Integr. equ. oper. theory 28 (1997) 261 - 288 0378-620X/97/030261-28 $1.50+0.20/0 9 Birkh~iuser Verlag, Basel, 1997

I IntegralEquations and OperatorTheory

INTERACTION BETWEEN THERMOELASTIC AND SCALAR OSCILLATION FIELDS

L. JENTSCH and D. NATROSHVILI

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.

Introduction Problems connected with the interaction between vector fields of different dimension have received much attention in the mathematical and engineering scientific literature and have been intensively investigated for the past years. They arise in many physical and mechanical models describing the interaction of two different media where the whole process is characterized by a vector-function of dimension k in one medium and by a vector-function of dimension n in another one (for example, fluid-structure interaction when a streamlined body is an elastic obstacle, scattering of acoustic and electromagnetic waves by an elastic obstacle, interaction between an elastic body and seismic waves, etc.). Quite many authors have considered and studied in detail the interaction between an elastic isotropic body, which occupies a bounded region ftl and where a three-dimensional elastic vector field is to be defined, and some isotropic medium, say fluid, which occupies the unbounded exterior region, the complement of f/1 with respect to the whole space, where a scalar field is to be defined. The time-harmonic dependent unknown vector and scalar fields are coupled by some kinematic and dynamic conditions on the boundary 0f~l, which lead to various type of non-classical interface problems of steady state oscillations for a piecewise homogeneous isotropic medium. An exhaustive information in this direction concerning theoretical and numerical results can be found in [1-5], [7-10], [12], [15-16], [21-22], [24],

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lentsch and Natroshviti

[32-37]. The case, where the elastic body under consideration is anisotropic, has been treated in [30]. In the present paper we generalize the results of the above cited works into two directions: first, the both interacting media are assumed to be anisotropic and. second, in a solid body occupying the bounded region fh we consider the four-dimensional thermoelastic coupled field instead of the three-dimensional elastic one. Moreover, we consider the most general interface conditions with arbitrary coefficients on the boundary 0fL which cover the natural coupling conditions of real physical fields in particular cases. We are mainly interested in the existence and uniqueness of solutions to the interface problems of steady state oscillations which involve a frequency parameter aJ. When studying these problems there arise difficulties due to the above-mentioned anisotropy property of media in question. In this case for both fields we need non-trivial analogues of the classical Sommerfeld principle and the well-known Rellich's lemma [38]. We apply the potential method developed recently in [18] for the thermoelasticity theory of anisotropic bodies. In the non-resonance case. i.e. when there are no Jones modes (see subsection 1.5), we reduce the original interface problems to equivalent systems of boundary pseudodifferential equations (ODE) on Oft1. The ellipticity, index problem and invertibility of the matrix boundary pseudodifferential operators (~DE) are studied on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies [18]. From these results follow the existence and uniqueness theorems for the original interface problems in question. The solutions are represented in the special fbrm of potential type integrals. Moreover, in the resonance case, i.e. when there exist Jones modes, we analyse the cokernels of the corresponding tgDOs and establish efficient conditions of solvability for the non-homogeneous transmission problems. In particular, we have shown that the so-called direct scattering problems are solvable for arbitrary values of the frequency parameter w.

1

Formulation

of Problems

and Uniqueness

Results

1.1. T h e r m o e l a s t i e field. Let f~l = f~+ C /i~3 be a bounded region (diamf~l < +oo) with smooth, connected, non-self-intersecting boundary S = 0fh; f12 = f~- = /~3\fll, fll = g/1 U S. Later we shall state the smoothness conditions required for S more precisely. The region ~1 is supposed to be filled up by a homogeneous anisotropic medium with the elastic coefficients ckjpq= c~q~j= cjkpq, k, j, p, q = 1,2, 3, and the density pl = c o n s t > 0. The system of equations of linear thermoelastodynamics reads ([31], ch.V)

A(D, Dt)U(x,t) = X(x,t),

(1,1)

where U = (ur,u4) T, u = (ul, u2, us) T is the displacement vector, u4 is the temperature , X = ( - F T, _ Q ) T F = (F1, F~, Fa) T is the bulk force; Q is the heat source, z = (x~ x2, z3) denotes the spatial variable, while t is the time variable; D = V = (D1, D2, Da), Dp = D ~ =

O/Oxp, D, = O/Ot; A(D, Dt)

:=

[ [C(D)- ptlaD2t]3x3 [-/3k~Dj]3xl ] , [- ToflkjDjDt]~x3 A(D) - coD~ 4x4

(1.2)

Jentsch and Natroshvili

C(D) := [ckjpqDjDq]a•

263

A ( D ) : = )b+D~Dp,

(1.3)

)tvq = Avp are heat conductivity coefficients, co is the thermal capacity, To is the temperature of the medium in the natural state, /3pq = /3qp are expressed in terms of the thermal and elastic constants; here and in what follows the summation over repeated indices is meant from 1 to 3, unless otherwise stated; the superscript T denotes transposition; for the sake of simplicity the notation [N],~x~ is also used for the m • n matrix [Nkp]mx~; I,~ : [6~j]~• stands for the unit m • m matrix (m _> 1). In the sequel without restriction of generality we consider the homogeneous version of equation (1.1) (i.e., X = 0). In the thermoelasticity theory the stress tensor {r the strain tensor {ekj} and the temperature field are related by Duhamel-Neumann law crkj = Ckjpqepq -- flkju4,

r

= 2-1(Dl~uj + Djuk),

k , j = 1,2,3.

The k-th component of the vector of thermostresses acting on a surface element with the unit normal vector n = (ni, n2, n3) is calculated by the formula

o'kjn j = CkjpctCpqnj - /3kjnju4 = ekjpqnjDqup -/~kjnju4,

k = 1, 2, 3.

(1.4)

Further we introduce the classical stress operator

T(D, n) = [T~p(D, n)]3•

: [ckjpqnjDq]ax3

(1.5)

and the thermoelastic stress operator

P(D, n) = [IT(D, Tt)]3x3, [--flkjnj]3xl]3x4.

(1.6)

Clearly, due to (1.4) we have

cr~jnj = [T(D,n)ulk-/3kjnju4 = [P(D,n)U]k,

k = 1,2,3.

If all functions involved in (1.1) are time-harmonic dependent, i.e., U(x, t) = U(x) e x p { - i ~ t } , then we get the so-called steady state oscillation equations of the theory of thermoelasticity

A ( D , - i w ) U ( x ) : 0,

(1.7)

where w > 0 is the oscillation (frequency) parameter and U(x) is a complex-valued vector function of the real variable x. The operator A(D,-ico) is defined by (1.2), where Dt is to be changed formally by -iw. From the physical considerations it follows that (see [11], [31]): a) the matrix I := [tpq]3xa is positive definite, i.e.,

i(. ( = ~ ( ~ p _> ~01(C

~0 = co~st > 0;

(1.s)

b) Ckjpqekjepq is a positive definite quadratic form in the real symmetric variables ekj = ejk, which implies the positive definiteness of the matrix C(~) for ( E /R3\{0} (see (1.3)), i.e.,

c(~)(, ( = Ck~(~)(~(k _> 511~l~I(I~, % = const > o, where ( is an arbitrary three-dimensional complex vector ( C ~3 and the upper bar denotes complex conjugate.

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Jentsch and Natroshvili

Here and throughout the paper a-b = ~r~=l akb~ denotes the scalar product of two vectors in era. We will also use the following notations (when this causes no confusion): a) if all elements of a -vector v = ( v l , " ' , vm) (matrix N = [Nkj]mx~) belong to one and the same space X, we will write v E X (N E X) instead of v C [X] ~ (N ~ [X]~• b) if K : X1 x .-- x X,~ --+ Y1 x .-. x Y~ and X1 . . . . . X , ~ , Y1 . . . . . Y~, we will write I( : X --+ Y instead of K : [X] "~ -+ [Y]L 1.2. S c a l a r field. We assume that the exterior domain f~2 is filled up with a homogeneous anisotropic medium with :density P2- Further, let some physical process;in f~2 be described by a scalar function (scalar field) w(x,t) being a solution of the homogeneous "wave equation"

A(D, Dt)w(x, t) := %qDpDqw(x, t) - p2D~w(x, t) = O, where apq = aqp are real constants defining a positive definite matrix, i.e.,

for arbitrary ~ C Ca. As in the previous subsection, if w(z, t) = w(x) exp{-iwt} is time-harmonic dependent, we get the steady oscillation equation for the complex-valued function w(x):

A(D,-iw)w(x) := a(D)w(x) + p2agw(x) = 0 with

a(D) := apqDpDq,

(1.10)

where aJ > 0 is again a frequency parameter. 1.a. R a d i a t i o n c o n d i t i o n f o r t h e s c a l a r field. In this subsection we shM1 apply results of [38] to derive Sommerfeld type radiation condition for the scalar field defined bY equation (1.10). To this end let us introduce the characteristic function corresponding to the operator el(D,-iw)

~(~,-i~)

:= - A ( - i ~ , - i ~ )

= a ~

- p~,

~c

~.

