± )rpp (qp± ) χ(0,n) (|ω  |)gs± (x; ν)v(ω) ⊗ [˜

(8.19)

qp± ; ν)] + 0ω (|x|−1−κ ), Lp3 (˜ Lp1 = Lp2 = K1p , Lp3 = K2p , and also from (8.3) mp (x; (ν ± i0)2 ) = ∓ ± (ω)v(φ(ω))M1p (ω; ν) ∓ rsp (ω)rpp (ω) = gp± (x; ν)ω ⊗ [rsp

v(φ(ω))M2p (ω; ν)] ± 2 (qp± )v(ω) ⊗ v(ω)M3p (ω; ν) + χ(0,n) (|ω  |)gs± (x; ν)rsp

0ω (|x|−1−κ ), M1p (ω; ν) M2p (ω; ν)

=

2−1 c−2 p

=

2−1 c−2 p

(8.20)

exp[iν(∓ω  , ±n−1 φ3 (ω  ))/cp ]f (y)dy, exp[iν(∓ω  , ∓n−1 φ3 (ω  ))/cp ]f (y)dy,

M3p (ω; ν) = 2−1 (cp cs )−1 ω3 (1 − n−2 |ω  |2 )−1/2 92

exp(∓iνωy/cs )f (y)dy.

(8.21)

Note that in the third term above rsp contains the factor (1 − n−2 |ω  |2 )1/2 which cancels the factor (1 − n−2 |ω  |2 )−1/2 , so that there is actually no singularity. From (8.18)-(8.20) vp (x; (ν ± i0)2 ) = gp± (x; ν) t ω Fp± (ω; ν) + gs± (x; ν)v(ω)χ(0,n) (|ω  |) ± (ω; ν) + 0(|x|−1−κ ), Fsp

(8.22)

± can be read off where κ ∈ (1/2, 1), the values of Fp± and Fsp from (8.18)-(8.20), and we point out that we have written 0 in place of 0ω , since the latter is equal to a sum of bounded functions of ω defined for all ω ∈ S2+ .

9

The Uniqueness Theorem

The purpose of the present section is to formulate a uniqueness class for the solutions v± (x; ν) = v(x; (ν ± i0)2 ) of [M (D) − ν 2 I]v± (x; ν) = f (x) ∈ D(R3+ ), B(D)v± (x , 0; ν) = 0 constructed in §§5-8. To do so, we first recall the properties of the smooth, bounded v± (x; ν) constructed and their properties and asymptotics. We present this summary as Theorem 9.1 From (5.3), (6.12), (7.7), (7.45), (8.22) with Fj± = Fj± (ω; ν; f ) a bounded function of (ω, ν), j ∈ {sh, sv, sp, p, ps, R}, κ ∈ 93

(1/2, 1), κ ∈ (0, 1/2), a > 0 v± (x; ν) = vs± (x; ν) + vp± (x; ν) + vR± (x; ν), ◦±

◦±

vs± (x; ν) + vp± (x; ν) = v s (x; ν) + v p (x; ν) + 0(|x|−1−κ ), ◦±

± ± ± v s (x; ν) = gs± (x; ν){h(ω)Fsh + v(ω)[Fsv + χ(0,n) (|ω  |)Fsp ]}, ◦±

± v p (x; ν) = gp± (x; ν)t ω(Fp± + Fps ),

(9.1)

vR± (x; ν) = γ± (x)|x|−1/2 FR± + R± (x; ν), ± (x; ν) = (4π|x|)−1 exp(±iν|x|/cp,s ), gp,s 

|R± (x; ν)| ≤ const (1 + x3 ) exp(−ax3 )|x |−1−κ . These components have the property that [M (D) − ν 2 I][vp± (x; ν) + vs± (x; ν)] = Ψ∗p Ψp f + Ψ∗s Ψs f is orthogonal in H to [M (D) − ν 2 I]vR± (x; ν) = Σ∗ Σf . Furthermore, t

