Dynamical problems of the theory of elasticity and thermoelasticity

281. 282. 283. 284. 285. 286. 287. 288. 289~ 290. 291. 292. 293. 294.

T. P. Lukashenko, "Integrals and n-dimensional conjugate functions," Izv. Akad. Nauk SSSR, Ser. Matem., 38, No. 3, 546-582 (1974). Kh. Sh. Mukhtarov, "On some multiplieative inequalities and their application to linear singular integral operators," Dokl. Akad. Nauk SSSR, 182, No. 4, 764-767 (1968). F. Tricomi, On Linear Second Order Partial Differential Equations of Mixed Type [Russian translation], Gostekhizdat, Moscow--Leningrad (1947). L. HSrmander, Estimates for Translation-Invariant Operators [Russian translation], IIL, M o s c ~ (1962). V. Yu. Shelepov, "Boundary value problems and integral equations," Candidate's Dissertation, Donetsk (1968). P. G. Yurov, "Riemann's homogeneous boundary value problem with an infinite index of logarithmic type," Izv. Vyssh. Uchebn Zaved., Matematika, No. 2, 158-163 (1966). A. Browder, Introduction to Function Algebras, W. A. Benjamin, New York (1969). A. Devinatz, "Conjugate function theorems for Dirichlet Algebras," Rev. Union Mat. Arg., 23, No. i, 3-30 (1966). ,~ . ,, G. Giraud, 'Equatmons ~ int~grales principales, ~tude suivie d'une application, Ann. Scient. Ecol. Norm. Super., Ser. 3, 51, Nos. 3-4, 251-372 (1934). J. Kr~l, "On the logarithmic potential," Comment. Math. Univ. Carolinae, ~, No. i, 3-10 (1962). J. Kr~l, "Some inequalities concerning the cyclic and radial variations of a plane path-curve," Czechoslovak Math. J., 14(89), 272-280 (1964). J. Kr~l, "On the logarithmic potential of the double distribution," Czechoslovak Math. J., !!(89), 306-321 (1964). W. Rudin, Fourier Analysis on Groups, Interscience, New York--London (1962). J. Wermer, Seminar Uber Funktionen Algebren, Lecture Notes in Math., i (1964).

THE DYNAMICAL PROBLEMS OF THE THEORY OF ELASTICITY AND THERMOELASTICITY V. D. Kupradze and T. V. Burchuladze

UDC 539.3:517.53

INTRODUCTION In the classical theory of elasticity and thermoelasticity one encounters two types of dynamical problems; on the one hand, there are the problems in which the laws of motion as functions of time are known in advance and usually have a sinusoidal character; on the other hand, there are the problems in which the character of the dependence of time is unknox~n and has to be determined from the solution itself. The problems of the first type describe the steady-state or the stationary motions and the problems of the second type describe the nonstationary motions, unrestricted with respect to the time. Both, especially the latter ones, are more difficult problems than the problems of statics and have been solved only lately in the general form. A large number of investigations have been devoted to the dynamical problems of elasticity and thermoelasticity for domains with special boundaries. A sufficiently complete bibliography for the most important investigations of this kind is given in [25]. A significantly smaller number of papers are devoted to problems with domains having boundaries of an arbitrary configuration [20, 26, 42, 44, 46, 51, 54]. A detailed investigation of the fundamental dynamic problems of elasticity and thermoelasticity for such domains has been given, apparently for the first time, in [6, 11-13, 15, 50]. The present survey contains a presentation of these results. We give information, most of them nonstandard, about the singular potentials and multidimensional singular integral equations, which here play an essential role and have an important significance in Translated from Itogi Nauki i Tekhniki (Sovremennye Problemy Matematiki), Vol. 7, pp. 163-294, 1975. This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, iV. Y. 10011. N o part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, w i t h o u t written permission o f the publisher. A copy o f this ar~cle is available f r o m the publisher f o r $ Z 50.

415

many problems of mathematical physics; we make use also of the fundamental results of the theory of the statical problems (see the references in [14, 15] on the papers by T. G. Gegeliya and M. O. Basheleishvili). The history of the development of the mathematical theory of elasticity has always been mainly the history of its statical problems and, first of all, of its first boundaryvalue problem, i.e., the determination of the elastic equilibrium when the boundary values of the displacements are given, and of its second boundary-value problem, i.e., the determination of the state of equilibrium when the boundary values of the stresses are given. The brightest fluorishing period of the elasticity theory is related with these problems, coinciding with the penetration into this theory of the ideas and methods of the theory of potentials and integral equations. The classical investigations of Fredholm and Lauricella, Korn's investigations with the application of the integral equations and "Korn's inequalities," the prototypes of the contemporary coerciveness conditions, Weyl's fundamental investigations in the foundations of the theory of elastopotentials, and some others are closely related to the theory of potentials and integral equations. Unfortunately, in the initial period the application of these ideas was not equally successful in all cases; if they have allowed to investigate the first problem in an exhaustive manner within the classical framework, the investigation of the second problem was not completed in any of the works of that period. At one time this caused some relaxation in the interest for general methods and a corresponding increase for special methods, better adapted to the solution of particular subclasses of problems. Since then the situation has changed and the general methods have occupied their proper place. In 1947 Friedrich's paper [45] was published with a first rigorous proof of Korn's inequality and of the existence of solutions of the first and second problems on the basis of ideas which were original for the modern variational approach to the solving of boundary-value problems. To this same period belonged the well-known work of S. L. Sobolev [37] on the functional-analytic method in mathematical physics, with applications to the polyharmonic equation and to Cauchy's problem for a linear and quasilinear equation of hyperbolic type. The development of the variational methods have been reflected in the important works of S. G. Mikhlin [22, 23]. In 1950 Fichera's paper [43] was published, in which by combining the method of potentials with the methods of functional analysis one investigates the fundamental problems of the elastostatics, including the third fundamental problem (on one part of the boundary the displacements are prescribed, on the remaining part the stresses are prescribed). Fichera introduces the class V of vector-functions t(~) ~ (t1(~), ta(~), t3!~)), defined on the closed surface S, such that there exist vectors t(1)(x), t(2)(x), t ( ) ( x ) , defined in the domain D bounded by the surface S and satisfying the conditions:

t(~

, t(i~(~)n~ti( ~)' I t(~) ( ~ ) n ~ d ~ = O , S'

i [xj (ttOn~)--xt (tltlne)l d~= =0,

i, 7 = 1, 2, 3;

S t

w h e r e S' i s t h e b o u n d a r y o f an a r b i t r a r y r e g u l a r domain D ' , D ' c D , n~ i s t h e n o r m a l a t t h e point ~S'. I t i s p r o v e d t h a t a n e c e s s a r y and s u f f i c i e n t condition for the existence of a regular solution of the second problem is that the boundary value of the stress vector t(~) s h o u l d b e l o n g t o t h e c l a s s V. R e l a t i v e t o t h e t h i r d p r o b l e m , when t h e d i s p l a c e m e n t s a r e p r e s c r i b e d on $I and t h e s t r e s s e s t ( ~ ) on S 2 ~ S \ S ~ one c o n j e c t u r e s t h a t a n e c e s s a r y and s u f f i c i e n t condition for solvability is the possibility of an extension of t(~) from Su to S so that the extended vector should belong to the class V.

In 1953 the paper [7] was published, in which one has investigated the problems of statics and of the stationary oscillations by the method of elastopotentials and of the multidimensional singular integral equations. These investigations have signified a turn to the general methods in the boundaryvalue problems of the theory of elasticity. To a certain extent, the works [37, 43, 45] have anticipated and outlined the contemporary development of the theory of weak (generalized) solutions, based on the methods of functional analysis; the paper [7] was one of the pioneers in the application of the potential methods and of the multidimensional singular integral equations.

