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43(2007), 471–504

Modified Elastic Wave Equations on Riemannian and K¨ ahler Manifolds

By

YoshiyasuYasutomi

Abstract

We introduce some geometrically invariant systems of differential equations on any Riemannian manifolds and also on any K¨ahler manifolds, which are natural exten- sions of the elastic wave equations onR3. Further we prove the local decomposition theorems of distribution solutions for those systems. In particular, the solutions of our systems on K¨ahler manifolds are decomposed into 4 solutions with different prop- agation speeds.

§0 Introduction and Results

Introduction

The elastic wave equation onR3 is written as follows:

P u:=ρ∂2

∂t2u−(λ+µ)grad div u−µ∆u

=ρ∂2

∂t2u−(λ+ 2µ)grad divu+µrot rotu=f,

whereuis a 3-dimensional vector field of the displacement of an elastic body,ρ is the density constant andλ, µare the Lam´e constants. It is well-known that any distribution solutionuofP u= 0 is decomposed into a sumu=u1+u2of solutions u1, u2 satisfying the following additional equations:

rotu1= 0, divu2= 0.

Communicated by H. Okamoto. Received May 9, 2003. Revised August 17, 2005, Febru- ary 1, 2006.

2000 Mathematics Subject Classification(s): 58A10, 35L10, 32C38.

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-0041, Japan.

c 2007 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

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We call u1, u2 a longitudinal wave solution and a transverse wave solution, respectively.

The elastic wave equations on Euclidean space are well studied in the Scattering theorem and propagation problems by Kawashita [6], Shimizu [7], and so on.

Real elastic waves (earthquakes) propagate through many layers which do not necessarily lie in good order. Therefore, more general study, we extend the elastic wave equations on Euclidean space to ones on Riemannian manifolds. In a physical generalization of this system of equations to Riemannian manifolds we replace div, rot with some covariant differentiations. However covariant differentiations do not commute with each other in general. Hence we consider the new elastic wave equations which do not depend on the choice of coordi- nates, and find the natural model of hyperbolic equations with multi-values and multi-modes extended from the theory of the elastic wave equations as the model to apply the polarization theory.

To begin with, in Chapter 2, we consider the physical generalizationPorgu

=0 of the elastic wave equation on a Riemannian manifold. In Chapter 3, we introduce a new differential equationPR u= 0 which is a modification on the lower order term of the original equationPorgu= 0. Then we show that; the new differential equation admits a decomposition of any solutions into longi- tudinal wave solutions and transverse wave solutions. However, the original equation does not admit any similar decompositions in general. Moreover we generalize PR and Porg to operators on p-differential forms. In Chapter 4, we deal with the differential equations PCu= 0,PCu= 0 on complex manifolds and PK u= 0 on K¨ahler manifolds. We show any distribution solutions of the differential equations PC u= 0 andPCu= 0 admit some decompositions into 2 solutions with different propagation speeds. In the same way, we also show that any distribution solution of the equationPK u= 0 admits a decomposition into 4 solutions with 4 different propagation speeds.

Results

Definition 0.1. Let M be an n-dimensional Riemannian manifold and M = Rt×M. Let u =

uii be a contravariant vector field on M with parameter t; precisely, a contravariant vector field on M with dt, u = 0.

We assume the density constant ρ and the L`ame constants λ, µare positive.

Because the Riemannian metric tensorgij (and the inverse metric tensorgij of gij) and the covariant differentiation j are commutative on the Riemannian manifolds (cf. [2] Section 15), we define the original elastic wave equation as

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follows:

Porg ui:=ρ∂2

∂t2ui−λgijjkuk−µgikjkuj−µgjkjkui

=ρ∂2

∂t2ui−λ∇ikuk−µ∇jiuj−µ∇jjui

=ρ∂2

∂t2ui−λ∇ikuk−µ∇kiuk−µ∇kkui=fi, where we denote i=gijj according to the custom (cf. [1] Section 26).

In this paper, we often omit

by Einstein’s convention.