◦±

◦±

A(ω)E0−1 A(D)v p,s (x; ν) = ±iνcp,s v p,s (x; ν) + 0(|x|−2 ),

ω = x/|x|, ∞ ±

γ± (x)A(α, 0)E0−1 A(D)γ± (x)dx3 = const > 0,

t¯

0 ± (x , 0; ν) = 0. independent of x , α = x /|x |. Also, B(D)vp,s,R

94

Our task is to define classes Rν± of functions with the properties listed above and to show that any solution of [M (D) − ν 2 I]w± = 0, w± ∈ Rν± , is zero. In so doing, we shall not strive for maximum generality but rather simply get the job done. Definition 9.1 For ν > 0 w± ∈ Rν± iff ± 1 w± ∈ C 2 (R3+ , C3 ), w± = ws± + wp± + wR , [M (D) − 2 ± ± ν I](ws + wp ) ∈ H is orthogonal to [M (D) − ν 2 I]vR± , ± and B(D)ws,p,R (x , 0) = 0;

◦±

± 2 Dβ ws,p = ws,p + 0(|x|−1−κ ), κ ∈ (1/2, 1), |β| ≤ 2, ◦±

ws

◦±

wp

= gs± (x; ν)[h(ω)S1± (ω; ν) + v(ω)S2± (ω; ν)], = gp± (x; ν)t ωP ± (ω; ν),

± (x; ν) = (4π|x|)−1 exp(±iν|x|/cp,s ), and S1 , S2 , P where gp,s are bounded scalar functions of ω, ν;

± 3 Dβ wR (x; ν) = c(ν)γ± (x)|x|−1/2 + R± (x; ν), |β| ≤ 2,

γ± (x) = exp(±iν|x |/cs R0 )[γ1 (±α) exp(−a1 x3 ) + γ2 (±α) exp(−ia2 x3 )], a1 , a2 > 0, γ1 , γ2 smooth functions of α = x /|x |, 95

±

γ± (x) t A(α, 0)E0−1 A(D)γ± (x)dx3 = const > 0,

t¯



|R± (x; ν)| ≤ h(x3 )|x |−1−κ , κ ∈ (0, 1/2), h ∈ S(R+ ). Remark 9.1 It is clear that the solutions v± (x; ν) of Theorem 8.1 are contained in Rν± . Theorem 9.2 Let v ± ∈ Rv± be solutions of [M (D) − ν 2 I]v ± = f 1D(R3+ ). Then v ± are the only solutions in Rν± . We suppose that w± ∈ Rν± are solutions of [M (D) − ν 2 I]w± = 0. Then by property 1) of the class ± =0 Rν± [M (D) − ν 2 I](wp± + ws± ) = 0 and [M (D) − ν 2 I]wR individually. To simplify the notation we drop the indices ±. For 0 < A ∈ R let BA = {x ∈ R3+ : |x| ≤ A}. Denoting by dS the element of surface area of S 2 , because of the boundary condition B(D)wp,s (w , 0) = t A(n)E0−1 A(D)wp,s (x , 0) = 0, n = (0, 0, 1), we have Proof.

0 = (wp + ws ), [M (D) − ν 2 I](wp + ws )BA − [M (D) − ν 2 I](wp + ws ), (wp + ws ) t¯ (wp + ws ) t A(ω)E0−1 A(D)(wp + ws )A2 dS = −i {|x|=A}∩R3+

96



t¯ ◦

◦

◦

◦

(wp + ws ) t A(ω)E0−1 A(D)(wp + ws )A2 dS + 0(A−κ )

= −i {|x|=A}∩R3+



t¯ ◦

= ±ν

◦

◦

◦

(wp + ws )(cp wp + cs ws )A2 dS

{|x|=A}∩R3+

+0(|A|−κ ) = ±ν {cs |h(ω)|2 |S1 (ω)|2 + cs |v(ω)|2 |S2 (ω)|2 + cp |P (ω)|2 }dS + 2 S+