416

The theory of weak solutions, in the application to the boundary- and initial-value problems, is distinguished by the minimality of the requirements from the given data; hence, its known generality and elegance. However, when passing from the weak solutions to the classical ones, these requirements become as stringent as the requirements of the classical theory or even more so. Recently, a new work of Fichera [44] has been published, in which the power and the refinement of this method is proved with the greatest compldteness; one also gives a presentation of the "difficulties" in passing from the weak solutions to the classical ones. In our opinion, the method of singula r potentials and singular integral equations differs in a greater efficiency in comparison with other general methods. Under sufficiently general conditions, this method allows us to obtain effective solutions, making use of electronic computers (see Chap. II, Sec. i, Subsection 7, Sec. 2, Subsection 8; Chap. III, Sec. i, Subsections 6, 7), and almost always, at the proof of the existence theorems, it allows to indicate a concrete analytic construction of the solutions (see Chap. II, Sec. i, Subsections 5, 6, Sec. 2, Subsections 6, 7; Chap. III, Sec. i, Subsections 4, 6, 7). Above, in connection with [43], we have mentioned necessary and sufficient conditions for the solvability of the second and the third problem. It is important in practice that these conditions be formulated directly in terms of the given data contained in the assumptions of the problem. This is achieved very easily with the aid of the integral equations [9, 1 0 ] . T h i s m e t h o d a l l o w s us t o s o l v e a s e r i e s o f new p r o b l e m s ; among t h e s e a r e t h e c o n t a c t problems of the theory of elasticity f o r p i e c e w i s e - h o m o g e n e o u s b o d i e s , t h e b o u n d a r y and c o n tact problems of thermoelasticity, boundary-value problems for Cosserat mediums, dynamic p r o b l e m s , and o t h e r s [14, 1 5 ] . Progress in the investigation of equations in the multidimensional case t u r y ; i t was s t a r t e d b y t h e w e l l - k n o w n and o t h e r s [ 3 , 1 6 - 1 8 ] , and l a t e r on i t In [44], using Fourier's method Fichera solves the mixed problem for the analytic case. The e q u a t i o n s o f adjoint equations; t h e p a p e r s [ 6 , 42] Basic sources

for this

initial--boundary (mixed) problems for hyperbolic is completely related to the second half of our ceninvestigations o f O. A. L a d y z h e n s k a y a , M. I . V i s h i k , w a s c o n t i n u e d i n V. A. I I ' i n " s paper [5].

in conjunction with the methods of functional analysis, a self-adjoint system of equations of higher order in thermoelasticity do n o t b e l o n g t o t h e c l a s s o f s e l f a r e d e v o t e d to them.

survey have been

[14,

15].

CHAPTER I INTRODUCTORY REMARKS. I.

FUNDAMENTAL STATEMENTS

Terms and Notations

We introduce the following notations: ~ is the set of all ordered triples of real numbers, i.e., the three-dimensional Euclidean space, Oxlx2x3 is the Cartesian system of c o ordinates, x =

(xi), y = (yi), i = i, 2, 3, are points in ~3,

r Ix-YI=|S~ (x~-y' )~|a~=_J1 12 is the

distance between the points x and y. S - ~D is a two-dimensional, connected, bounded, compact manifold in ~ , bounding the domain D; S will be called a closed surface. The surface S bounds two domains in ~3: a finite (bounded) one Ifr and an infinite (unbounded) one D-, i.e., D - = ~ \ D + , where D + = D + U S , Qr----{(X, t):x~O, rE(O, T)} is a four-dimensional cylinder, Sr----((x, t):xCSD, t~[0, T)} is the lateral surface of QT, n(y) = (ni(y)) is the unit normal vector at the point yGS , pointing outside D+; dx = dxldx2dx~ is the element of volume, dyS is the element of area of the surface S at the point y, D' is a bounded, strictly interior subdomain of the domain D, ~ ) ' ~ D . We make the following conventions. Each n-dimensional vector f = (fl,

-, fn) = (fi) (real or complex)with the norm

,fI~_~r-- ,fiI21U2 is con[ i=I ]

sidered as a one-column matrix

417

f=

/. ll---il/, n

if

~--(?i ..... %z), then the product

f?:= "?f

denotes the bilinear form

f?--=~ f~-?~; the k=l

Product of the matrix a~lla~I[~n• of dimension m • p by the matrix sion p • n is defined as the matrix c=[[c~k[imy.n , where

O=!Ibi~r215

of dimen-

P

c~ = ~ aub]~;

]=1 in particular, t h e p r o d u c t o f a m a t r i x o f d i m e n s i o n m x n (p = n) b y an n - d i m e n s i o n a l i s an m - d i m e n s i o n a l v e c t o r

(1.1) vector

H

Z

a~kPk

k=l

(1.2)

k=l

m•

The sign * at a matrix a denotes the transposed matrix a*=I1a~kHp• ., p, denotes the k-th (vertical) column vector of the matrix a

a*i~=aki; a,

.~I, 2,

m/r

and s o m e t i m e s we s h a l l w r i t e a = a ..... , i . e . , we r e p r e s e n t a m a t r i x b y t h e a g g r e g a t e o f its p vertical column-vectors. The square of the norm of the matrix a is defined by the equality

[Sl2 -~-

k I2 = [a

k=l

E laik 12. k~l

n ( a b e ) k = E ~z~akibU~ d,

Obviously, . . .

.

.

ab

=~

=

~.~e~k

and for m----p:

i=l

(aO*)*9 =]{[(ab*)*~]kJ [,

[(ab*)*9] k = ~

E ai,bk/?i.

I n t h e s e q u e l we s h a l l

consider matricial differential operators. Such an o p e r a t o r i s a m a t r i x whose e l e m e n t s a r e scalar differential operators. If aik is a differential o p e r a t o r and b k j i s a f u n c t i o n upon w h i c h t h e o p e r a t o r a i k may a c t , t h e n a i k b k j i s t h e f u n c t i o n o b t a i n e d by t h e a c t i o n o f t h e differential o p e r a t o r a i k upon t h e f u n c t i o n b k j ; i f a i k and b k j a r e d i f f e r e n t i a l operators, then aikbkj is the differential o p e r a t o r o b t a i n e d by t h e a c t i o n o f t h e o p e r a t o r a i k upon t h e o p e r a t o r b k j . Thus, i f a=IIctmIlm• is the matricial differential operator and b=Ib~kl[p• is a matricial function, then ab is the matricial function c~-[]cmII~• , where the function Cik is defined by (i.i). In particular, if a-~-!laikil;,• is a matricial differential operator and ~-----(~..... ~p) is a vector function, then ag~ is the vector function defined by (1.2). If a~]Iaik[lmxp, tricial differential

If operator, by ( 1 . 1 ) , 2.

b=

are matricial differential operators, then ab is the mallb.kl ~ ! pXn operator c=[[ci~ljmx n , where Cik a r e d e f i n e d b y ( 1 . 1 ) .

a==]laikIJ,nxp

i s a m a t r i c i a l f u n c t i o n and b-----Ilbikllpx n i s a m a t r i c i a l d i f f e r e n t i a l t h e n ab i s t h e m a t r i c i a l d i f f e r e n t i a l o p e r a t o r c==t]ci~H~xn, w h e r e Cik a r e d e f i n e d i n w h i c h a i k o c c u r as t h e c o e f f i c i e n t s of the differential operators bkj.

Some Function Spaces and Surfaces of Class Jib(a) Let ~ be a function defined in the domain D c ~ ~.

Definition. We say that ? belongs to the class C~ (~GC ~ (O)), if ~ is continous in D; TGCk(D), k being a positive integer, if at each point x @ D there exist all par-

418

tial derivatives of 4, up to the order k, with respect to the Cartesian coordinates of the point x and if these derivatives are continuous in D. Definition.

We say that the function ~ can be extended continuously at the point g60D, H m 4(x).

3D being the boundary of D, if there exists the finite limit

DDx ~ l!