Because of the duality between contravariant vector fields and covariant forms, and the fact that differential operators attach covariant vector (tensor), we consider a new differential equation

Porgui: =ρ∂2

∂t2ui−λ∇ikuk−µ∇kiuk−µ∇kkui

=ρ∂2

∂t2ui(λ+ 2µ)ikuk+µ∇kiuk−µ∇kkui2µRliul

=fi,

where Rli is the Ricci tensor (cf. [1] Section 26).

When we put PR ui :=ρ∂2

∂t2ui(λ+ 2µ)ikuk+µ∇kiuk−µ∇kkui,

the differential equation PR ui = fi on M is a modification on the part of order 0 of the differential equation Porgui = fi, and we can rewrite PR u = ρ∂t22u+ (λ+ 2µ)dδu+µδdu = f for a 1-differential form u = uidxi (cf. [1]

Section 26). Here, d, δare the exterior differential operator and the associated exterior differential operator on M, respectively.

The differential operatorsd,δoperate onp-differential forms for allp, then we extend the equations naturally to equations forp-differential forms.

Let(p)

TM be a vector bundle ofp-differential forms onM. LetEM(p)be a sheaf of p-forms onM withC coefficients, andDb(Mp) a sheaf ofp-currents on M; that is, p-forms with distribution coefficients. In this article, we do not mean distributions the dual space ofC0(M). Our distributions behave as

“functions” for coordinate transformations. Further we defineEM(p) andDb(Mp). Definition 0.2. We denote byEM(p), Db(Mp)the sheaves of sections ofEM(fp), Db(fp)

M which do not include the covariant vectordt. That is, setting the projec-

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tionπ:Rt×M →M, we define EM(p):=EM(0)f

π−1EM(0)

π1EM(p), Db(Mp):=Db(0)f

M

π−1EM(0)

π1EM(p).

For

u=

1≤i1<···<ip≤n

ui1···ip(t, x)dxi1∧ · · · ∧dxip∈Db(Mp),

we define an operatorPRforDb(Mp)onM (1≤p≤n−1), where the coefficients {ui1···ip} are supposed to be alternating with respect to (i1· · ·ip).

Definition 0.3. We define sheaf-morphismsPR:Db(Mp)−→Db(Mp)by PR u:=ρ∂2

∂t2u+ (λ+ 2µ)dδu+µδdu.

Forp= 1, this equation is the covariant form ofPR ui.

When p = 0 or n, PR u= 0 reduces to a wave equation. Therefore we suppose 1≤p≤n−1.

Foru∈Db(Mp), we define equationsMR, MR1,MR2,MR0 as follows:

MR : PR u= 0, MR1 :

PR u= 0, du= 0, ⇐⇒

(∂t2+α∆)u= 0, du= 0,

MR2 :

PR u= 0, δu= 0, ⇐⇒

(∂t2+β∆)u= 0, δu= 0,

MR0 :







PR u= 0, du= 0, δu= 0,

⇐⇒







t2u= 0, du= 0, δu= 0.

Here, α = (λ+ 2µ)/ρ, β = µ/ρ and = +δd : Db(Mp) Db(Mp) is the Laplacian onM.

Further we define subsheavesSol(MR;p),Sol(MRj;p), (j= 0,1,2) ofDb(Mp) as follows: For NR=MR,MRj,

Sol(NR;p) :=

u∈Db(Mp)usatisfies NR .

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Then, we have the following theorem.

Theorem A(Theorem 3.1). For any germu∈ Sol(MR;p)

(t,x), there exist some germs uj∈Sol(MRj;p)

(t,x)(j= 1,2) such thatu=u1+u2. Furthermore, the equationu=u1+u2= 0impliesu1, u2∈ Sol(MR0;p)

(t,x). Equivalently, we have the following exact sequence:

0−→ Sol(MR0;p)−→ SF ol(MR1;p)⊕ Sol(MR2;p)−→ SG ol(MR;p)−→0,

where F(U) =U⊕(−U),G(U1⊕U2) =U1+U2.