0(A−κ ). which implies (A → ∞) that 0 = S1 (ω) = S2 (ω) = P (ω) for a.e. ω ∈ S+2 , and so ws,p ∈ H. Further, let CAL = {x ∈ R3+ : |x | ≤ A, x3 ≤ L} : 0 = wR , [M (D) − ν 2 I]wR CAL − [M (D) − ν 2 I]wR , wR CAL =

wR (x , L)B(D)wR (x , L)

t¯

−i |x |≤A

L −i

wR t A(α, 0)E0−1 A(D)wR Adφdx3 .

t¯

0

Letting L → ∞ gives ∞ 0 =

|c(ν)|2 t γ(x) t A(α, 0)E0−1 A(D)γ(x)dx3 dφ + ¯

S1 0

97

∞



c¯(ν) t γ(x) t A(α, 0)E0−1 A(D)R(x, ν)A1/2 dφ + ¯

dx3 |x |=a

0

∞



R(x, ν)c(ν) t A(α, 0)E0−1 γ(x)A1/2 dφ +

t¯

dx3 |x |=A

0

∞



R(x, ν) t A(α, 0)E0−1 A(D)R(x, ν)Adφ.

t¯

dx3

¯

|x |=A

0

Letting A → ∞ and integrating on x3 , we have ∞ 0 = |c(ν)|

2

γ(x) t A(α, 0)E0−1 A(D)γ(x)dx3 dφ = const |c(ν)|2 .

t¯

S1 0

Hence, w± ∈ H which implies that ν 2 > 0 is an eigenvalue. Since this is impossible, w± ≡ 0. The proof is complete.

10

The Principle of Limiting Amplitude

We now show that the unique solutions of (5.2) are the limits as t → ∞ of exp(iνt)u(x, t), where u(x, t) is the solution of (5.1) with f ∈ D(R3+ ), u(x, 0) = ut (x, 0) = 0. This constitutes the principle of limiting amplitude and is the justification for the assumption often made in the applied literature that u(x, t) = exp(−iνt)v(x) where v(x) is a solution of (5.2). 98

From Theorem 4.2 and (4.22) by Duhamel’s principle   t 0 u(x, t) = [U (t − τ )F ]1 (x) exp(−iντ )dτ, F = , f 0 t [U (τ )F ]1 (x) exp(iντ )dτ =

exp(iνt)u(x, t) = 0

t 

˜ ∗ exp(ijcp | · |t)Ψ ˜ jp F (x) {Ψ jp

j=±1 0

˜ js F (x) ˜ ∗js exp(ijcs | · |t)Ψ +Ψ ˜ j F (x)}1 exp(iντ )dτ ˜ ∗j exp(ijR(·)t)Σ +Σ ≡ vp (x, t) + vs (x, t) + vr (x, t). We consider, for example, vp (x, t). It can be shown by integration by parts [16] that Ψ∗p (cp | · |−1 exp(±icp | · |t)Ψp f (x) ∈ L1 , so that by (4.22) (taking Ψp = Ψ+ p to be specific) limt→∞ vp (x, t) = −1

∞

= −i2

Ψp (x, η)(cp |η|−1 ) exp[iτ (cp |η| + ν)]Ψp f (η)dηdτ +

0 R3−

−1

∞

i2

Ψp (x, η)(cp |η|)−1 exp[iτ (−cp |η| + ν)]Ψp f (η)dηdτ =

0 R3−

99

−i2−1 lim

∞

→0 R3− 0

Ψp (x, η)(cp |η|)−1 exp[iτ (cp |η| + ν + i1)]

Ψp f (η)dτ dη + −1

i2

∞ lim

→0 R3− 0

Ψp (x, η)(cp |η|)−1 exp[iτ (−cp |η| + ν + i1)]

Ψp f (η)dτ dη = lim Ψp (x, η)[cp |η|2 − (ν ± i1)]−1 Ψp f (η)dη →0 R3−

= vp± (x; ν), where vp± (x; ν) is the function of §8. In an altogether similar way vs (x, t) → vs± (x; ν) and vR (x, t) → vR± (x; ν), which establishes the advertised principle.