Definition. We say that ~ C U ( D ) , if 4~Cu(O) and if in addition ~ and all its derivatives up to order k can he extended continuously at each point of the boundary ~D. The following statement holds:

if 9~C (D) , then the function ~, defined by the equal-

ity

[e (x),

x6D, xfiOD,

q,(x) = ] lira ~(z), tD[Dz -~ x

is continuous on the set D. We n o t e that the uniform continuity of 4 in D is a sufficient condition (and if D is bounded, then also necessary) for its continuous extendability at each point of 3D. Definition. If ,~ is a vector or a matrix defined in D, then that the components belong to the class C k ( D ) [ C k ~ ] . Definition. A matrix

96Ck(D)[4EC~(D)]

A function 4, defined in D is said to be regular if

4----]I?i~[[ is regular if all

means

4@CI(D)AC2(D).

4ik are regular.

If the function ~, defined in D+(D -), can be extended continuously at the point y60D+(0D -) , then by 4+(y) or {9 (y)}+ [?- (y) or {~(y)}-] we shall denote the limit

~+ (Y) ~: {4 (Y)}+ ~ lira 4 (x) [?- (y)~. {4 (Y)}-~- lira ,~ (x)]. D+Ox-+Y

D-~x ~ y

Definition. We shall say that 4 belongs to the class C~ (~fiC~ where D is a bounded domain, if for any x and x' from D we have the inequality 14(x)-4(x')]-~ 0 , then ~ can be extended continuously at each point of the surface 3D and ~, defined by the formula

[4 (x), x D, d?(x) = ] lim q~(z), x 6 O D , tDDz ~ x

satisfies the inequality

i@(x)--~.(x')I ~0, is called a Lyapunov surface and a surface of class ~2(0) is called a surface with continuous curvature (see [4]); if SEek{0) for every natural number k, then we write SEJ]= and we say that S is infinitely differentiable. Definition. Let SGJ/I(0), let f be a function defined on S, let xGS(z,d) and let ~ , $2, ~3 be the coordinates of the point x in the system (z). We define on T(z, d) the function fz by the equality fz(~1, ~2) = f(x). By definition,

fCC~'~(S),

if for any point

z~S,fzECk'~(~(z,d)),f:IIf~il[~C~'~(S), if

all

f u ~ C ~' ~ (5:). The following assertions hold I.

(see [4]):

If the function f, defined on S~d]~(O), satisfies for any x, N@S the inequality where c = const > 0, ~ = const, 0 < e < i, then f@C~

if(x)--f(y)l.~< c[x--fjl~ ,

2. If 3~3]~(~), k>/l, tor normal to S.

=>0,

then

I~@C~-~'=(S), where

n - n(y) = (ni(y)) is a unit vec-

3. The Fundamental Equations of Motion in the Stress Components. Hooke's and Duhamel--Neumann's Laws We consider the dynamical problems for isotropic deformable rigid bodies with and without the influence of the interactions of the mechanical and thermal processes; they contain the problems of the classical theory of elasticity and thermoelasticity. It should be remarked that thermoelasticity is a comparatively new area of mechanics synthesizing two, previously independently developing, disciplines: the theory of elasticity and the heat-conduction theory. The concept of the state of stress formed as a result of the interactions of elastic and thermal deformations, which is the essence of thermoelasticity, conclusions that are satisfactorily consistent with the observed facts. By the assignment of an elastic medium, in the mathematical the assignment of a domain in the Euclidean space, occupied by a sume that a homogeneous, isotropic, elastic medium, with density thermal coefficients ~, x, ~, occupies a domain D C ~ s . Here y = cient of linear heat extension, ~ - f ?To - , z-----~ is the coefficient

of the fields leads to

sense, one can consider medium at some time t. As0, Lam~ constants %, ~ and (3% + 2~)a, a is the coeffiof temperature conductiv-

ity, k is the coefficient of heat conductivity, ~ is the specific heat capacity of a unit volume, To is the initial temperature of the medium taken to be equal to the temperature of the undistorted state. Generally, it is considered that [14, 15, 27, 54]

p>Oi ~>0, 3ZH--2~>O, ~>0, 71~>0.

(3.1)

In the linear theory the elasticity property is expressed by a special dependence between the quantities which characterize the stress and strain states, called Hooke's law. The state of stress is characterized by the stress tensor ~(x! t)-----[[zik(X,t)I13• ~i~(X,Y) being the stress components, while the state of strain is characterized by the strain tensor e(x, t)--=]iel~(x, t)ll3• e~(x, t) being the strain components. If v(x, t) = (v:, va, v3) is the elastic displacement vector,

elk=ll2\axk_axi ], i,

then

k=1,2,3.

The e q u a t i o n s o f t h e m o t i o n o f an e l a s t i c body, under the action t)) , h a v e t h e f o l l o w i n g f o r m i n t h e s t r e s s c o m p o n e n t s :

(3.2) o f mass f o r c e s

~(x,

t)--(,~i(x,

421

z.~

t=1

Ovm (x, t) O~ok(x, t) Oxi ~-9:~k(x, t)----O Ot~ , k = 1 , 2 , 3 .

H o o k e ' s law a s s u m e s t h e s i m p l e s t medium h a s t h e form

linear

(3.3)

d e p e n d e n c e , w h i c h i n t h e c a s e o f an i s o t r o p i c

3

~ (X, t) ~- k ~ ~ eli (x, t) -t- 2~em (x, t),

(3.4)

j=l

where 6ik is Kronecker's

symbol, ~i~0,

i~k,

~kk=l.

Consequently,

according

~,Ox~ Oxi)"

to

(3.2),

( 3.5 )

F o r t h e d e d u c t i o n o f t h e f o r m u l a s ( 3 . 5 ) one has t a k e n i n t o a c c o u n t o n l y t h e m e c h a n i c a l p r o c e s s e s , t h e t e m p e r a t u r e o f t h e medium has b e e n c o n s i d e r e d ; c o n s t a n t ; i n f a c t , t h e t e m p e r a t u r e o f an e l a s t i c b o d y v a r i e s , i n g e n e r a l , i n t i m e and from p o i n t t o p o i n t . These v a r i a t i o n s a r e c a u s e d b y t h e a c t i o n o f h e a t s o u r c e s as w e l l as by t h e i n f l u e n c e of the deformation process itself. The work of the deforming forces is converted into heat, inducing a change in temperature, as a result of which there arise additional stresses, called thermal stresses. If, e.g., the action of the heat sources changes the temperature of the medium, then elastic strains will appear even in the absence of exterior stresses and mass forces. Already in 1835, i.e., shortly after the fundamental aspects of the theory of elasticity have been formulated, Duhamel and, somewhat later, Neumann have shown that in a heated homogeneous isotropic body the stress components depend on the strain components in the following manner [29, 41, 54]:

~i~(x, t ) = ( k d i v v - - 7 0 ) ~ k @ 2 ~ e i k ,

i, /~=1, 2, 3,

(3.6)

where 0 = ( T - To) is the deviation of the temperature and T is the absolute temperature (the temperature after deformation). The relation (3.6), establishing a linear dependence between the strain components, the temperature deviation and the stress components, is called the Duhamel--Neumann law. The function e(x, t), under the assumption tion equation 1 00

AO-- y ~ - - ~

I~To

0

0 ; then we have Green's formula

9

{,-

(,,.

D:

(6.1)

6"

where 3

aVp {aUp

, OUq'i.

(6.2)

p,q:l

as the positive direction of the normal to S we have taken the direction which is exterior with respect to D +. Indeed, one can easily verify the identity

(6.3) q=l

where rpq(U), p, q = I, 2, 3, are defined by (3.5). The integral over D+ of the left-hand side of (6.3) exists.