LetXbe ann-dimensional complex manifold with a Hermitian metric, and (q,r)

TX a vector bundle of (q, r)-type differential forms onX. LetEX(q,r)be a sheaf of (q, r)-forms onX withCcoefficients, andDb(Xq,r) a sheaf of (q, r)- currents on X. SettingX =Rt×X, we also defineEX(q,r), Db(Xq,r) similarly to EM(p),Db(Mp).

Definition 0.4. We define sheaf-morphisms PC,PC :Db(Xq,r) −→Db(Xq,r) onX which are similar toPR:

PC= 2

∂t2 +α1∂ϑ+α2ϑ∂, PC= 2

∂t2+α3∂ϑ+α4ϑ∂,

where α1, α2, α3 and α4 are positive constants. Here, ∂, are the exterior differential operator, the conjugate exterior differential operator onX, andϑ, ϑare the associated operators of∂,∂, respectively.

For u Db(Xq,r), we define equations MC, MC1, MC2, MC, MC3, MC4 as follows:

MC : PC u= 0, MC1 :

PC u= 0,

∂u= 0, ⇐⇒

(∂t2+α1)u= 0,

∂u= 0, MC2 :

PC u= 0, ϑu= 0, ⇐⇒

(∂t2+α2)u= 0, ϑu= 0,

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MC : PCu= 0, MC3:

PCu= 0,

∂u= 0, ⇐⇒

(∂t2+α3)u= 0,

∂u= 0, MC4:

PCu= 0, ϑu= 0, ⇐⇒

(∂t2+α4)u= 0, ϑu= 0.

Here, = ∂ϑ+ϑ∂ and = ∂ϑ+ϑ∂ are the complex Laplace-Beltrami operators.

Further we define subsheaves Sol(MC;q, r), Sol(MCj;q, r) (j = 1,2), Sol(MC;q, r),Sol(MCk;q, r) (k= 3,4) ofDb(Xq,r)as follows: ForNC=MC,MCj, MC,MCk,

Sol(NC;q, r) :=

u∈Db(Mq,r)usatisfiesNC . Then, we get the following theorems.

Theorem B(Theorem 4.1). For any germu∈ Sol(MC;q, r)

(t,z), there exist some germs uj∈ Sol(MCj;q, r)

(t,z) (j= 1,2) such thatu=u1+u2. Theorem B (Theorem 4.2). For any germ u ∈ Sol(MC;q, r)

(t,z), there exist some germs uk ∈ Sol(MCk;q, r) (

t,z) (k = 3,4) such that u = u3+u4.

Now we assume that X is a K¨ahler manifold; that is, for the Hermitian metrich, we have the equationd

hjk(z)dzj∧dzk

= 0, and we know that hjk can be described ashjk=jkφwith a smooth real functionφlocally (cf.

[3] Chapter 1, Section 7). Then the following equations for operators onDb(Xq,r) are well-known (cf. [4] Chapter 3, Section 2):







== 12∆,

∂ϑ+ϑ∂= 0, ∂ ϑ+ϑ ∂= 0,

∂∂+∂∂= 0, ϑϑ+ϑϑ= 0.

(0.1)

As for the relationship between the conditions (0.1) and the K¨ahler condition, we give a brief introduction and a proof of the equivalency in Appendix.

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Definition 0.5. We define sheaf-morphisms PK : Db(Xq,r) −→ Db(Xq,r) on X by

PK= 2

∂t2 +α1∂ϑ+α2ϑ∂+α3∂ϑ+α4ϑ∂.

Here,α1, α2, α3 andα4are positive constants.

Whenq, r= 0 orn, PK u= 0 reduces to a wave equation. When q= 0, n or r = 0, n, PK stands for PC or PC, respectively. Therefore, we suppose 1≤q, r≤n−1.