100

Appendix We verify (6.14). To this end, for α ∈ S 1 we write t

t

A(α, 0)E0−1 A(D , 0) + t A(α, 0)E0−1 A(0 , D3 ),

t



A(α, 0)E0−1 A(D) =

A(α, 0)E0−1 A(D , 0) =

α1 c2p D1 + α2 c2s D2 α1 λD2 + α2 c2s D1



0

  0  α2 λD1 + α1 c2s D2 α2 c2p D2 + α1 c2s D1  0 0 c2s (α1 D1 + α2 D2 )  t

  A(α, 0)E0−1 A(0 , D3 ) =  

0

0

0

0

α1 c2s D3 α2 c2s D3

α1 λD3



  α2 λD3  .  0

We set a = ν(1 − n2 R02 )1/2 /cs R0 , b = ν(1 − R02 )1/2 /cs R0 , 

α1 c2p + α2 c2s

  A ≡ A(α) =  α1 α2 (λ + c2s )  0

101

α1 α2 (λ + c2s ) 0 α22 c2p + α12 c2s 0



  0 ,  2 cs

  , 

   B ≡ B(α) =  

0

0

0

0

α1 λ



  α2 λ  ,  0

α1 c2s α2 c2s

γ˜± (x) = exp(∓iν|x |/cs R0 )(cs R0 /ν)1/2 γ± (x). Then γ± (x)A(α, 0)E0−1 A(D)γ± (x) = ± t γ˜± (x)A˜ γ (x) + t γ˜ (x)Bλ± (x),

t¯

¯

¯

λ± (x) = i(R02 − 2)π(±α) exp(−ax3 ) +

+ 2 (1 − R02 )σ(±α) exp(−bx3 ), t¯

γ˜± (x)A˜ γ± (x)

= (R02 − 2)(1 − n2 R02 )−1 t π(±α)Aπ(±α) exp(−2axs ) + ¯

¯

2i t σ(±α)Aσ(±α) exp(−2bx3 ) + 4(R02 − 2)(1 − n2 R02 )−1/2 Im t σ(±α)Aπ(±α) exp[−(a + b)x3 ]. ¯

= (R02 − 2)2 [c2p (1 − n2 R02 )−1 + c2s ] exp(−2ax3 ) + 4[c2s + c2p (1 − R02 )] exp(−2bx3 ) − 4(2 −

R02 )(1

−n

2

R02 )−1/2 [c2p

exp[−(a + b)x3 ], 102

(1 −

R02 )

+

c2s

(1 − n2 R02 )]

t¯

γ˜± (x)Bλ± (x) =

= ±(c2s − λ)[R02 − 2)2 exp(−2axs ) + 4(1 − R02 ) exp(−2bx3 )] ± 2(R02 − 2) exp[−(a + b)x3 ]{c2s (1 − R02 ) − λ

[

(1 − R02 ) ×

(1 − n2 R02 ) + (1 − n2 R02 )−1/2 ]}.

Hence, with a ˜ = (cs R0 /ν)a, ˜b = (cs R0 /ν)b ∞ t¯ γ± (x)t A(α, 0)E0−1 A(D)γ± (x)dx3 ± 0

= (R02 − 2)2 (2˜ a)−1 [c2p (1 − n2 R02 )−1 + 2c2s − λ] +12c2s (1 − R02 )(2˜b)−1 + 2c2s ˜b−1 + 2(R02

− 2)(˜ a + ˜b)−1 {λ

(1 − R02 )[(1 − n2 R02 )−1/2

2 2 2 − (1 − n R0 )] + cs [4 (1 − R02 ) (1 − n2 R02 )−1/2 ] + 4 − R02 } ˜−1 + 6c2s (1 − R02 )˜b−1 + 2c2s ˜b−1 > 2c2s (R02 − 2)2 a − 2(2 − R02 )[2c2s + c2s (2 − Ro2 )](˜ a + ˜b)−1 > 6c2s (1 − R02 )˜b−1 − 4c2s (2 − R02 )(˜ a + ˜b)−1 + 2c2s ˜b−1 > 6c2s (1 − R02 )˜b−1 + 2c2s ˜b−1 − 2c2s ˜b−1 (2 − R02 ) = 4c2s (1 − R02 )˜b−1 103

> 0.