The characteristic surface (ellipsoid) in/i~ a defined by the equation

9 ~ ( ~ , - i ~ ) = a~. ~ - p ~

= 0,

~ e ~3,

(1.11)

we denote by SA. It is evident that for an arbitrary vector r / E / R 3 with Ir/l = 1 there exists on!y one point ~(r/) C S.~ where the outward unit normal vector n(~(rl)) has the same direction as r/, i.e.,

~(~(~)) = ~.

(1.12)

Note that the point -~(r/) belongs to S.a as well and n(-~(r/)) = -r/. It can be easily checked that the solution of the system of equations (1.11) and (1.12) is given by the formula

~(~) = ~ 4 N ( a

',-~)-'/~ a-%

(1.13)

Jentsch and Natroshvili

265

since the exterior unit normal of Sx at the point ~ E Sx is defined by the equation

Here 8 -t is the matrix inverse to •. Now we can define the class of functions S~(f~-) satisfying the generalized Sommerfeld type radiation conditions (see [3Sl): a function w belongs to S~(fl-) where either r = 1 or r = 2, if w E C l ( a - ) and for sufficiently large Ix[

w(39)=0(139[-1)'

0w(39)

039~ +(-1)~i4k(71)w(39)=O(1391-2)' k = 1 , 2 , 3 ,

(t.14)

where 71 = x/139] and where ~(71) E Sx corresponds to 71 (i.e., 4(71) is defined by (1.13)). Obviously, the conditions (1.14) are equivalent to the classical Sommerfeld radiation conditions for the Helmholtz equation if a(D) is the Laplace operator (see, for example, [6], [39]). In the sequel functions satisfying the conditions (1.14) will be referred to as radiating functions. Denote by 7 x ( x , w , r ) and %(39) the following fundamental solutions of the operators , 4 ( D , - i w ) and a(D), respectively: 7x(39,w,r) =

exp{(-1)~+1iav~(a-139' x)l/2} 4rclgt1112(gt-~39.39)1/2 ,

w,(39) = -[4~131i/~(a-139 9 39)1/~]-1,

r = 1,2,

~ e ~\{0},

(1.15)

(1.16)

where lal = det a. LEMMA

1.1 The fundamental solution 7x defined by (1.15) satisfies the conditions: %~(39,~,r) ~ c ~ ( ~ \ { o } ) n s~(~\{o}); in a neighbourhood of the origin

i) ii)

ID~,,/.dx,~,r) - D~%(39)1 _< ~13911-1~t, c = con~t > 0, where/3 = (/31,/32,/33) is an arbitrary multi-index and 1/31=/3~ +/32 +/33;

iii)

for s~Jycientty large Ixl

~(39-

y,~,r) :

1341 exp{(-1)"+ii(4 9 (x - y))} t- O(lxl-2), 4~(p~i~l)i/~ l~l

(i.17)

where !t varies in a bounded subset of licla and 4 = 4(rl) E SX corresponds to the direction 71 = 39/[xl; the asymptotic formula (1.17) can be differentiated any times with respect to 39 and y. Proof. The conditions i) and ii) follow directly from (1.15) and (1.16), while the item iii) is a consequence of equations (1.13), (1.15) and formulae (~--1(39 __ y). (39 -- y))l/2 = (~-lx.

X)1/2 __ (~-lx.

4(71 ) = Cd(p2)I/2(~-lx . 39)-1/2 ~-139,

39)--1/2 (~--lx. y) _[_ O(1391-1), T] = 39/1391,

~(p~)v~-(~-~(~ _ ~). (39 _ y))~/~ = (4.39) - (~. y) + o(1391-~) Now we formulate the analogue of Rellich's lemma.

266

Jentsch and Natroshvili

LEMMA

1.2 Let w e Sr(f~-) be a solution of (1.10) in ~Q- and let

lim Im f ,(D,n)w(x)w(x)dEn = 0,

R--+oo

(1.18)

J ER

where Et~ is the sphere centered at the origin and of radius R, u(D, n) = a~qn~Dq,

(1.19)

n(x) = x/]x] = rI is the exterior unit normal vector to ER at the point x. Then w(x) = 0 in ~ ' . Proof. The radiation conditions (1.14) along with (1.18) imply

nm [ Iw(x)l~(a%. ~)-l/~dr, R = 0.

(1.20)

R--+c,z , 1 ER

Note that w is a solution of the elliptic equation with constant coefficients and, therefore, it is an analytic function of the real variable x in the domain gt-. Moreover, since w C Sr(ft-), we can write the following integral representation formula (cf. [38])

w(x) = [ ~ ( ~ - y, ~, r)[~(D~, ~(V)>(~)]-d& S

- / [ - ( D ~ , n ( V ) ) ~ ( ~ - V,
9 e a-,

(1.21)

S

where the operator u(D,n) is defined by (1.19) and n is again the outward unit normal vector to S. Due to Lemma 1.1, from (1.21) we then conclude

w(x) = lxl-lb(~) e x p { ( - ! ) l + q { x } + O([x[-2),

(1.22)

for sufficiently large Ix[, where { = {(r/) E SA corresponds to r~ = x/ix I and

b({) = -[5{[[4~rw(p=]hl)l/=] -1 I f [u(Dy, n(y))w(y)]- exp{(-1)~i{y} dSy ~,S

- ( - 1 ) ~ i f u({, n(y))[w(y)]- exp{(-1)'i~y} dE v } . S

Further, (1.20) a~d (1.22) yield f Ib(~(~))I~(a-~. ,7)-~/~d~ = O, E1

whence b({) = 0. Now by (1.22) we get w(x) = O([x]-~). This in turn implies that w(x) has a compact support (see [38]) which, together with the above-mentioned analyticity of w in fl-, completes the proof. |

Jentsch and Natroshvili

267

R E M A R K 1.3 It is obvious that for the functions of the class S~(f~-) the conditions (1.i8) and (1.20) are equivalent. 1.4. F o r m u l a t i o n of i n t e r f a c e p r o b l e m s . We are now in the position to formulate the basic mathematical transmission problems corresponding to the interaction of the fields introduced above. Problem (p(~,r)), l = 1, 2, 3. Find a regular vector U E C2(~ +) N C1(~-T) and a regular radiating function w C CZ(f~ - ) N Cl(f~ -) 0 S r ( ~ - ) which are solutions of equations (1.7) and (1.10), respectively, and satisty the following coupling conditions on S = OFt• [P(D,n)U(x)]

+

~.

[A(O(D,n)u4(x)] + = [u(x). n(x)] + =

dl[w(x)]-n(x ) + f(x), f = (fl,f2,f3) r, f4(x), d2[u(D,n)w(x)]- + fb(x);

(1.23) (1.24) (1.25)

here and throughout this paper n(x) denotes the exterior unit normal vector to S at the point x E S, the symbols [-]+ denote limits on S from ~+, fj (j = 1, ..., 5) are the given functions on S; ~(1) := irl,

A(2)(D,n) = )~(D,n) := ~pqnpDq, A(a)(D,n):= d'a,~(D,n ) q-d~I1,

(1.26)

where d~- d~ r 0, and I1 stands for the identical scalar operator, while dl, d2, d~ and d~ are complex constants. Note that ~(1) corresponds to the prescribed Lemperature on S, ~(2) _ to the prescribed heat flow through the surface S, and A(3) - to a free heat exchange over the surface S. Conditions (1.23) and (1.25) in the real physical problems describe a dynamic and a kinematic coupling of the fields in question on the interface S (cf. [22], [24], [15]). A pair (U, w) constructed by the regular vector U and the radiating regular scalar function w, occurred in the above formulation, in the sequel will be referred to as a solution of Problem

(p?,T)).

1.5. U n i q u e n e s s results. We denote by J the set of values of the frequency parameter a; > 0 for which the following boundary value problem

C(D)u(x) + p,co2u(x) = O, flkjDkuj(x) = O, x E ~+, [T(D,n)u(x)] + = 0, [(u(x). n(x))] + = 0, x e S,

(1.27)

admits a nontrivial solution. Such solutions (vectors) are called Jones modes. Clearly, J is at most countable [27]. In general J ~ {0 ([21], [23]), however there exist domains for which J = ~ ([13], [23], [19]). For example, in [19] it is shown that if g/+ is a Lipschitz domain whose boundary surface oc = 0 ~ + contains two plane regions (submanifolds) with non-parallel normal vectors, then the homogeneous problem (1.27) for isotropic bodies admits only the trivial solution. THEOREM

1.4 Let S be C2-smooth, w (L J, and let

(-1)rdld2 > 0,

Re{~d~} > 0.

(1.28)

Then the homogeneous Problems (PLY#)) (1 = 1,2,3) (fj = 0,j = 1, ...5) have only the trivial solution.