For u∈Db(Xq,r), we define equationsMK, MKi (i = 1,2,3,4), MKjk, MKjk0 (jk) = (13),(23),(14),(24)

as follows:

MK : PK u= 0, MK1 :

PKu= 0,

∂u= 0, MK2 :

PKu= 0, ϑu= 0, MK3 :

PKu= 0,

∂u= 0, MK4 :

PKu= 0, ϑu= 0,

MK13 :







PKu= 0,

∂u= 0,

∂u= 0,

⇐⇒







t2+α1+α3

2

u= 0,

∂u= 0,

∂u= 0,

MK23 :







PKu= 0, ϑu= 0,

∂u= 0,

⇐⇒







t2+α2+α3

2

u= 0, ϑu= 0,

∂u= 0,

MK14 :







PKu= 0,

∂u= 0, ϑu= 0,

⇐⇒







t2+α1+α4

2

u= 0,

∂u= 0, ϑu= 0,

MK24 :







PKu= 0, ϑu= 0, ϑu= 0,

⇐⇒







t2+α2+α4

2

u= 0, ϑu= 0,

ϑu= 0,

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MK130:











PKu= 0,

∂u= 0,

∂u= 0,

2u= 0,

⇐⇒













t4u=2u= 0,

t2+α1+α3

2

u= 0,

∂u= 0,

∂u= 0,

MK230:











PKu= 0, ϑu= 0,

∂u= 0,

2u= 0,

⇐⇒













t4u=2u= 0,

t2+α2+α3

2

u= 0, ϑu= 0,

∂u= 0,

MK140:











PKu= 0,

∂u= 0, ϑu= 0,

2u= 0,

⇐⇒













t4u=2u= 0,

t2+α1+α4

2

u= 0,

∂u= 0, ϑu= 0,

MK240:











PKu= 0, ϑu= 0, ϑu= 0,

2u= 0,

⇐⇒













t4u=2u= 0,

t2+α2+α4

2

u= 0, ϑu= 0,

ϑu= 0.

Further we define subsheaves Sol(MK;q, r), Sol(MKi;q, r)

i= 1,2,3,4 , Sol(MKjk;q, r), Sol(MKjk0;q, r)

(jk) = (13),(23),(14),(24)

of Db(Xq,r) as the sheaves ofDb(Xq,r)-solutions, respectively.

Then, we have the following theorem.

Theorem C(Theorem 4.3). For any germu∈ Sol(MK;q, r)

(t,z), there exist some germs uij ∈ Sol(MKij;q, r)

(t,z) ((ij) = (13),(23), (14),(24)) such that u=u13+u23+u14+u24.

Further, we find thatu=u13+u23+u14+u24= 0implies ujk∈ Sol(MKjk0;q, r)

(jk) = (13),(23),(14),(24) . Equivalently, we have the following exact sequence:

0−→

(ij)

Sol(MKij0;q, r)−→G

(ij)

Sol(MKij;q, r)−→ SH ol(MK;q, r)−→0.

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Here,

(ij)

Sol(MKij0;q, r) :=

(uij)

(ij)

Sol(MKij0;q, r)

(ij)

uij= 0

,

G(U13⊕U23⊕U14⊕U24) =U13⊕U23⊕U14⊕U24,H(U13⊕U23⊕U14⊕U24) = U13+U23+U14+U24.

§1. Preparation from Riemannian Geometry

In this section, we recall some notations and terminologies in Riemannian geometry used in this paper according to [1] (Chapter 2,5), [2] (Chapter 3), and [5] (Chapter 1,4).

We assume that M is oriented. Then, there is a global section Ω of EM(n) onM, which never vanishes on M.