~O--~(Z'-Opzpq(tt)) D+ (q) " ',(p)

integral

dX

On the other hand, to the

we can apply the Gauss--Ostrogradskii formula (see [48]).

which, due to the identity

{(~q) ((~,~p~Vfipq (tt.))nq}+= {(p~)~op((~q)r produces

nq)}+--~-{OTU}+,

(6.1).

Remark. The proof of the applicability of the Gauss-Ostrogradskii formula under our assumptions has a nonelementary character (see [48]). The proof is simplified if instead of the assumption SEmi(a) we take $6~2(0). In this case, constructing inscribed parallel surfaces and applying formula (6.1), we obtain the proof of (6.1) by an obvious limiting process (see [35, 38]). Finally, if we replace the requirement of the integrability of A(~/~x) over D+ by the requirement =6C2(D-+), leaving the other assumption unchanged, or even weakening somewhat the requirement imposed on S, then the proof can be further simplified. In this form this theorem is given in most books on potential theory (see [21, 24]). We rewrite the expression (6.2) for E(v, u) in the form

/a% clvq\ /aUp , auq~ , p ~ /avp ovql {oup auq~

3~,+2~ . p:/,q

p, q

(6.4)

from where

E (v, u) = E (~, v).

(6.5)

In particular, if v = ~ (the upper bar denotes complex conjugation), then

3k + 2~

[aup . aUq 2

~ [oup auq [2.

e(L u)=-- T- d~v~b"-+~Z i~+o~,, +~~log--~ p-7~q

p ,r

(6.6)

We note that if u is the real elastic displacement vector, then E(u, u) coincides with the specific energy of the elastic deformations [14, 15]. If zL, ~6CI~)+)AC~-(D+), Az~, Av6 LI(D+), S@J]I(=), then from (6.1), according to (6.5), we obtain Green's formula

S["'-' (b " - " "

D+

2.

Green's

]d.,: =I 3 I ,'+

F o r m u l a s in T h e r m o e l a s t i c i t y .

(6.7)

Let

u=(ttl, U2, tt3)6C'(D -+) n C ~(D+), "o=(v,, v2, v3)EC'(/D+),

Iz,EC'(D-+), A ( ~ . ) u E L , ( D " ) ,

Ss

a>O,

427

then

I [v (Art--'( grad tt,) + E (v, tt) -- 7tt, div v] dx = f ~ (rtt--,Vttt,)+dg, D+

(6.8)

S

which is obtained from (6.1) and from the obvious identity 3

a

v g r a d u 4 = ~, ~

(v~tt4) -- u4 div v,

k=l

with the subsequent application of the Gauss--Ostrogradskii formula. In particular, if U = (u, u~) is a regular four-dimensional vector, solution of the homogeneous equation of the stationary motion of thermoelasticity

and [ = ( ~ , equality

~4)

is the complex conjugate vector, then from (6.8) for v = ~ and from the

f (tt4A~4+gradtt4grad~4) dx-

l

D+

O-U-4dS ~4a~-

(6.9)

S

there follows at once

Dr

icoq 4

S

Taking in (6.10) the complex conjugate expression, account (6.5), we obtain

2? "--

,

forming the difference and taking into

9

9

ii

(6.11)

o~-n i - - "

'S

D+

If U = (u, u~) is a regular solution of the homogeneous equation of thermoelastic pseudooscillations [15], B(~/~x, iT)U = 0, T = O + i~, O > 0, which has an important auxiliary significance in the study of the nonstationary problems of thermoelasticity, then, in analogy with formula (6.10), one establishes the identity

an] dS.

(6.12)

D+

We give one more of Green's formula that will be useful in the sequel. If U = (u, u4) and = (~, ~4) are four-dimensional regular vectors in D + and BU, BU are absolutely integrable in D +, then

D,'-

S

The proof is obtained from the identity 0

-- iOJT~E a~ (LL41Lk)'

with the subsequent application of the Gauss--Ostrogradskii

formula and formulas

UBU

--

UBU

:

(uAtt -- nAn) +

(u4A~4- u4A~4)+ 7 ~, ~ (/s (k)

a

(~)

Since by definition

U~U--U3?U=ttPU--tt43~

--ttPU--tt4f~

\

the surface integral in (6.13)

i ( u ~ U - tY~U)+dS "s 428

]

~

~

,

(6.7),

(6.8).

can be replaced by the integral

\ (~U,~L/-- ~:U.3~U):-dS. s 3. Energy Identities. We s h a l l t r a n s f o r m G r e e n ' s f o r m u l a ( 6 . 1 ) , ( 6 . 8 ) , in w h i c h now we s h a l l a s s u m e , r e s p e c t i v e l y , u = ( u ~ , u s , u s ) , U = ( u , u , ) r e a l and d e p e n d i n g a l s o , i n addition to the point x, on the time t. Assume that in (6.1) u(x, t) is a regular solution of the homogeneous equation of the nonstationary motion of the theory of elasticity A(3/~x)u p(~au/~t =) = 0, while v = 3u/3t. On the basis of the identity { O__'i ll _ .

Ou O~ll

p O lOu ~

/ Ou

~

" 1 c3

vA\ox.~ --PS~Sg--2 ~l~ ' E~gi' u j - g o T E ( U , U). one can rewrite (6.1) in the form

DL2 0t -t-gE(It,

(37) (TU) dS,

(6.14)

or

a (E~)_l_E(p)) =_,,~,o._.. ~[ Ou +(Tu)+ d3," atS

where E(k) is the kinetic and E(P) is the potential energy of the motion: E ( k ) = I p Ou '~

ax, E(P)=-ff1 ,)E (u, u)dx.

D§

D+

In order to obtain the energy equation in thermoelasticity, we s h a l l s t a r t w i t h i d e n t i t y ( 6 . 8 ) i n wh• v = ( v : , v a , v3) a n d U = ( u , u~) a r e a r b i t r a r y regular vectors. L e t U = (u, u,) be a regular solution of the homogeneous equation of the nonstationary thermoelastic mot ion

and v = ~u/~t. By virtue of the identities

( A u - - ~ grad u4) = y 9 ~0[ ~0ul , E (v, u) = -~1 ~0 E (u, u), (6.8)

takes

the following

form:

D+

We t r a n s f o r m

-(i+(PU)WS ( 9

.

ot - . L21o l- 2 e(,,

div,dx= f

D+

I O --'~ u4~divudx

the integral

;

(6.15)

&"

1 au~ O AN4=U~/. 0? + ~ q ~ d i v u

since

, we h a v e

D+

'

0

D+

Substituting

OU4 Ou,. --- V !u4Au4dx =2_~I ~[u4[ 0 2dx + -~ V iigrad u412dx_ ~ ! U4On_ dS.

~

.

D+

into

O

(6.15),

p O. ~

D+

we o b t a i n

1

the desired

S

energy identity

D+

side,

D+

~I[gradu41edx=l[~PU+-~u4anJ

D+

In the left-hand

D+

in addition

to the derivatives

ao.

(6.16)

S

with respect

to time of the kinetic

and potential energy, there occur the derivatives of the thermal energy function ~ vq 3 ~ 1ru 4112dx D+

and the dissipation function

?-IIgradu412dx (see [54]). O+

429

Remark. Formulas (6.14), (6.16) remain valid also in the infinite domain D-, provided the solution satisfies asymptotic conditions, sufficient for the vanishing for R § ~ of the surface integrals over C(0, R). 7.