268

Jentsch and Natroshvili

Pro@ Let some pair (U, w) be a solution to the homogeneous Problem (Pl(~#)). We set Bn = {x E /Ra : [z[ < R} and En = OBe, where R is supposed, to be a sufficiently large positive number such that f~+ C Be. Further, let f ~ = f~- N BR. Clearly, Ofl~ = S O ge. Since w and the components of the vector U are regular functions in the bounded domains f ~ and ~2+, respectively, we can write the following Green's formulae [18]:

f {A(D, "ico)w(z) w(x) - w(z) A(D, ~ i~ ) w ~x ~~ dx = f (u(D, n)w(x) w(x) - w(x) u(D, n)w(x)} den ER

/{[u(D,

n)w(x)]-[w(z)]- - [w(z)]-[~,(D,n)w(x)]-}dS,

S

f {[A(D,

i

--ia~)U(x)]kuk(x) -- ~ o [A(D, -~w)U(x)]4[u4(x)l}dx i

= - f {ckjpqDpuq(x)Dkuj(x)-w2lu(x)l 2- ~o~kjDku4(x)Dju4(x) ~+

Co 2 +~o0[U4(X)I }dx "OYf

{[P(D,T~)U(z)]+k[u-~] "{"

S

+ } dS,

(1.29)

where P(D, n), u(D, n) and ;k(D, n) are boundary differential operators given by (t.6), (1.19) and (1.26), respectively. Taking into account the conditions of the homogeneous Problem (Pt(~'r)) and separating imaginary parts in (1.29), we get

Im f u(D, n)w(x) w(x)dER = Im f [u(D, n)w(z)]-[w(x)]-dS, ER

S

f

~fToe AkjD~u4(x). Dju4(x)dx+

5az(Re da)

w~oo sf [[u4(x)]+12dS

S

where 5kl is Kronecker's symbol and da = These equations, due to (1.28), imply

d~/d'a.

1 a f+ w-To )~kjDku4(x)Dju4(x)dx+ (~al(Red3) w ~ sf i[u4(x)]+12ds

+d,3=Imfu(D, Tl)w(x)w(x)dER = O, En

~ = x/{x I.

(1.30)

Jentsch and Natroshvili

269

For sufficiently large R and x E 23, by (1.14) we have

D~w(x) = (-1)~i~k(rl)w(x) + O(R-2),

where as above ~(~) e S~ corresponds to '7 = z/lx[ (see (1.13)). Therefore v( D, rl)w(x) = apq~pDqw(z) = i(-1)r(arl . ~(rl))w(z) + O( R -2) = i(-1)~w(pff/~ (a-lr]. rl)-l/~w(z) + O(R-2). Consequently, taking in (1.30) the limit as R -+ +oo, we obtain if ~-To

AkjDku4(x)Dju4(x)dx +

6adReda ~ o ~ f I[u4(x)]+l~dS S

f Iw(x)l~(a-'~,7)-'/~d~

+(-])rdld2Cv (p2) 1/2 lim R-++oo ZR

O,

(1.31)

where in the left-hand side we have only non-negative terms, due to the assumptions (1.28) of the theorem and condition (1.8). As a result, we see that each summand in (1.31) is zero. Remark 1.3, Lemma 1.2 and the inequality (1.8) then yield

w(x)=O in s

u4(x)=~=const

in ft +.

(1.32)

Further, we note that datA(D,-iw)u4(x) = 0 in f~+ which readily follows from (1.7). This equation for u4(x) = ~ = const implies datA(D, -ia~)~ = (wol)3(iaJco)~= 0 and, consequently, we get fi = 0. Thus, u4(z) = 0 for x C fi+. Due to (1.32), the homogeneous Problem (Pt(~'~)) will be converted then into the problem (1.27) for the vector u, which completes the proof. | R E M A R K 1.5 From the above uniqueness results it follows that the homogeneous Problems (Pl(~'~)) admit only nontrivial solutions (% u4; w) where the exterior scalar field w and

the interior temperature field u4 vanish in f~- and f~+, respectively, while the nontrivial displacement vector u solves the boundary value problem (1.27), i.e., u is a Jones mode. In the next section we collect some auxiliary material which will be background to derive and analyse boundary integral (pseudodifferential) equations corresponding to the interface problems formulated above.

2

Thermoradiation

conditions.

Potential

type

opera-

tors 2.1. R a d i a t i o n conditions in anisotropic t h e r m o e l a s t i c i t y . When studying the solvability questions of the above interface problems by the BIE methods there arise, as an intermediate step, boundary value problems for the thermoelastic field in the exterior domain f~-. These problems for anisotropic elastic bodies were investigated quite recently in [18]. Mainly, we need the uniqueness results which are closely related to the radiation conditions at infinity (the so-called generalized Sommerfeld-Kupradze type radiation conditions

270

Jentsch and Natroshvili

[23]). To formulate these conditions let us introduce the characteristic polynomia! of the operator A( D, - i w ) M({,-ico) := det A ( - i { , - i c o ) , { e / R a, w > 0. The corresponding characteristic surface is defined by the equation M(~, - i w ) = A(~)~((, c o ) ,

icoc0r

(2,1)

= 0, ~ e ~ ,

where

A(-i~,-ico) = [| [co~p,I3-C(e)]3• L [coT0~Aj]l• ~5(~,co) := det[C(~) - co2pl/3], C(~) = [ckjpq~j~q]axa,

[i~Xj]~• 1i , -a(~) + icoc0 j4•

~)(~,co) := det[O(~)

A(~) -- Ap,~q,

- co2pl/3] ,

C(~) = C(~) + [colr0/3kj~pq~j~q]a•

Clearly, (2.1) is equivalent to the system of equations ~([,co)--0

and

~(~,co)=0

for

~ e / R a.

(2.2)

In what follows we consider

the so-called regular case (cf.[38], [27], [29]): I ~ Vr 7~ 0 at real zeros of the polynomial r I I ~ The Gaussian curvature of the surface, defined by the real zeros of the polynomial ~([,cr does not vanish anywhere. Moreover, we assume the system (2.2) (i.e. the characteristic equation [2.1)) either to be inconsistent in/i~ 3 or to define a two-dimensional manifold. From the above requirements it follows that in the case under consideration the real zeros of the system (2.2) form nonself-intersecting, closed, convex two-dimensional surfaces $1,.... S~, 1 < m < 3, enveloping the origin. Therefore, for an arbitrary unit vector r / = ~/l~l with z E /Rs \{0}, there exists exactly one point on each Sj (say ~J = ~J(r/) C Sj) at which the exterior unit normal ~(~J) to Sj has the same direction as 77. The point fJ = fJ(r/) E Sj will be referred to as the point which corresponds to r1. Note that if fJ corresponds ro 7/, then f3 E Sj and n ( - f -i i = -r]. If the system (2.2) is inconsistent i n / R a, then we have no real characteristic manifold and in this case we put m - 0. Now we can formulate the thermoradiation conditions (for details see [18]). A function v, which may be vector- or matrix-valued, belongs to the class SK~(~Q- ), r 1, 2, if it is C k s m o o t h in fl- and for sufficiently large x the following relations hold (no summation over j in the last equation):

v(~) = E?=~ <~)(~), <~) (~) = o(1~I-~), Dp(~)(x)+i(1)'~r162

p=1,2,3,

j=l,...,m,

(2.3)

where ~J C Sj corresponds to the vector 7/= ~/Ixl, Clearly, this definition is essentially related to the operator A(D,-ico) and to its characteristic equation (2.1). It can be shown that in the isotropic case equations (2.3) are equivalent to the well-known radiation conditions in the thermoelastic oscillation theory of isotropic bodies (see [23]). Therefore (2.3) are called the generalized Sornrnerfeld-Kupradze type radiation conditions in anisotropic thermoelasticity [18]. A four-dimensionM vector g = (m, ..., u4) r is said to be an (m, r)-thermo-radiating vector if it satisfies (2.3).

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271

T H E O R E M 2.1 [18] Let (-1)~+~aJ > 0 and let U be a regular (rn, r)-therrno-radiating solution of equation (1.7) in f~- satisfying the following boundary conditions on S = OfF : a) [u(x)]- = 0 or [P(D,n)U(x)]- = O, and b) [ u 4 ( x ) ] - = 0 or [A(D,n)u4(x)]-=O, where P(D, n) and t(D, n) are given by (1.6) and (1.26), respectively. Then U(x) = 0 in ~ - . 2.2. F u n d a m e n t a l solutions. Let r e ( x ) be the homogeneous (of order - 1 ) fundamental matrix of the operator C(D) (see [271, [29]), 2~

re(~) = 7~-s

= -(s~i~l) -~ ]C-l(a~)d~,

9 9 //~3\{0},

0

where G = [gkjax3 is an orthogonal matrix with the property a r x T = (0, 0, I~1)~, while r/= (cos ~, sin ~, 0)v; here )r and ? - 1 stand for the generalized Fourier transform ~nd its inverse operators, respectively, which for smooth functions are defined as follows: ?~_~[f] = f f(x)exp{ix~}dx,

~'(/~[g] = (27r) -~ f g(()exp{-ix~}d~.