Definition 1.1. The inner products·,·:(1)

Tx(1)

TxM R,

·,· :(p)

Tx(p)

TxM R, are defined as follows. We choose a local positive orthonormal system (ω1, . . . , ωn) of C sections of TM concerning the Riemannian metric; that is, there is a positive valuedC functionαsuch that ω1∧ · · · ∧ωn =αΩ>0, and forωi=n

j=1aijdxj(i= 1,2, . . . , n) with a local coordinate system (x1, . . . , xn), we havegij =n

k=1akiakj. Then for

σ=

1≤i≤n

σidxi, τ =

1≤i≤n

τii ,

we define

σ, τ:=

1≤i≤n

σiτi, and for

φ=

1≤i1<···<ip≤n

φi1···ipdxi1∧ · · · ∧dxip,

ψ=

1≤i1<···<ip≤n

ψi1···ipdxi1∧ · · · ∧dxip, we define

φ, ψ:=

1≤i1<···<ip≤n

φi1···ipψi1···ip

:=

1≤i1<···<ip≤n 1≤j1<···<jp≤n

φi1···ipgi1j1· · ·gipjpψj1···jp.

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Definition 1.2. We denote byd:Db(Mp)→ Db(Mp+1) the exterior differ- ential operator which acts on Db(Mp) as a sheaf-morphism. Then the following formulas are well-known:















d(φ±ψ) =dφ±dψ (φ, ψ∈ Db(Mp)),

d(φ∧ψ) =dφ∧ψ+ (1)pφ∧dψ∈ Db(Mp), ψ∈ Db(Mq)),

d(dφ) = 0∈ Db(Mp)),

forf ∈ Db(0)M,df:= ∂f

∂xjdxj∈ Db(1)M. Here 0≤p≤n. Ifp=n,dφ= 0 holds.

Definition 1.3. The isomorphism:

TM

TM of vector bun- dle is defined as follows:













:(p)

TxM (n−p)

TxM is a linear map,

i1∧ · · · ∧ωip) =δ1···n i1···ipj1···jn−p

ωj1∧ · · · ∧ωjn−p

= (1)(i11)+···+(ip−p)ωj1∧ · · · ∧ωjn−p, for any permutation (i1, . . . , ip, j1, . . . , jn−p) of (1, . . . , n).

Here (i1· · ·ip) and (j1· · ·jn−p) are indices satisfying

(i1· · ·ipj1· · ·jn−p) is a permutation of (1· · ·n), 1≤i1<· · ·< ip≤n, 1≤j1<· · ·< jn−p≤n.

Remark. The definition above does not depend on the choice of the positive orthonormal system1, . . . , ωn}.

Proposition 1.1. We setφ, ψ∈(p)

TxM. Then we obtain











φ∧ ∗ψ= (∗φ)∧ψ=φ, ψ ω1∧ · · · ∧ωn,

1 =ω1∧ · · · ∧ωn=

g dx1∧ · · · ∧dxn,

∗φ= (1)(i11)+···+(ip−p)√g gi1j1· · ·gipjpφi1···ip dxj1∧ · · · ∧dxjn−p

(n−p)

TxM . Here g= det(gkl).

Let U ⊂M be an open subset. Let α(p) ∈ Db(Mp)(U), β(p) ∈ EM(p)(U) be sections. We suppose thatβ(p)has a compact support inU. Then the following integral is well-defined:

(p), β(p)) :=

Mα(p), β(p) ω1∧ · · · ∧ωn.

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Definition 1.4. Let α(p) ∈ Db(Mp), β(p−1) ∈ EM(p−1) be sections. We suppose β(p−1) has a compact support. Then the sheaf-morphismδ:Db(Mp) Db(Mp−1)is defined as

(δα(p), β(p−1)) = (α(p), dβ(p−1)).

Hence we have

δ= (1)n(p−1)+1∗d∗. Definition 1.5. LetXrsbe the sheaf ofr

TxM⊗s

TxM-valuedC functions, andDbrsthe sheaf ofr

TxM⊗s

TxM-valued distributions. Then, the sheaf-morphisms :XrsXrs+1, Dbrs→ Dbrs+1 are defined as follows:























fora(x)∈X00, we have ∇a(x) = ∂a

∂xjdxj, for

∂xj X10, we have

∂xj

=Γj ki

∂xi ⊗dxk, fordxjX01, we have

dxj

=−Γi kjdxi⊗dxk, fore∈Xrs, f Xrs, we have (e⊗f) = (∇e)⊗f+e⊗ ∇f.