Fundamental and Singular Solutions

In order to solve the stationary and nonstationary boundary- and initial-value problems of the theory of elasticity and thermoelasticity, we apply the properties of the generalized potentials and the theory of multidimensional integral equations, both regular and singular. The potentials are constructed with the aid of the fundamental or other singular solutions of the corresponding differential equations. In spite of the fact that the differential equations under consideration belong to different canonical types and some of them do not belong to any of the standard types, in all cases a fundamental role is played by the fundamental and other singular solutions of the elliptic-type equations with a parameter (real or complex). A distinctive peculiarity of such solutions is the presence of a point singularity of the form Ix[ -~ and the simple mechanical sense: in the theory of elasticity and thermoelasticity they express, respectively, the displacements, the displacements and the temperature in the infinite medium ~s, which appear under the action at the origin of a concentrated unit force, force and heat source, respectively. We construct the matrices of the fundamental and other singular solutions of the operators A(3/3x, m) and B(3/3x, ~) and we indicate their basic properties. i. The Matrix F(x, ~). The matrix of the fundamental solutions for the operator A(~/ ~x, ~) or the fundamental solution of the homogeneous equation of the oscillations A(3/~x, ~)u = 0 has the form

r (x, 2

'P~J

p=l

~T

'

(7.1)

where i is the imaginary unit, while k~ and ka are nonnegative numbers defined by the equalities

%_~2p(2~F)-l,

~ o ~ ( _ 1)P(2~p~2)-I.

(7.2)

We can represent (7.1) in the following form: I

2

3

)

F(x, ~o)=I[F, V, vii, F(x, r

i, V2], F~i), j = 1 , 2 , 3 .

(7.3)

] /0 ) The column v e c t o r r ( x , ~) s a t i s f i e s t h e e q u a t i o n A , ~ , ~ f F~-0 f o r j = 1, 2, 3 a t each p o i n t o f t h e s p a c e ~3, e x c e p t a t t h e o r i g i n . Indeed, taking into account the formulas (hq-k~)Vki(x,~)=--(kq-F)[2~F(k+2F)] "~ O= exp(m,!x3 OxkOx] !x! ' 1 mv [' (x, ~0) = [ 2 ~ (~+2~)] -~ 0 exp (ikllx!) Oxi Ixl ' for

the k-th

component of 3

r0

the vector

A ' t ~0- , ~

we o b t a i n

!

,

0

(x,

o~) = 0.

p=[

The m a t r i x r ( x , ~) i s sometimes c a l l e d the Kupradze m a t r i x , who it systematically to elasticity t h e o r y problems [8, 9]. I t can = F ( x , c,~)= = F ( - - x , e0 and f o r any xGU, -- oo < ~ < oo, we have t h e F)lx1-1, k , j = l , 2 , 3 , c@., p) b e i n g a p o s i t i v e number which depends easily where

430

established

that

I~(x)--iiP~i(x)!13•

f o r x6~3 .,{0}, 1

2

lira ( x , ~P,

r

(x),

c o n s t r u c t e d i t and a p p l i e d be e a s i l y s e e n t h a t F*(x, o~,) i n e q u a l i t y IF~](x, ~){ 0 , xES.

where

The f i r s t one c o n v e r g e s a b s o l u t e l y i n t h e u s u a l i m p r o p e r s e n s e , w h i l e a s i n g u l a r i n t e g r a l w h i c h e x i s t s o n l y i n t h e Cauchy p r i n c i p a l - v a l u e sense. L e t us a g r e e t o c o n s i d e r as t h e p r i n c i p a l Jo=lim -

~

value of the integral ?(Y) 0

1

~-i

lx--yl

~-~o s\s~'~,~)

the limit

t h e s e c o n d one i s (if

it

exists)

ays,

(8.1)

where, as before, S (x, e ) = l-l, (x, n (x), z) N S, e~0, then each solution of class Lp(S), p > i, of Eq. (8.27) belongs to the class cr,B(s). In this case the statement: "if ~ELp (S) , then ~Ecr'P(S)'' is understood in the following manner: ? of class Lp(S) is equivalent in the sense of Lp(S) to some function ~', which belongs to the class cr,8(S). *The kernels K i and Kj are mutually conjugate if so are the operators

Li(~)

and

LI(cD. 445

From Theorem 28 there follows THEOREM 31,

Cr'~(S).-+Cr'~(S)

If the conditions of Theorem 29 hold and the operator L(?)--• is invertible, then [~ll(s,~,~) 0).

(the

From (1.2) we have the representation

Nz, o ~(z, y , = - - ( O

n)P(z--y)--!R(z,t)N(~-{,n)P(t--v)d,S, N(a~

where R(z, t) is the resolvent of the kernel

(1.3)

and, consequently,

IR(z, t)i < -iz__tl2-~" -

(1.4)

On the basis of (1.3) and (1.4) one proves that

~EG~(2, ~'),

O 0 being a sufficiently small number [7]. Here ~ (t, y) and h(x, y) are unknown matrices satisfying the conditions: TEC~ yy6D +, O < ~ < a , h(x,y) is a regular solution in D+ of the equation equation

A(~

It is known [7] that

Au--px~u-~-O, v x 6 D +, viES

F(x--t, ixo) is

and belongs to the class

a regular solution of the

T(05,/zirCG(2--a,j ~, I)

on S x S

and to the class G(2, ~, 1) on S •

oaF(x--t, i•

[

c

I a.[',a~.a.~ ,< j.-t p+m, ml+m2+ma=tn,

(a~,n~jP ('x - t ,

r '0 Bx, Ba a r e m a t r i c e s

of class

i%)=BI

m=0,

l, 2 . . . . .

(1.12)

0 1 (x, 0 On(x) l x - - t l + & ( x , O + & ( x , t),

G(0) on S • S w h i l e

B, i s

a matrix

of class

G(1,

~, 1) on S •

S.

For the determination of with a weak singularity

448

?(t, y) one obtains a system of Fredholm integral equations

+I

(f,

'"o> (',

..o)-:,,

(1.13)

I +,

S

and for h(x, y) we arrive at the boundary-value problem

Vx, y~D+:A ( O )h--ox~h=O,

_j=l *

.

where W ( / ) ( y ) = ! [ T ( o , n ) P ( z - - y ,

)

1

2

3

.

,Xo)] ~(1 (z)dzS; {~(/)(Z)}n/=,={#(Y), @(1), '~(')}7='

iS a complete o r -

s thonormal system of solutions of the homogeneous system associated with (1.13). Since f~ C O, ~(S) and ~ > 0 , this problem is uniquely solvable and one has the representation

h (x, y) = t F (x -- t, ix0) ~ (t, y) diS, S

where

v(t,y)EC~

is the solution

of the uniquely solvable

singular

integral

equation

8

These representations, by the application of the corresponding theorems on the composition of kernels, similar to the already considered scheme, allow us to obtain the estimates:

[(7(=~)(x, y; --x~)I~< !x--~,r s ' V:c, flCD+,

(1.14)

!x_:~, "rx, y~D +, x-@S

being the nearest point to x on the surface

(1.15)

S:[x--x[=minlx--y I (we

note that x lies on ~Es the line n(x-~). For a detailed derivation of the estimates (1.7), (1.9), (1.14) and (i.15), see [2]. By the generalization of certain results from [47], making use of the methods of [14, 15] and applying the techniques of [2], one can show that

a%) (x' ':; -~2~ I Ox~ [~ ~ ,

Vx, y~D +, i = 1,2, 3.

(1.16)

We also note that the Green tensors G(i), i = i, 2, possess a symmetry property of the form

Ou)(x, y)=O~,)(y, x), O(~_)(x,y; --x~)=G*(2)(y,

x; --Xo2).

(1.17)

This is proved, as usually, by the application of the Green formula (6.7), Chap. I, in the corresponding situation. The estimates (1.7), (1.9), (1.14), and (1.16), in the case when the matricial differential operators A(3/~x) and T(~/~x, n) are replaced by the scalar operators h and 3/~n(x), have been obtained in [36, 39, 40]. 2. Interior Problems. The Spectrum of the Proper Frequencies. We consider the fundamental boundary-value problems for the finite domain D+, the interior oscillation problems. (o

Problem

(|)~.

I0

v xE D § : A ~OY' F~C o, ~.(D+), f 6 C 1, o(S)

uECI(D+~NC2(D+), regular

Determine a vector

~)u (x) = --pF (x),

in D+, from the conditions

Vz~S:u+(z)=f(z),

being given vectors.

o)

Problem

(ll)~.e .