References [1] K. Aki and G. Richardson, Quantitative Seismology, W.H. Freeman, San Francisco (1980). [2] L.M. Brekhovskii, Waves in Layered Media, Izd. AN SSSR, Moscow (1957). [3] D. Eidus, The limiting absorption principle for the diffraction problem with two unbounded media, Comm. Math. Phys., 107, 29-38 (1986). [4] D.S. Gilliam and J.R. Schulenberger, On the structure of surface waves, Adv. Appl. Math., 4, 212-243 (1983). [5] D.S. Gilliam and J.R. Schulenberger, Electromagnetic waves in a three-dimensional half space with a dissipative boundary, J. Math. Anal., 89, No. 1, 129-185 (1982). [6] D.S. Gilliam and J.R. Schulenberger, The Propagation of Electromagnetic Waves Through, Along, and Over a Three-Dimensional Half Space, Verlag Peter Lang, Frankfurt am Main, Bern, New York (1986). [7] D.S. Gilliam and J.R. Schulenberger, Steady-state Solutions of Maxwell’s Equations Over a Three-dimensional Conducting Half Space, (to appear in Differential and Integral Equations). [8] T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, New York (1966).

104

[9] J. Keller and F. Karal, Elastiv wave propagation in homogeneous and inhomogeneous media. J. Acoustic Soc. Am., 31, No. 2 (1959). [10] V.D. Kupradze, Potential Methods in Elasticity Theory, Gozizdat. Fiz.-Mat. Lit., Moscow (1963). [11] M. Matsumura, Asymptotic behavior at infinity for Green functions of first order systems with characteristics of nonuniform multiplicity, Publ. RIMS Kyoto, 12, No. 2, 317-377 (1976). [12] S.G. Mikhlin, Multidimensional Singular Inegrals and Integral Equations, Pergamon Press, Oxford (1965). [13] J.R. Schulenberger, Boundary waves on perfect conductors, J. Math. Anal. Appl., 66, No. 3, 514-549 (1978). [14] J.R. Schulenberger, Elastic waves in the half space R2+ , J. Diff. Eq., 29, No. 3, 405-438 (1978). [15] J.R. Schulenberger, Isomorphisms of hyperbolic systems and the aether, Comm. Part. Diff. Eq., 5, 109-148 (1980). [16] J.R. Schulenberger, Time-harmonic solutions of some dissipative problems for Maxwell’s equations in a threedimensional half space, Pacific J. Math. Vol. 142, No. 2, 313-345, (1990). [17] J.R. Schulenberger and C.H. Wilcox, The limiting absorption principle and spectral theory for steady-state wave propagation in inhomogeneous, anisotropic media, Arch. Rat. Mech. Anal., 41, No. 1, 46-65 (1971).

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[18] J.R. Schulenberger and C.H. Wilcox, The singularities of the Green matrix in anisotropic wave motion, Indiana Univ. Math. J., 20, No. 12, 1093-1116 (1971). [19] B.R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Izd. Mosk. Univ. (1982). [20] C.H. Wilcox, Sound Propagation in Stratified Fluids, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo (1984). [21] K. Yosida, Functional Analysis, Springer-Verlag, BerlinHeidelberg-New York (1966). [22] A.A. Vinnik, Radiation conditions for domains with infinite boundaries, Izv. Vyssh. Uchebn. Zaved., Mat., No. 7 (182), 37-45 (1977).

106