By 75@) we denote the homogeneous fundamental solution of the operator A(D) (cf. (1.16)) z)l/2] -1 ,

~,A(x) = - [47rjili/2(i-'x,

Z ~ /~3\{0},

where ] = [lpq]3xa and I~1-- act A. Applying the limiting absorption principle, in [18] there have been constructed the fundamental matrices FA(X, w, r) e SK~(/Ra\{0}), r = 1, 2, for the operator A ( D , - i w ) . They have the following properties (for details see [18], Section 2): a) In a neighbourhood of the origin (Ix I < 1/2)

ID~rAkj(~,.~,r) -- D~rkj(x)l _< c~}~)(x),

c = coast > 0, k , j = 1,...,4,

(2.4)

where

r(x) = [rkj(x)]4• =

[ [re(x)]3• [o]1•

[o]3• ] vA(~) ~•

here ~(okJ)(x) = 1, ~kJ)(x) = - l n t z ] , for l _ < k , j < 3 ,

andk=j=4,

~IkJ)(x) = Ix] 1-1, I _> 2,

while

~O(Ok4)(Z) = ~9(04k)(,T) = -- in Ix[,

~}k4)(x) -~- ~}4k)(x)

= IX[ -l

l_>1,

for k = 1,2, 3; fl = (ill, f12, fla) is an arbitrary mu]ti-index. b) For sufficiently large Ix[ (and 1 _< rn <_ 3) m

rnfx - y,.~, r) = }2 cr j) Ix1-1 exp{(-1)r+li( x - Y)~J} + O(Ix[-~), j=l

(2.5)

272

Jentsch and Natroshvili

where y varies in a bounded subset of/R 3 and the asymptotic coefficients C(~) = L, , y p q j 4 x 4 do not depend on Ixl (they depend on the thermoelastic constants, the frequency parameter w and the direction, = x/Ixl); the point ~J E Sj corresponds to the direction r] = x/Ixl; the equation (2.5) can be differentiated any times with respect to x and y. Note that if m = 0, then the elements of the corresponding fundamental matrix

CA(X, w) = } ' / ~ [ A - l ( - i ~ , -iw)] of the operator A(D, - i ~ ) together with all its derivatives decrease more rapidly than any negative power of Ixl as Ix] -+ oc. Therefore, in this case any solution of the equation (1.7) in f~-, which increases at infinity no more than some polynomial, actually decrease more rapidly than any negative power of Ixl as Ixl --+ oc. Note that PA(X, ~) has again the matrix P(x) as a dominant singular part in the vicinity of the origin (Ixl < 1/2) (cf. (2.4)). In what follows we will mainly consider the case 1 < m _< 3, although all the results obtained remain valid for m = 0 as well. 2.3. P o t e n t i a l s . In what follows VN(g)(x) and WN(g)(x) denote the sing!e and double layer potentials, respectively, which are constructed by means of the fundamental solution (matrix) of the differential operator N. In particular,

v~(g)(~) = f % d z - y,~o,~)Kv)d&,

(2.6)

S

w~(g)(x)

=

f ~,(D~, ~(y))~A(x - ~, ~, ~)g(~)d&,

(27)

S

VA(h)(~) = f r ~ ( x , y,~,~)h(y)d&,

(2.S)

S

WA(h)(x) = f [Q(G,n(Y), --b)rAV(Z - y,~,r)]Th(y)dG,

(2.9)

S

where g is a scalar function, while h = (h~-, h4) r is a four-dimensional vector with h = (hi, h2, ha) T ,

[ [v(o,n)]s•

: [

[-i~V0Zkj~j]s•

t01,x3

] ] 4X4 '

T(D, n), ,(D, n) and )~(D, n) are given by (1.5), (1.19) and (1.26). Lemma 1.1 and inequalities (2.4) imply that the dominant singular parts of the potential operators of oscillations (2.6) and (2.7) are given by the potential operators of statics V~(g) and W~(g), respectively, constructed by the fundamental solution %(x):

V~(g)(x) = f"/a(Z -- y)g(y)dS~, W:(g)(x) = fu(Dy,n(y))%(y - x)g(y)dS ~. S

(2.10)

S

Analogously, the dominant singular parts of the potential operators of thermoelastic oscillations 12.8) and (2.9) are given by the corresponding static potentials (Vc(h) s, VA(h4)) s and (We(h) T, WA(h4)) r, respectively, where

v~(b(x) = f r~(~ - y)~(~)e&, S

VA(h~)(~)= f ~(~ - y)h~(y)dX~, S

(2.1~)

Jentsch and Natroshvili

273

f

(2.12)

Wc(f~)(x) = [[T(Dy, n(y))Vc(y - x)]r[z(y)dXy, S

Wh(h4)(z) = f ,~(Dy, n(y))%(y - x)h4(y)dXy.

(2.13)

S

Due to Lemma 1.1 and the relations (2.4) it is obvious that the oscillation potentials (2.6) (2.9) and their dominant static approximations (2.11)-(2.13) have the same smoothness properties and jump relations. To establish the asymptotic behaviour of the oscillation potentials at infinity we have to apply the asymptotic formulae (1.17) and (2.5). L E M M A 2.2 [I8] Let S E C k+l+~' with integer k > 0 and 0 < a < a' < 1, and let N be one of the differential operators .4, a, A, C, A. Then i) the operators VN : c k + ~

--+ c k + l + c ~ ( ~ ) ,

W N : C k + a ( S ) ---} C k + a ( - ~ ) ,

are bounded; moreover, V.a(g), WA(g) C Sr(f~-) and VA(h), WA(h) C S/(m(~-); ii) for arbitrary z E S, g C Ck+~(S) and h = (,~v h4)-c E Ck+~(S) the following jump relations hold on S:

[V~(g)(~)] •

fv~(z-y,~,r)g(y)dS~

=

=:~g(~),

k > 0,

S

[.(D., ~(z))V~(g)(z)] •

=

~:2-1g(~) + f . ( D . n(z))vA(~ - y, ~, r)g(V) d& S

=: [W~(g)(~)] •

[~:2-111 +.~(J)]g(z),

k _> o,

= i2-1g(z) + f . ( D ~ , n ( y ) ) ~ ( y - z,
[u(D., n(z))Wa(g)(z)] •

[V~(h)(z)] •

=: [+2-111 + Jc(9)]g(z), k _> o, =: s k>_l, =

frA(x-

=: nAb(z),

(2.14) > 0,

S

[B(D~, n(z))VA(h)(z)] •

=

T2- h(z) +

=:

[T2-'% +*C~)]h(z),

+

f [ g ( D ~ , n ( y ) , - i ~ ) P ~ ( z - y,~,r)]Th(y) dSy

S

[W~(h)(z)y = •

k > o,

S

[B(D~,n(z 0w~(h)(z)] •

=: [~:2-'h + ~C(~)]h(z), =: Z:Ah(z), k > l ;

~ _> 0,

here

U(D,n) =

[T(D,n)]axa

[Oh•

[t3kjW]a• ] )~(D,n)

,' 4•

(2.15)

274

Jentsch and Natroshvili iii) the operators

nN : ck+~(s) ~ ck+~+~(S), ]~N : ck+l+a(~) ~ ck+c*(~),

where N is either .4 or A, are bounded.

LEMMA 2.3 [I8], [29], [28] Let k, a, a', g, h and S be the same as in Lamina 2.2. Then: i) the dominant singular parts of the scalar operators of steady oscillations 7"IAI :f2-111 + ~(~), • + t2(~), s are the corresponding static operators defined by the equations, respectively:

nog(~)

:= [vJg)(~)] ~ = [ %(x - y)g(y)dS~,

(2.16)

S

(~=2-'I, + K~)g(z) := [u(D~,n(z))V~(g)(z)] •

7:2-'g(~) +/~,(D~,r4~))%(~

=

(::L2-1[1 -1.-I~*a)g(z )

S :-~- [ W a ( g ) ( Z ) ] 4- =

+

~og(z)

- y)g(y)dS~,

(2.1r)

:J::2-1g(z) (2.18)

f u(Dy, n(y))%(y - z)g(y)dSv, S

:= [.(D~,n(z))WJg)(zU;

(2.19)

ii) the dominant singular parts of the matrix operators of thermoelastic oscil&tions =F2-~I4 + ~(~), 7-tA, :k2-114 + ~2(2), and s are the following static operators, respectively:

,o,3x ]

[ [:F2 -lh + ,1Cc]3x3

l

[0],•

=F2-111 + t:A

[011•

•

+ ]C;

[011•

4X4

L

4X4

[011X3

,o13xl1 7/n

s

4X4

4X4

where

nc~(z)

:= [vc(~)(z)V = f rc(z - y)~(v)dS~, S

[:F 2-'h + tcc]~(z)

:= [T(D~,n(z))V~(~)(z)] ~ =

,2-,a(z) +

fr(,,,n(z))rc(zS

[+ 2 -l& + ~:~]~(z)

:= [wc(~)(z)] ~ =

+2-'~(z) + f T(D~, ~(y))r~(y - z)~(v)dSy, S

z:~(z)

:= [T(D~, n(z) )Wc(~)(z)]~;

Jentsch and Natroshvili

275

the boundary operators 7-ta, T2-111 ~- ]~A, -[-2-1[1 + ] ~ and s are defined by (2.16)-(2.19) with a and ~,(D, n) replaced by A and A(D, n), respectively; iii) n N , (~=2-1I~ + K:N), (+2-xI~ + ~ v ) and s (n = 1 for N = A , a , A , n = 3 for N = C and n = 4 for N = A) are elliptic pseudodifferential operators (gADOs) of order - 1 , O, 0 and 1, respectively, with the index zero. Moreover, ~lv and K.*y are formally mutually adjoint singular integral operators ( SIO), while --7-iN and s are formally self-adjoint non-negative operators with positive definite principal homogeneous symbols for N = C, a, A; iv) the operators 74N, 2-11~ + K.N, --2-11~ + ICN + 7-IN, s + (2-1/n + ]~N) and their adjoints have the trivial null-spaces for N = a, A and N = C. The principal homogeneous symbols (symbol matrices) of the above operators can be written explicitly in terms of curve integrals (for details see [18], Section 5).