Determine a vector u(x), regular in D +, from the conditions

vxEO+: A ( a ,

r

The c o r r e s p o n d i n g homogeneous problems

v z e S : [Tu(z)]+--f(z): ( f - F =- 0) a r e d e n o t e d by

feC~ (1)~,o, (lI)~, 0" For t h e

homogeneous problems (I)0+0, (II)+0 t h e u n i q u e n e s s theorems do n o t h o l d ; s p e c t r u m o f e i g e n v a l u e s r e l a t i v e to t h e p a r a m e t e r ~.

t h e y admit a d i s c r e t e

449

{0\ v~L~(D~) . Let v be an arbitrary regular vector in D+ and let A L~x)

Applying Green's

formula (6.7), Chap. I, to v(y) and F(td--X , c0), k-----1,2,3,V GGD § x6j9 + being a fixed point, excluding x by a small sphere llJ(x, ~), -0>0, and then taking the limit for e + 0, we obtain the formula of the general representation of a regular vector in the form

8

S

D+

I t i s c l e a r from the method by which general formula.

(1.18) has been o b t a i n e d

t h a t we a l s o have the more

(x)77 (z)= ir ( x - ~, ~o)(rv (y))+d.yS8

(0 S

t~)'

D+

0,

~ > 0 , F~C~

F--O([xl-2-e), e >0,

then the

r

problem resented

(I)~ip i s u n i q u e l y s o l v a b l e f o r any v a l u e o f t h e p a r a m e t e r a . The s o l u t i o n i s r e p by a linear combination of double layer and volume potentials if a is distinct r

from the proper frequencies of the problem

(II)~,0 , and by a linear combination of simple layer,

double layer and volume potentials if ma is a proper frequency of the problem (I~)+0. ]?roof. We seek the solution in the form

tt (x) -~6C l'!~(S), ~6C ~ (S)

where

~

(,b)"(x) -~-V (7) (x) + ~/2U (F) (x),

( 1.48)

are unknown vectors.

For the density ~, according to the boundary condition, we obtain the integral equation

'

.

\Oj ' .

,(g)

=f(z)--V(?)(z)--p/2U(F)(z).

(1.49)

S

I f c02 i s n o t a p r o p e r f r e q u e n c y o f t h e p r o b l e m (ll)0+.0 , t h e n , by Theorem 6, t h e nonhomogeneous e q u a t i o n ( 1 . 4 9 ) , i s s o l v a b l e f o r an a r b i t r a r y r i g h t - h a n d s i d e a n d , t a k i n g ~ = 0 , we h a v e t h e proof of the theorem.

6,

Let 2 be a proper frequency of the problem (ll)0+.0. In this case, according to Theorem t h e n e c e s s a r y and s u f f i c i e n t conditions for the solvability o f Eq. ( 1 . 4 9 ) h a v e t h e form (k}

~ V (,~) (z) ~(z)clS=A~,

I~=I, 2 . . . . . ,~,

(1.5o)

S

where Ak=

S

(k)

f(z)?(z)dS--~/2

~

(k)

k

U(F)(z)?(z)dS,

v {~}k=1 is a complete system of solutions of the

s associated homogeneous equation (1.37).

But (k)

iv

~

S

(k)

(k)

v

('4 a s =

dS

S

(k)

(k)

and, by virtue of Theorem 6, V ( ? ) ( z ) = @ ( z ) , where {4(Z)}k= Iv is a complete system of orthonormal solutions of the homogeneous equation (1.36). Therefore, the solvability conditions (1.50) take the form (~) ~,(z)~(z)dS=Ak, k = l , 2 . . . . . ,~. (1.51)

l

S

(l)

It is clear now that in (1.51) the vector ~ has to be taken equal to

~=~A~,

and

i=!

the theorem is proved. It is important to note that the obtained solution is unique, in spite of the fact that integral equation (1.36) has nonzero solutions; this is a consequence of the fact that, according to Theorem 5, the double layer potentials with the indicated densities are equal to zero in D-.

THEOREM 10.

If

S@JI2(,r a>O, fEC~

FEC~

~>0, F=O(]x[-2-s),

e>0,

t h e n the p r o b -

(o

lem (II)~,F i s u n i q u e l y s o l v a b l e f o r any v a l u e o f t h e p a r a m e t e r 2 . The s o l u t i o n i s r e p r e s e n t e d by a linear combination of simple layer and volume potentials if a is different ~+ from the proper frequencies of the problem (I)0,0 , and by a linear combination of simple layer, ~ double layer and volume potentials if 2 is a proper frequency of the problem (1)0,0. Proof.

We seek the solution in the form

u (x) = V (V) (x) + W @) (x) + p/2U (F) (x), where

".?~C~

(1.52)

~,~CI"~(S) are unknown vectors.

For the density T we arrive at the integral equation

457

- ~ (~) +

~) r (~-- v, ~0)]~ (v) a,S = f (~)--[rw (,) (z)l-- p/2ru (P) (z).

T

( l . 5 3)

o+ If a a is not a proper frequency of the problem (I)o,o, then, by Theorem 7, the nonhomogeneous equation (1.53) is solvable for an arbitrary right-hand side and, taking ~ - O, we obtain the proof of the theorem. r

Let m~ be a proper frequency of the problem bility condition has the form

l

(I)0+0. By virtue of Theorem 7, the solva-

(k)

[TW(~)(z)l-.~(z)dS=B~,

k=l,

2 . . . . . ~,

(1.54)

S

(k)

where

Bk:~.f f(z),(z)dS-P/2 (k) i TU(F)(z),~(z)dS (k) S

{'~ (z)}~'=l is a complete system of linearly

and

S

independent solutions of the associated equation, in the class C:'B(S),

:o.

(1.55)

S

On the basis of Theorem i0, Chap. I, (1.54) takes the form

(k) ~[TW(~)(z)]+~(z)dS:-B~,

k=l,

2 . . . . . ~.

(1.56)

8

(j)

On the o t h e r hand,

the d o u b l e l a y e r p o t e n t i a l

W(~)(x) s o l v e s t h e p r o b l e m

(71

(j)

(k) ~

(1)0~o and, (k)

,

according to Theorem 7, we can assume that [TW('~)]+~---p. Clearly, if {?}k=~, {@};=I are biorthonormal, then condition (1.56) is automatically satisfied for

Bi r (z).* j=l

The uniqueness of the obtained solution is proved in the same manner as above. Remark. Theorem i0 can be proved without applying the biorthonormality property of the complete systems of linearly independent solutions of the associated equations. This will be proved below on the example of the exterior problems of the theory of thermoelastic oscillations. 7. The Representation of the Solution in Generalized Fourier Series. H e r e w e present a variant of the Fourier method, based on the idea of series expansion with respect to orthogonal functions. We shall see that as a computational method, this variant has an efficiency advantage over the usual Fourier method. The fact is that the application of the Fourier method to the solving of boundaryvalue problems leads, in general, to the necessity of expanding the desired function with respect to the elements of some basis system, which itself is the solution of a new boundaryvalue problem, not simpler than the initial one. Therefore, the numerical realization is possible only if we know the eigenvalues and eigenfunctions of this problem. The method presented here allows us to construct in all cases, except for some characteristics (see the remark to Theorem ii, Chap. II), the basis system in an explicit form directly from the given data of the problem. Nevertheless, the method possesses a great generality, allowing the application to very diverse problems, including to those which

*On the basis of Theorem 8, one proves [9, 14, 15] that the complete systems of linearly ink k v dependent solutions {(pv }k~1,{~}~=i of the associated integral equations can be considered, without loss of generality, to be orthonormalized

I (i)(~) 8

458

[0, i:~k,

cannot be solved by some of the known methods

(see details in [14, 15]).

e+

We consider the problem the other problems. First Method.