Here,

Γi kj =gjlΓilk=gjl·1 2

∂gil

∂xk +∂glk

∂xi −∂gki

∂xl are the Riemann-Christoffel symbols.

Proposition 1.2. We set e=eji1···jr

1···isdxi1⊗ · · · ⊗dxis

∂xj1 ⊗ · · · ⊗

∂xjr Xrs. Then we have

∇e=

keji1···jr

1···is +eji1···jp−1qjp+1···jr

1···is Γq kjp−eji1···jr

1···ip−1qip+1···isΓiq

pk

×dxi1⊗ · · · ⊗dxis

∂xj1 ⊗ · · · ⊗

∂xjr ⊗dxk. Hence we call the followingthe covariant differentiation:

ke=

keji1···jr

1···is +eji1···jp−1qjp+1···jr

1···is Γq kjp−eji1···jr

1···ip−1qip+1···isΓiq

p k

×dxi1⊗ · · · ⊗dxis

∂xj1 ⊗ · · · ⊗

∂xjr.

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§2. Elastic Mechaniques on Riemannian Manifolds

Let M be an n-dimensional Riemannian manifold with metric g. We consider an elastic body G in M. A motion of G is identified with an open subsetG ofMwith one parameter family of diffeomorphismsht(·) (tR):

hs:G∩ {t= 0}−→ G∩ {t=s}.

Then the elastic wave equation for G is formulated as a time development equation for small motions ofG; that is,htis close to the identity map and it is expressed as

hit(x) =xi+ui(x, t) in a local coordinate system (x1, . . . , xn), whereu=

ui(x, t)∂i is the small displacement vector field.

Then, the differential dh of the map hand its dual map dh are given as follows:

dh:TxM ξi −→ηi=ξj ∂hi

∂xj

=ξi+ ∂ui

∂xjξj∈Th(x)M,

dh:Th(x)M ηi−→ξj=ηi ∂hi

∂xj

=ηj+ ∂ui

∂xjηi∈TxM.

Let us calculate the difference between the line element ofM ds(x)2=gij(x)dxi⊗dxj

and its pull-back ds(x)2 byh:

εij(x)dxi⊗dxj :=1 2

ds(x)2−ds(x)2

=1 2

gkl(h(x))∂hk

∂xi

∂hl

∂xjdxi⊗dxj−gij(x)dxi⊗dxj

. Here εij is called the strain tensor. By ignoring the non-linear terms of um,

kum,lum, we have εkl =1

2(gkmlum+gmlkum+ummgkl). On Riemannian spaces, equations

mgkl=mgkl−Γmln gkn−Γmkn gnl= 0

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hold (cf. [2] Section 15), so we have εkl= 1

2{gkmlum+gmlkum+ummln gkn+Γmkn gnl)} (2.1)

= 1

2{gkm(∂lum+unΓnlm) +gml(∂kum+unΓnkm)}

= 1

2(gkmlum+gmlkum).

In physics, we assume that the stress tensor σji has a linear relationship with the strain tensor at each point. Hence there exists the elastic coefficient tensorEjilk such that

(2.2) σji

√g =Ejilkεkl.

As a physical assumption forEijkl, we have the equations:

Eijkl=Ejikl=Eijlk.

In particular, it is well-known that the elastic coefficient tensor of an isotropic elastic body has the following form:

(2.3) Eijkl=λgijgkl+µgikgjl+µgilgjk,

where λ and µ are the two L´ame constants. Therefore, from (2.1), (2.2) and (2.3), we get

σji

√g =Eijklεkl (2.4)

= (λgijgkl+µgikgjl+µgilgjk) 1

2gmklum+1

2gmlkum

=λgijlul+µgjllui+µgilluj.

Hence the equation of power-balance between the stress of the elastic body and the external force is written as follows:

dfi= σji

√gdSj,

where dfi is the external force vector for the surface elementdSj.

In order to introduce the elastic wave equation, we consider a small neigh- borhood V of a point x in M, whose boundary is given by a smooth closed

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