3

Existence results

3.1. R e d u c t i o n to P I E s . We start with Problem (P@'~)). It is obvious that by virtue of (1.24) and (1.26), if we formally put d~ = 0 in the formulation of Problem (p@,r)), we get then Problem (PI(~'~)), while d~ = 0 gives Problem (P@'r)). From now on, without loss of generality, we assume that w > 0 and we look for a solution (U, w) in the form of single and double layer potentials V(x) = VA(h)(x),

x C ~+ = ~1,

w(x) = W~(g)(x) + p0V~(g)(x),

(3.1) 9 e a - = a2,

(3.2)

where h = (~n-, h4)-c, /~ = (h,, h2, ha) and g are unknown densities, and p0 = Pl --4-ip2 is a complex number with p2 r 0. We note that the single layer potential (3.1) is constructed by the fundamental matrix FA(X, a~, 1) (since the case a~ > 0 is treated), while the potentials in the representation (3.2) are constructed by means of the fundamental solution 7~t(x, w, r). Due to the interface conditions (1.23)-(1.25) and Lemma 2.2, we obtain the following system of ~DEs for the unknown densities on S: {[-2-1/4 + ]C(~)]h}k - dlnk[-2-111 + IC(~) + poT-t~4]g = fk,

k = 1,2, 3,

d~{[-2-~I4 + E(~)]h}4 + d~{TiAh}4 = f4(z), 3 ~-~{7-lAh}~nj - d~{s + P012-111 + E(~)I}g = fh. j=l Put ~ = (~-r h4, g) r and ~b = (fl, ..., fh) v, and rewrite (3.3) in the matrix form Paw = r

(3.3)

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Jentsch and Natroshvili

where

[(--2-114 -~- ]~(~))pq]3X4 =

d~"~A}4q]lx4 lgj(Ol-~A)jqllx4

[{d~}(-2-1/4 + ~(~)) +

0

[E3=1

_d2 S (1)

(3.4)

5x5

with s~O)__ __2-1/1 ~_ ]~2) _}_po'~-~4 alld

S~1)--- ~A ~- p0(2-!/I ~- K~(~))9

(3.5)

It is evident that because of Lemma 2.2, item iii) 113 possesses the mapping property 113:[Ckq-lq-c~(s)]5 ~

[ck+i+c~(s)] 4 x [Ckq-c~(s)]

(3.6)

. i , ! provided S E C k+2+~ with 0 < c~ < ~ < 1. Denote by 111 and 112 the boundary ~DOs generated by the representations in the case of Problems (1P(~'r)) and (P~(~'r)). From the above arguments it follows that 791 and 792 are represented by (3.4) with d~ = 0, d~ = 1, and d~ = 1, d~ = 0, respectively. Therefore ;02 possesses the mapping property (3.6) as well, while we have for 111

(3.i)-(3.2)

111 : [ck4"l-F~

[ckT~(S)].

5 -+ [ck+l+c~(s)] 3 X [ck+2q-~(S)] X

3.2. T w o a u x i l i a r y l e m m a t a . We present now two !emmata which will essentially be used in the study of properties of the boundary operators ?l. L E M M A 3.1 Let g E CI+~(S) with S E C 2+~', 0 < a < c~' < 1, and let w be given by formula (3.2) with Po as above. Moreover, let w vanish in f~-. Then g = 0 on S. Pro@ Due to Lemma 2.2, item ii), we have

[w(z)] + - [ w ( z ) ] -

= g(z), [v(D~,n(z))w(z)] + - [ . ( D . , n(z))w(z)]- = -pog(Z),

(3.7)

whence, by assumption of the lemma, we derive [v(D~,n(z))w(z)] + + p0[w(z)]+ = 0,

z ~ S.

(3.8)

Since w is a regular solution of equation (1.10) in f~+, by applying the divergence theorem for w and N, we obtain

f[a.J).~,D,~- p~'~lwl'l & fb,(D, s =

as.

a+

This equation together with the conditions (3.8) and (1.9) implies

4

0

Now the proof immediately follows from the condition p2 :~ 0 and equation (3.7).

II

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277

R E M A R K 3.2 In fact, we have proved that the homogeneous interior boundary value problem (1.10) and (3.8) with po = p~ + ip2, p2 ~ O, has only the trivial solution in the class of regular functions. The next lemma gives the explicit form for the equivalent pseudodifferential scalar lifting operators in the case of an arbitrary surface S. L E M M A a.a [28], [19] Let

7e(_~)g(z)_ 2~

g(y)dS~,

(3.9)

S 1

3 [rj(D~,

"7"~.(l)g(z) ~ ~--~Sf j~l +

{

n(z), ~)IZ-Z--~] [7j(ay,n(y),~)g(y)]dSy

,/o,

- g ( z ) + ~ ~ O~(y~ Iz - y~ g(y)dS~

}

, z e S,

(3.1o)

where O/On denotes the usual directional normal derivative and rj( Dz , n( z ), S), j = 1, 2, 3, stand for the directional derivatives in the tangent plane to S at the point z : 7"1 = n2Da - naD2, r2 = naD1 - nlD3, ra = nlD2 - n2D1.

Then i) the operators 7r : Ck+I+~(S) --+ Ck+I-'~+~(S) with S E C k+2+~' and their formal adjoint ones are isomorphisms between Ck+l+~(oc) and Ck+l-m+~(S); ii) the principal homogeneous symbol of the operator T~(,~) at any point z E S is I~1"~, where ~ E 1t~2 \ {0}; moreover, ~(-1) is formally seIf-adjoint; iii) ~(,~) is an equivalent lifting pseudodifferential operator of order m on S. 3.3. P r o p e r t i e s of operators Pt. Here we will analyse the solvability of the boundary pseudodiferential equation Pz~ = ~

(3.11)

which corresponds to Problem (Pz(~'r)) due to the representations (3.1) and (3.2). Throughout the subsection we assume that the conditions S E C 2+k+~', 0 < a < a ' < l ,

k_>0,

w~

or,

(3.12)

are fulfilled. L E M M A 3.4 The homogeneous version of equation (3.11) (r = 0) has only the trivial solution in the class [C~(S)] 4 x [CI+~(S)]. Proof. Let ~ = (hq-,h4,g) -c E [Oct(S)] 4 x [C14-c~(S)] be some solution of the homogeneous equation (3.11) (~b = 0). Then the pair (U,w), constructed by the solution ~ via the representations (3.1) and (3.2), is a regular solution to the homogeneous Problem (p[~,,r)). Therefore, by the uniqueness Theorem 1.4, we deduce U(x) = VA(h)(x) = 0 in f~+ and

278

lentsch and Natroshvili

w(x) w a(g)(x) + poV.a(g)(x) = 0 in 9-. By Lemma 3.1 in view of the last equation we derive g = 0 on S. Further, we have U(x) = VA(h)(x) e Cl-]-c~(~-) N S/%z?(a,) and [~ZA(/Lt)]"= 0 o i l s (due to Lemma 2.2, item ii). Theorem 2.1 then implies VA(h)(x) = 0 in f~-, and, eonsequentl:/, h=0onS. Now let us introduce the following diagonal matrix operators =

[/313x3 [0]axl [0]axl ] Q1 ~---~ [011• ~(1) 0 [0]1•

0

"]~'(- 1)

[ ,

]

[oh•

[h]4x4 | , [0]1• 74(-i) ] ~x5

O:=Oa=

(3.13)

5X5

with T{(.~)defined by (3.9) and (3.10). From (3.4) and (a.13) we have:

]

[(--2-1/4 + ]~(~))pq]3X4 [--dlnk Si~ QlPl =

[StY(l)~'~A4q]1x4 [7~(_~)E~=~ nj(nA)j~]~•

Qfl~z :

[(d~(-2-~I4 -4- ~(~)) + d~gr~A)4v]lx4

(3.14)

0 J , -<7~(_1)Sl ~) ~

[n(,1) 3Ej--I nJ(~-~A)jq]lx4

0

]

,

(3.15)

-dz'~(-1)S~ 1) 5x5

where 1 : 2,3 (d~ : 1 and dg = 0 for I : 2); here $(0) and $(1) are defined by (3.5). L E M M A 3.5 Operators @P~ (1 = 1,2,3) are singula r integral operators of normal type with the index equal to zero. Moreover, i) equations Q~P~ = Qlr and Plgo = • are equivalent under the following assumptions on ~: e [c~+~(s)] ~ • [c~+~+%s)] • [c~+%s)] for ~ : t, e [Ck+~(S)] ~ for l

=

2, 3;

ii) null-spaces of the operators Q{Pt are trivial. Pro@ The fact that QzT~z(l = 1, 2, 3) are singular integral operators (i.e., qDOs of order 0) is a ready consequence of Lemma 3.a and formulae (3.13)-(3.15). For a scalar (matrix) pseudodifferential operator N" on S we denote by O-(H)(z, ~) the corresponding principal homogeneous symbol (symbol matrix) with z E S and { 6 /R2 \ {0}. Further, we show that Oz#l are elliptic qDOs, that is, their principal (homogeneous of order 0) symbol matrices are not singular.