([)f,~ and we restrict ourselves to short indications

According to the results of Subsection 2, Sec. i, if a 0)

regarding

is not a proper o+

frequency of the homogeneous problem (1)0+0, then the nonhomogeneous uniquely solved for arbitrary f and F from well-defined classes.

problem

(I)f,? can be (~

,

We make this assumption and we construct the approximate solution of the problem (1)~,F. ~+

Let u(x) be the exact solution of the problem

(l)f,r. According to (1.18') we have

(x) u (x) = ! r ( x - v, ~,) ~ (v) dyS-- H (x), VxCD + U D-,

(1.57)

"s

where

l o-~,

( v - ~,

a,s

i r ( x - v, ~) e (v) av

S

is a given and

D+

?(y)=[Tu(y)] +

an unknown vector.

%

L e t S be an a r b i t r a r y , smooth, closed surface,~containing S and h a v i n g no coma-non p o i n t s with it. The finite domain bounded by the surface S will be denoted by D. Let

{xe}~_l be a c o u n t a b l e

set

of points

on ~,

distributed

i n an e v e r y w h e r e d e n s e man-

ner. THEOREM ii. set

If ~= is not a proper frequency of the problem (|)~0, then the countable

{F(x'~--y,

~)}k~_-l~

i=l,

Proof.

a) Linear independence.

2, 3

is

linearly

3

i n d e p e n d e n t and c o m p l e t e i n t h e s p a c e L a ( S ) .

Let

N

i

~ ci~['(x~-y, ~o)=o, vg~S, i~1 k = l 3

w h e r e Cik a r e c o n s t a n t s

and N i s an a r b i t r a r y

natural

number.

The v e c t o r

3

i

v ( x ) = ~ ~cikF(x k i~l

X,~>) ,Vx~D" is the solution of the problem (I)0,0 and since 2 this p r o b l e m , t h e n v ( : z ) = 0 , YxGD +. Because of the analyticity v (X)=0, V x ~ b or, in components

N

k=l

is not a proper frequency of i n ~, from h e r e we o b t a i n

N

Y~ Y, e~rji(x~-x, ~)=0, ]=1, 2, 3, vXG0.

(1.58)

~=I k=l %

If we make t h e v a r i a b l e

point

%

x from D to approach the a r b i t r a r y

point

x k*, 1-O,

(2.38)

presents interest as an essential instrument for the investigation of the problem of nonstationary motion. Eq.

In Subsection 3, Sec. 2 it has been proved that the interior and exterior problems for (2.38) admit unique solutions under the condition Re T > 0. Now we have the possibility to prove existence theorems for these problems.

We establish some properties of the numbers Xk, k-- i, 2, 3, as functions of the parameter ~ = ~ + i~, o > 0; these properties will not be used only here. Equations (3.6), which determine Xk, have now the following form:

Xl+X2------

(1-}-:.)_

~+29J'

)~1X2=~ )~+2,u'

a-~-

~ '

k+2"---~"

(2.39)

The parameter e plays an important role; its values are contained in the interval (0, i) and for the majority of actual bodies we have s 0.

)'~ --n~-,~ X~2 __

p~§ 2,u ' ~,3'~

p~ ,u

A detailed examination shows that the parameters

%k, k = I, 2, 3, have the following properties:

if :s~- p~-')u - - ~ (I---:)>0, then for s > 0 in the

complex half-plane ~ > os we have a) k k (~) ----~k -P i,Sk, /e = 1,2, 3, ~e > b) kk = k k ( ~ ) , (=) = 0 (l~l), k = 1,2, 3.

k = 1, 2,

O, 1,~@ t,~,

3, a r e

analytic

functions

o f T and f o r

large

values

of

lr ),k

We p r o c e e d now w i t h t h e p r o o f o f t h e e x i s t e n c e theorems for the pseudooscillation problems. The matrix of the fundamental solutions of the homogeneous equation B(~/3x, iT)U = 0 is obtained from the matrix ~(x, m) by the substitution m = iT; these solutions contain expressions of the form exp(i%k[x --y I) and by virtue of the property a), in the half-plane Re T > oc, the elastopotentials in which the solutions of the boundary-value problems are expressed will satisfy at infinity stronger damping conditions than those which were used in Subsection 23, Sec. 2 for the proof of the uniqueness theorem. Obviously, the integral equations of the boundary-value problems are now singular, but with principal parts coinciding with the principal parts of the corresponding integral equations for B(3/~x, ~)U = 0, more exactly, they differ only by completely continuous terms. Therefore, the integral equations that solve the boundary-value problems of the pseudooscillations belong to that type for which the Fredholm and alternative theorems remain valid. But the homogeneous equations which correspond to the nonhomogeneous equations under consideration, for Re ~ > os, according to the uniqueness theorem, have only the zero solutions and, consequently, the nonhomogeneous equations are solvable for Re ~ > ~ for arbitrary values of the right-hand sides. Thus, we have

472

(IV)?,H for the equation THEOREM 24. If R e ~ > o e > 0 , the problems (1)f,H,(H)~,H, (ll)P,H, [+ B(3/3X, iT)U = --H are uniquely solvable and the solutions are represented in the form: for the problem (1)fi,H--U (x)= W @)(x)-1-~- U (H)(x), for the problem

~ ~ (II)Y,H-U (X) = V (?)(x) + 1 U ([-I)(x),

for the problem

(III)},n--U(x)=L ( k ) ( x ) - ~ U(H)(x),

for the problem

(IV)},s--U (x)=M(,u,)(.'r - 1 U (H)(x),

l~

1

w h e r e ~, ~, X, ,~ a r e t h e s o l u t i o n s of the integral e q u a t i o n s o b t a i n e d by t h e a c t i o n upon t h e thermoelastopotentials, W, V, L, M a r e s o l u t i o n s of the homogeneous equation B(~/3x, iz)U = 0 c o r r e s p o n d i n g t o t h e o p e r a t o r s E, J?, 5~ G, w h e r e E i s t h e 4 x 4 i d e n t i t y matrix 9 8. The R e p r e s e n t a t i o n of the Solutions by Series. The a p p l i c a t i o n of the approximate m e t h o d s d e s c r i b e d i n S u b s e c t i o n 7, S e c . 1 i s r e a l i z e d i n t h e p r o b l e m s o f t h e r m o e l a s t i c i t y without essential alterations. Obviously, the analogy with the oscillation problems of the theory of elasticity is m a i n t a i n e d and w h a t h a s b e e n s a i d t h e r e a b o u t t h e o s c i l l a t i o n frequency 2 remains valid also here. I n t h i s c a s e t h e f u n d a m e n t a l m a t r i c e s h a v e d i m e n s i o n 4 x 4 and i n o r d e r t o emp h a s i z e some p e c u l i a r i t i e s w h i c h f o l l o w f r o m h e r e , we c o n s i d e r one o f t h e f u n d a m e n t a l p r o b l e m s i n a more d e t a i l e d m a n n e r . Assume, f o r e x a m p l e , t h a t one s e e k s t h e a p p r o x i m a t e s o l u t i o n o f t h e p r o b l e m (~l)+,n, whose exact: s o l u t i o n we d e n o t e b y U = ( u , u 4 ) . Let 2 be a real number, distinct from the p r o p e r f r e q u e n c i e s o f t h e p r o b l e m ( 2 . 1 5 ) and ScJI2(a), a > 0 . Then, f o r U = ( u , u , ) , a c c o r d ing to the formula (2.21), the general representation o f a r e g u l a r v e c t o r , we s h a l l h a v e

VxED+:2U(x)=---

~ -~-~,n O(y--x,r

T(y)d~,S-q-2(x),

(2.40)

S

(2.41) S

where

(y) = u § (y), v y c S

is an unknown vector with four components,

(x) = f r ( x - v, ~) e (y) d,S + ~ r (x-- v, ~1 H (y) dr. S

THEOREM 25

D+

The countable set ! ~

,a, ~(y--xk,~) /

9

i---1,2,3,4, is linearly indek=l'

pendent and complete in the space Lz (X).* Proof..

a) Linear independence. 4

Let

N

l

y

(2.42)

t=1 k = l

where cik are constants

and N i s an a r b i t r a r y

natural

number 9

We consider the vector 4

N

i

~(x)=Y, Y,c~(x-x'~,oO, x~g. /=1

~'=I if)

According to (2.42) this vector solves in D + the problem

co *Here {xk}k_lcS

(II)~,;

is the same as in Subsection 7, Sec. i.