Jentsch and Natroshvili

279

Applying Lemmata 2.3 and 3.3, equations (3.14) and (3.15), and the theorem on a principal symbol of composition of ~DOs [14], we arrive at the formulae

0"(Q1~1) = [{0-(QlPl)}kJ]sxs [0"(--2-1/3 -I'- ]~C)13X3

[013xl

[--dlnk(z)0"(--2-1[1 '~ K:a*,)]3xl

[011x3

I~ I0" (']"'~A )

0

0

-d2 I,,zl-10.(,co)

=

[0h•

,

o ( Q A ) = [{o(QA)}~,]s• [0"(-2-1/3+~:c)]~x~ =

[011•

[0]lx3

[013xl [-dl'll, k(z)0"(-2-111 -t- )~*)]3xl 40"(--2-1 [1 "~- ]~A) 0 0 -d2 j~l-10.(,Ca)

where I = 2, 3. Due to the homogeneity of the above matrices it is sufficient to consider only the case I~1 = 1. The latter equations yield det 0" ( ( ~ l P l ) = det 0"(Q2P2) = det 0"(QaPa) =

--d20"(7-/A)0.(s det 0"(-2-'/3 + KJc), -d2 O"(-2-11, +/C,) 0.(s det O"(-2-1/3 + K:c), -d2 d~ 0.(-2-1Ii +/CA) O'(s det O'(-2-1/3 + Kc).

(3.16) (3.17)

From Lemma 2.3, item iii) it follows that 6r(QtPl), (l = 1, 2, 3) are non-singular matrices, since all the multipliers in the right-hand sides of the above equations do not vanish anywhere for z ~ S and I~1 = 1. Thus, the operators QlPl are the SIOs of normal type [23], [25]. To study the index problem, let us note that the dominant singular parts of the operators QlPl and N'I, where

[-2-1/3 + K:c + 7/c]3x3 "~F1 :=

[013x1 [-dlnk(z)(-2-'I1

[0]1X3

'~ (1) "]"~A

[011x3

0

+

K:*)]3xl

[0] -d2~P~.(_l)[~ ~

+ (2-1/1 -}- .~*)]

,

5x5

are the same, i.e., O'(Af,) = O ' ( Q l P l ) . Moreover, they have the same index since the index is invariant to a compact perturbation (see [231, [25]). Now we will prove that the homogeneous equation Aflp = 0 with

= (P,, .-.,~s)L

(3.18)

and its adjoint one N'~*#* = 0 with *

I * =

have only the trivial solution.

*\T

tfl,'",fs)

,

(3.19)

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lentsch and Natroshvili

Applying the embedding theorems (see, e.g. [23], Oh.IV) we conclude that the solutions # and #* of (3.18) and (3.19), respectively, belong to the space CI+~(S). Further, the equation (3.18) is equivalent to the following system: [--2-15 + K:o + 7-/c1~ - dln(Z)(-2-111 + ~*=)#s = O,

n,~

= 0,

[& + (2-'I, + ~;:)1,~ = 0,

where/~ = (#,,#~,#a) T, since 7~0) and R(-1) have the trivial null-spaces (see Lemma 3.3, item i). From Lemma 2.3, item iv) it then follows that #s = 0, #4 = 0 and t2 = 0, i.e., equation (3.22) possesses only the trivial solution. It is also obvious that (3.i9) is equivalent to the system of equations ( - 2 - ' I a + J C b + n c ) ~ * = O, ~aai~)i**, = O, - d , ( - 2 - 1 h + K.,)(fi* . n) - d2[s + (2-1/1 + )~Ta)l~(_l)]~; = 0, where ~* =

(p~, ~ , a~)~.

Now Lemmata 2.3, item iv) and 3.3 imply that/~* = 0. Thus, equation (3.19) has only the trivial solution as well. Therefore, IndN',= Ind(Q~7~) = 0. The equalities Ind(Qffr = 0 (1 = 2, 3) can be shown by quite the same arguments. To finish the prove we note that the operators O , : [C~+~

~ • [C~+'+~(S)] • [C~+~(S)] -+ [C~+~(S)? • [C~+I+%S)], Q~, Q~: [C~+~(S)]s -+ [c~+o(s)? • [c~+l+o(s)],

are isomorphisms due to Lemma a.3. item i). Moreover, Qt~b E [Ck+~(S)] 4 • [Ck§ implies that qo E [Ck+"(S)] 4 x [Ck+*+~(S)]. Now, the assertions i) and ii) of the lemma follow from Lemma 3.4. The proof is completed. | C O R O L L A R Y 3.6 Let the conditions (3.12) be fulfilled and let ~b be as in Lemma 3.5. Then the qIDE (3.11), which is equivalent to the singular integral equation Q~79~ = Qi~, is uniquely solvable in the class [C~+~(S)] 4 x [C~+I+~(S)]. Proof. It easily follows from Lemma 3.5 and Fredholm theorems for StEs of normal type on closed manifolds [23], [25]. II 3.4. Existence theorems. We now can formulate the main results of this paper concerning the existence and regularity of solutions to the original interface problems.

T H E O R E M 3.7 Let the condition (3.12) be fulfilled and let f,, f2, f3, fs E Ck+O(S), f4 e Ck+l+~(S). Then Problem (p}~,r)) has a unique regular solution (U,w) representable in the form of surface potentials (3.i) and (3.2), where the densities h and g are defined by the uniquely solvable ODE (3.3) with d'a = 0 and d~ = 1. Moreover, U C Ck+~+~(~ -g) and

w c Ck+l+%~ =) n S~(fF).

Jentsch and Natroshvili

281

T H E O R E M 3.8 Let the conditions (3.12) be fulfilled and fj 9 Ck+~(S), j = 1,5. Then Problem (P~(~'~)) (l = 2, 3) has a unique regular solution (U, w) representable in the form of surface potentials (3.1) and (3.2), where the densities h and 9 are defined by the uniquely solvable ODE (3.3) (with d~3 = 1 and d~ = 0 for l = 2). Moreover~ U 9 Ck+1+~(~-T) and

w 9 c~+'+%g :-) n s;(fl-). Proof of these theorems readily follows from the results of the previous Subsection 3.3. 3.5. A n a l t e r n a t i v e a p p r o a c h . If we look for a solution (U, w) to Problem (P~@'~)) in the form of potentials

U(x)

=

WA(h)(x),

~(~)

=

W~(g)(~) + p0V,(g)(~),

x 9 a -l- = a l ,

(3.20) 9 9 a - = a~,

(3.21)

we arrive then at the following system of ODEs on S for the unknown densities h = (hi,..., h4) T and g:

[s

-- dlnk[-2-*I1 + K.(~} + poT-t~]g = Ck,

d;[C~h]4 + 4'{[2-*I4 + ~ ) ] h h 3 E nJ{[ 2-1/4 + ~(A2)lh}j -j=l

d2{f~A + p0(2-1/rl

-t-)~(A))}g

k = 1,2, 3,

= r =

r

(3.22)

hered~=0andd~=lforl=l,d~=landd~=0forl=2, andd~a~r Denote the matrix operator generated by the left-hand side of (3.22) by

[--dlnk S(~

[(cA),,]3• tt -1 h + ]~(A2))}4q] 1 • 4 [{d~A + d~(2

0

[21=1 nj(2-iI4 + JCA (~))/q]1•

-d2S(1)

x1

(3.23) 5X5

where S (~ and 8 (1) are defined by (3.5). Applying the arguments, quite similar to above, we can prove that the operator 15l is an elliptic (in the sense of Douglis-Nirenberg) ODO with the index equal to zero; moreover, if the conditions (3.12) are fulfilled, then the operators "J~l : [ck+l+ce(s)] 5 --+ [ck+a(S)]3 X

[ck+14"a(S)] X [ck-[-~

T'l : [Ck+t+~(S)]s --~ [Ck+'~(S)]5 for

l = 2, 3,

are isomorphisms. Therefore, one can easily formulate the analogues of Theorems 7 and 8 (with only very slight modifications related to the representation of a solution by formulae (3.20) and (3.21)). R E M A R K 3.9 Note that the so-called direct boundary integral equation approach leads to the system of boundary ODEs of type considered above but with the extended spectral set J1 with respect to w: Jl = J U Jo, where J is the same as above, while Jo is the set of eigenvalues of the interior Dirichlet problem for equation (1.10) in ft +.

282

Jentsch and Natroshvili

3.6. D i s c u s s i o n on e x c l u d e d e i g e n f r e q u e n c e s . Let us examine what happens when '~ E J. It is evident that the homogeneous Problem (Pl (~'~)) in this case has a nontrivial solution (U, w) with U = (u, 0) r in f~+ and w = 0 in f F , where u solves the homogeneous boundary value problem (1.27) (see Theorem 1.4 and Remark 1.5). The boundary conditions of the problem (1.27) together with the equation u4 = 0 in f F imply that [B(D, n)U] + = 0 on S (see (2.15)). Therefore from the general integral representation of a solution of the equation (1.7) (see [18], Section 3)

u(~) = w~([u]+)(~) - v~([Bu]+)(~),

z c a§

it follows that the four-dimensional vecwr U = (u, 0) -c. where u is an arbitrary solution of the problem (_1.27), is representable as a double layer potential U(x) - WAth)(x) with the density h = (h T, 0), where h = (u) + and (h- n) = 0. It can also be proved that ifw E J and (h, g)V is an arbitrary solution of the homogeneous version of system (3.22), then 9 = 0, h4 = 0 and h = (hl,h2,h3) T possesses the same properties as above. Note that if (h, 9) T is an arbitrary solution of the homogeneous system (3.3), we can conclude only that g = 0. Therefore the alternative approach described in the previous subsection seems to be more natural for spectral frequences. Thus. if w E J and the non-homogeneous system (3.3) [(3.22)] is solvable (that is. w is orthogonal to the null-space of the corresponding adjoint operator), then the density g of the exterior field w is defined uniquely, while the density h of the interior thermoetastic field U is determined within a summand defining a nontrivial solution of the problem (1.27). In the wave scattering and fluid-solid interaction problems the vector-function f in the condition (1.23) has the form fon, where )Co is a scalar function and n is the unit normal vector of S at the point x C S (see [6], [24]). In what follows we will show that for the vector-function ~b = (fon, f4, fs) v, where f0 and n are as above, and f4 and f5 are arbitrary functions, the interface Problems (P~(~'~)) are solvable for all w E J by the both approaches described above. To this end we have to prove that the vector-function ~b is orthogonal to the null-spaces of the corresponding adjoint operators. Denote by P~ and 75/~ the operators adjoint to T'z and 751, respectively (see (3.4), (3.23)):

(T'l*)pq = - 2 - ! 1 4 + E(~))pq, P = 1,4, q = 1 - ~ , (Pz*),4 = d~a(-2-ah

+

)~(~2)p4 "~- d-~("~A*),4, p

(~*)~ = E j~= l ( n A * ) P J n J , (~?)~, = - ~ , ( - 2 - ~ h

(v,*)54 = o,

p.= (;57)pq = (s

-* (Pl)p5

=

~3

z~=l~

p "~- 1 , 4 ,

+ Jc(~2 + ~on~,)~q, q = 1,3,

(p,)5~ = - ~ [ ~ .

(~5~)p4 = - 4 ( ~ . ) , 4

~--- 1,4,

+ po(2-% + ~:(22)];

-. [(~ )pq]5• p = 1,4, q = 1,3,

2 _ ~ f 4 + ~c(2.))p4, p = 1,4, + ~ 3~ ~~

4

+ JC~?)~jnj,

p =

1-:-&

(3,24)

Jentsch and Natroshvili

N ('~l)Sq

=

283

- d 1 ( - 2 - 1 1 1 - ~ - ](~(~2 --~p0"~A*)nq, q

(~'z)~s = - % [ ~ .

(Pz)~4 = 0,

z

1,3,

+ ~o(2-1I~ +

here the following notations are used: (-[-2-1/4 -I- ~ ( ~ ) ) h ( z ) := [ W A * ( h ) ( z ) ] :t: = +2-~h(z)

+

f{B(D~,

~(y))r].(~

- y,~, ~)}~h(y)a&,

S

n~.h(z)

:= [V~.(h)(~)] • = [ r A . ( z - y, ~:, r)h(y)dSy, S

[Q(D~,~(~),

(T2-~ f~ + ~c(2?)h(~) := :F2-1h(z) +

[{Q(n.,.(z),i~)r~.(~

i~)v~. (h)(z)] ~ - y,~,

~)}h(u)dS~,

S

s

:=

Q(D~,n(z),iw)WA.(h)(x),

. lim

with (3.25)

w~.(h)(x) = f {8(D~, ~(v))r~.(z - v, ~, ~)} ~h(u)dS~, S

VA*(h)(x) = f FA*(X -- y, w, r)h(y)dSu,

(3.26)

S

where ]~A*(z -- y,a.~, ~') = P ~ ( y - z,o2, r) is a f u n d a m e n t a l m a t r i x of the adjoint o p e r a t o r

A*(D,-iw) = AV(-D,-iw) and h = (~v, h4)V with /z = (hi, h2, ha) v. Since the scalar operator ~4(D,-iw) is formally self-adjoint (see (1.10)), we have ~.

= ~,

~2 = ~),

~ 2 = ~(3 ), ca. = c-~,

and they are generated by the potentials w~.(g)(x) =

-w~(g)(~) = / ~(D~,n ( y ) ) % d x

- y,~o,

~)g(v)dS~,

S

v~.(g)(x)

= -v ~ ( g ) ( x )

=

/

~(x

-

v,~:,~)g(y)dS~.

S

Note that

~A. = (~A)*,

~(17 = (~:~2)).,

~27 = (~:(1)).

where (.hi)* denotes the operator formally adjoint to Ad.

cA. = (cA)',

284

Jentsch and Natroshvili

L E M M A 3.10 Let ~* = (h*V,g*) T, where h* = (h*T,h~)T and h* solution to the equation either 7~o * = 0 or ~ * = O. Then h*'n=0,

h;=0,

( h * h*

g*=0,

h%T be a

(3.27)

provided the conditions (1.28) are fulfilled. Proof. Let ~* = (h*r,g*) v be an arbitrary solution to the equation Pl*~* = O. Equation (3.24) then implies r

P/*~* =

l

S T , + VA.(ng ,-)W 9 {WA.(h" , -,d3h4) ,d3h4) }

7

]

{[u(D, n) + ~0][V.4(-dlh*. n) + W.4(-d2g*)]} +

= O.

(3.28)

sxl

In turn, (3.28) yields that the vector-function

U*(x) = (u *r, u~) T = WA. (h*,-d~3h*4)+ VA*(rig, dab4)

(3.29)

and the scalar function w*(x) --~ V.A(-31h* 9n) -~-Wfl(-d2g*)

(3.30)

solve the following BVPs:

d*(D,-iw)U*(x)=O in ~ - , [U*(z)]-=0 on S; A*(D,-iw)w*(x) = 0 in ~+, {[u(Dz,n(z)) +Po]W*(Z)} + = 0 on S.

(3.31)

It can be easily checked that if U* is a regular solution of BVP (3.31), then the vector U = (u *v, iwT0~44) is a regular (m, r)-thermo-radiating solution to the equation (1.7) in fland [U]- = 0 on S. Therefore U(x) = 0 in f~- due to Theorem 2.1. Consequently, U*(z) = 0 in gt-. We can also show that w*(x) = 0 in ~t+ (see the proof of Lemma 3.!). Further, the properties of the potentials involved in (3.29) and (3.30), together with the results just obtained, lead to the equations

[Q(D,n, iw)U*]+ = -(ng*, 75dsh4 T, [w*]- =-d2g*, [u(D,n)w*]- = - a l l ( h * 9 n),

[U*] + = (h*V,~3h])T,

(3.32)

Hence we get the following transmission problem for the vector U* (defined by (3.29) in Yt+) and the scalar function w* (defined by (3.30) in ~)-):

A*(D,-iw)U*(x)=O

in ~2+, in

(u*. n) + = - ( d , ) - l [ u ( D , n)w*l-

[(T(D,n)u*)~ + iwTo/3~jnju*4]+ = -(d2) -

-1

n
,

-

on S.

(3.33)

[dr3A(D, n)u*4 + d~u*4]+ = 0 If we introduce the vector U = (u *r, icoT0u-~4)and the function w = w*, we can reduce the transmission problem (a.33) to the homogeneous Problem (PiP'r)) with constants d~ = -d~ -~

Ientsch and~latroshvili

285

and d~ = -d{ -1 instead of d, and d2 (see (1.23)-(1.25)). Since the constants d~ and d~ satisfy the first inequality in (1.28), from (3.33) it follows that w*(x) = 0 in gt-, u~(x) = 0 in f~+ and u* solves the problem (1.27)_ (see Remark 1.5). Therefore equations (3.32) imply (3.27). The proof for the equation P/'~* = 0 is verbatim, i R E M A R K 3.11 In fact, we have proved that the only nontrivial part [z* of the nontrivial solution ~* to the homogeneous equation (3.28) coincides with the trace on the boundary S of some Jones mode u*. The inverse assertion is aslo valid. From the above results it follows that the vector-function ~b =(fon, f4, fS)T with arbitrarily given f0, f4 and f~, is orthogonal (even pointwise) to the null-spaces of the operators P~* and 75~*. Therefore the non-homogeneous interface Problem (pie,r)) with the above boundary data vector-function r is always solvable (but not uniquely for w C J). R E M A R K 3.12 In the present paper we have confined ourselves to the classical formulation of the transmission problems and obtained the uniqueness and existence results in Hb'Ider spaces. Applying the same analysis as in [17] and [20], we can study the above problems in the Sobolev W~, Bessel-potential H i and Besov B~,q spaces by the above approaches and obtain the corresponding uniqueness and existence results for weak solutions.

ACKNOWLEDGEMENT This research was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant number 436 GEO 17/4/96.

References [1] Bielak, J. and MazCamy, R.C., Symmetric finite element and boundary integral coupling methods for fluid-solid interaction, Quart. Appl. Math., 49(1991), 107-119. [2] Bielak, J., MacCamy, R.C. and Zeng X., Stable coupling method for interface scattering problems, Research Report R-91-199, Dept. of Civil Engineering, Carnegie Mellon University, 1991. [3] Brekhovskikh, L.M., Waves in Layered Medium, Nauka, Moscow, 1973 (Russian). [4] Bostrbm, A., Scattering of stationary acoustic waves by an elastic obstacle immersed in a fluid. J. Acoust. Soc. Amer., 67(1980), 390-398. [5] Bostrbm, A., Scattering of acoustic waves by a layered elastic obstacle in a fluid - an improved null field approach. J. Acoust. Soc. Amer., 76(1984), 588-593. [6] Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory. John Wiley, New York, 1983. [7] Ershov, N.E., Solution of a three-dimensional diffraction problem of acoustic waves by potential method. In: Mathematical Problems of Geophysics. Novosibirsk, 1985, 47-58 (Russian).

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[8] Ershov, N.E. and Smagin~ S.I., On Solving of a stationary diffraction problem by potential method. Soviet Doklady, 311, 2(1990), 339-342 (Russian). [9] Everstine, G.C. and Au-Yang, M.K., eds., Advances in Fluid-Structure Interaction- 1984, American Society of Mechanical Engineers, Neu York, 1984. [10] Everstine, G.C. and Henderson, F.M., Coupled finite element/boundary element approach for fluid-structure interaction. J. Acoust. Soc. Amer., 87(1990), 1938-1947. [11] Fichera~ G., Existence Theorems in Elasticity. Handb. der Physik, Bd. 6/2, Springer-Verlag, Heidelberg, 1973. [12] Goswami, P.P., Rudolphy, T.J., Rizzo, F.J. and Shippy, D.J., A boundary element method for acoustic-elastic interaction with applications in ultrasonic NDE, J. Nondestruct. Eval, 9(1990), 101-112. [13] Harg~, T., Valeurspropres d'un corps glastique, C.R. Acad. Sci. Paris, S~r. I Math., 311(1990), 857-859. [14] HSrmander, L., The Analysis of Linear Partial Differential Operators, III, Pseudodifferential Operators, Springer-Verlag, Berlln-Heldelberg-New York-Tokyo, 1985. [15] Hsiao, G.C., On the boundary-field equation methods for fluid-structure interactions. Proceedings of the 10.TMP, Teubner Texte zur Mathematik, Bd. 134, Stuttgart-Leipzig, 1994, 79-88. [16] Hsiao, G.C., Kleinman, R.E. and Schuetz, L.S., On variational formulation of boundary value problems for fluid-solid interactions. In: MacCarthy M.F. and Hayes M.A. (eds): Elastic Wave Propagation. IUTAM Symposium on Elastic Wave Propagation. North-Holland-Amsterdam, 1989, 321-326. [17] Jentsch, L. and Natroshvili, D, Non-classical interface problems for piecewise homogeneous anisotropic bodies, Math. Methods Appl. Sci., 18(1995), 27 49. [18] Jentsch, L. and Natroshvi|i, D., Thermoelastic oscillations of anisotropic bodies. Preprint 961. Technische Universit~t Chemnitz-Zwickau, Fakult~t fiir Mathematik, 1996. (See also 'Proceedings of Sommerfeld'96 Workshop, FreudenstMt, Schwarzwald, 30 September-4 October, 1996'). [19] Jentsch, L. and Natroshvili, D.~ Interaction between thermoelastic and scalar oscillation fields (general anisotropic case) i Preprint 97-5. Technische Unlverslt~t Chemnitz-Zwickau, Fakult~t fiir Mathematik, 1997. [20] Jentsch, L., Natroshvili, D. and Wendland, W., General transmission problems in the theory of elastic oscillations of anisotropic bodies. DFG-Schwerpunkt Randelementmethoden, Bericht Nr. 95-7, Stuttgart University, 1995. (To appear in 'Journal of Math. Anal. and Applications'). [21] Jones, D.S., Low-frequency scattering by a body in lubricated contact. Quart. J. Mech. Appl. Math., 36(1983), 111 137. [22] Junger, M.C. and Fiet, D., Sound, Structures and Their Interaction, MIT Press, Cambridge, MA, 1986.

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[23] Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O. and Burchuladze, T.V.,Three-Dimensiohal Problems of the Mathematical Theory of Elasticity and Thermoelasticity~ Nauka, Moscow, 1976 (Russian). [241 Luke, C.J. and Martin, P.A., Fluid-solid interaction: acoustic scattering by a smooth elastic obstacle. SIAM J. Appl. Math., 55, 4(1995), 904-922. [25] Mikhlin~ S.G. and Pr5ssdorf, S., Singular Integral Operators, Springer-Verlag, Berlin, 1986. [26] Miranda, C., Partial Differential Equations of Elliptic Type, 2-nd ed., Springer-Verlag, BerlinHeidelberg-New York, 1970. [27] Natroshvili, D., Investigation of Boundary Value and Initial Boundary Value Problems of the Mathematical Theory of Elasticity and Thermoelasticity for Homogeneous Anisotrapic Bodies by Potential Methods, Doct. Thesis, Tbilisi Math. Inst. of Acad. of Sci. Georgian SSR, Tbilisi, 1-325, 1984. [28] Natroshvili, D., Djagmaidze, A. and Svanadze, M., Some Linear Problems of the Theory of Elastic Mixtures, Tbilisi University Press, Tbilisi~ 1986. [29] Natroshvili, D., Boundary integral equation method in the steady state oscillation problems for anisotropic bodies. Math. Methods AppI. Sci. (to appear) [30] Natroshvili, D. and Sadunishvili G., Interaction of elastic and scalar fields. Math. Methods Appl. Sci.,19, No. 18(1996), 1445-1469. [31] Nowacki, W., Dynamic Problems of ThermoeIasticity. PWN-Polish Scientific Publishers, Warszawa, Poland, Nordhoff International Publishing, Leyden, The Nederlands, 1975. [32] Ohayon, R. and Sanchez-Palencia, E., On the vibration problem for an elastic body surrounded by a slightly compressible fluid, RAIRO Anal. Numfir., 17(1983), 311-326. [33] Sarchez Hubert, J. and Sanchez-Palencia, E., Vibration and Coupling of Continuous Systems, Springer-Verlag, Berlin, 1989. [34] Schneck, H.A., Improved integral formulation for acoustic radiation problems, J. Acoust. Soc. Amer., 44(1968), 41-58. [35] Seybert, A.F., Wu, T.W. and Wu, X.F., Radiation and scattering of acoustic waves from elastic solids and shells using the boundary element method. J. Acoust. Soe. Amer., 84(1988), 1906-1912. [36] Smagin, S.I., Potential method in a three-dimensional problem of the diffraction electromagnetic waves, Journal of Computational Mathematics and Mathematical Physics, 29, 1(1989), 82-92 (Russian). [37] Smagin, S.I., On a new class of integral equations lowering dimensionality of diffraction problems. Soviet Doklady, 316, 3(1991), 580-585 (Russian). [38] Vainberg, B.R., The radiation, limiting absorption and limiting amplitude principles in the general theory of partial differential equations. Usp. Math. Nauk, 21, 3(129)(1966), 115-194 (Russian).

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[39] Vekua, I.N., On metaharmonic functions. Proc. Tbilisi Mathem. Inst. of Acad. Sci. Georgian SSR, 12(1943), 105-174 (Russian).

LOTHAI%JENTSCH

Fakulffit fiir Mathematik~ Technische Universitiit Chemnitz-Zwickau Reichenhainer Str. 41, D-09107 Chemnitz, Germany e-mail address: [email protected] DAVID NATROSHVILI

Department of Mathematics, Georgian Technical University Kostava str. 77, Tbilisi 380075, Georgia e-mail address: [email protected] 1991 Mathematics Subject Classification: 31 B 10, 31 B 25, 35 C 15, 35 E 05, 45 F 15, 73 B 30, 73 B 40, 73 C15, 73 D 30

Submitted:

February 3, 1997