473

v x ~ O : :/~ ,'i,o-f o , o~',i ~7 = O, v z ~ , S : (~, 9 ) ~ = O, (0

and since the spectrum of the proper frequencies of the homogeneous problem

~~c~* ,)o.~ coincides

with the spectrum of the problem ([I)iL~ , while ~ , according to the adopted condition, is different from the latter, we have L~i~r)=0, Yx@f)~ and, due to analyticity, ~'(x)~O, vx6~); then, repeating the usual arguments we prove that all Cik = 0. b) Completeness. Let %(y) be an arbitrary four-dimensional vector from L=(S), satisfying the orthogonality conditions

.t ~("aT.'J'~zJ ~ ( Y - - x ~ q ~

i=1,2,3,4; k=1,2,3 .....

(2.43)

S

We introduce the double-layer potential

S

By virtue of (2.43) and because of the density of the distribution of the points x k on S we shall have V/ (},)(x)= 0, vxCS. At infinity W(l)(x) satisfies the thermoelastic radiation condition and is a regular solution of the equation B(~/3x, ~)W = 0, vx@S 9 Therefore, according to Theorem 14, W ( ~ ) ( x ) = 0 , u From here, by analyticity, W(),)(x):=O, vx6Oand, consequently, ~ Y = 0 , Vx6D-. Taking the limit from without, when x-+z6S, we obtain for ~(y) a homogeneous singular integral equation. According to Theorem 30, Chap. I, the solution of this equation in the class L2 (S) is included in the class C :, ~ (S), ~ > 0. Applying to W(%) (x) the Lyapunov--Tauber theorem (see Theorem 22, Chap. I) we obtain [~W(1)]+=0, and, since m2 is distinct from the proper frequencies of the problem ([~)+0,we have V/(}~)(x)=0, Vx~D +, and, finally, from the boundary properties of the potential W(X) (see Theorem 18, Chap. I) we obtain l(y)=0, VyGS; the completeness, and therefore, the theorem is proved. I

We enumerate the elements of the set {~(y--x~,~)}~=~ , i = I, 2, 3, 4, in the following manner :

,

. . . . .

k

and we orthonormalize the system have

{~(y)}k~=1 on S according to the Schmidt process; we shall k

k

{~,

s

(u)}~_-,, ~ (y) = ~ h~,~ (~), k = 1,2 .....

where h k , s are the o r t h o n o r m a l i z a t i o n c o e f f i c i e n t s . tern

(2.45)

Obviously, the complex conjugate sys-

(k) -

-

co

{p (y)}k=l.

(2.46)

is also complete. 9(y)

We represent the unknown vector (2.46)

in a Fourier series with respect to the functions

m

(y) ~ ~ r

(y),

whe re m

c~m =

i ~ (y) ~ (y) d y S ,

m = 1,2 .....

8

These coefficients are computed from the functional equation (2.41). (2.41), we have, in particular,

474

Indeed, from

f $

-

0

*

or

4

0

~-

S k=l

where jk, space.

k = I,

2,

3, 4,

are the unit

L e t m b e an a r b i t r a r y

natural

coordinate

vectors

of the four-dimensional

n u m b e r and s = [ (m+ 3 ) / 4 ] .

vector

We t a k e t h e i n n e r p r o d u c t o f

(2.47) with the vector C"i =/z,,,,4z_3j I +hm,41_oj2+hm,4~_~js+h,n,4~j 4 and we sum with respect to i from I to s, and taking into account that by definition we can assume that hm,k = 0, k > m,

S k = l 1=1

t=1

From (2.44) we have

~

0

) k

0j' g ~ ( y - x ' ' ~ ) =

4(i-1)+k

,

(y), k = 1 , 2 , 3 , 4 ;

i=1,2 ....

and (2.48) takes the form 4

+

4(/--1) + k

s

(2.49) S k = l /=1

t~l

But 4

8

rn

4(t-t)+k

h

n

+, (y)= n2]~ l ,.,.* (y)

k = l t=1

8

and on the basis of (2.46) we obtain from (2.49)

(Dra=XCrniQ(X l) , where m = i, 2, .

., s =

i=l

[(m + 3)/4], and the Fourier coefficients are found. We construct the vector

U (N)(x)=-l/2

! 0 n ~(V--x,o) s [ ~ kO~'

Applying the Cauchy--Bunyakovskii inequality,

* _ r we o b t a i n

v x E D +. as in Subsection

7, S e c .

1,

U (x)---- lira U (~) (x), v x f i D ' c D + N~

and the limiting process is uniform. The desired approximate solution is u(N). The application of the second and the third method is obvious. Other problems are solved in i similar manner, with obvious modifications. Remark.

If ~=i~, R e = > O , = 8 > 0 ,

then we have the pseudooscillation problems.

According to Theorem 24, they do not have eigenvalues and the approximate methods are applied without any additional assumptions. CHAPTER III NONSTATIONARY MOTIONS i.

Initial--Boundary-Value Problems of the Theory of Elasticity

i. The First Fundamental Problem. Conditions for the Given Data. The problems in which in addition to the boundary conditions it is necessary to satisfy also initial conditions, are called initial--boundary-value, or mixed, or nonstationary problems. Unlike the

475

stationary motions, in the nonstationary problems one investigates the propagation of the oscillations. At the investigation of this phenomenon there arise new mathematical obstacles which correspond to the complication of the physical situation of the dynamical state. The nonstationary three-dimensional problems for domains of an arbitrary configuration have been investigated very little in comparison with other problems of the theory of elasticity, at least by those methods by which the latter have been studied in a sufficiently detailed manner. A detailed investigation by classical methods of the mixed problems of the theory of elasticity in the three-dimensional case can be found in [12, 13, 50]. In this chapter we present these results. Problem ~)~F,~ We seek a vector u(x, t) (uECa(Q~)~C2(Q~)) , regular in the cylinder Q~ ={(x, t):xED", rE(O, oo)}, satisfying the conditions:

V(x, t)EQ~:A

(0)

a,~ (x, t) u(x, t ) - - p 7 = - - F ( x , t ) ,

vxED +: Iira u (x, t)-= ~(~ (x), t~+o

(1.1)

lira au (x, t) = ~o) (x),

'

t..+o

ot

( i 2)

'

v(y, t)ES~:u+(V, t)=/(y, t),

(l. 3)

for large values of t a~au

(x, t)

~, m, ., . . . .

at Oxx Ox2 ax 3 m=ml+tn~.nUm3-[-tn4, ink>O, re=O, 1,2, % > 0 S ~ = { ( y , t):yES,

F(x, t),

rE[0, or

v ( x , t)EQ~,

is the lateral

@~

@Z)(x),

[ -.< c exp (%0, i s a c o n s t a n t number,

s u r f a c e o f t h e c y l i n d e r Q~,

vx~D +, f ( y , t), v(y, t)~Soo

are given vector-functions.

Assume t h a t 1) t h e r e e x i s t

continuous derivatives

aV+qF(x.t) .

.

.

.

Vx~Z) +,

t>~O,

o f t h e f o l l o w i n g form:

q:+q2+qa=q,

O%>%.

~k(T) and

~(x, ~) are given functions and ~ = o + i ~ , a=const,

The uniqueness results from formula (6.14), Chap. I. Indeed, if u(x, t) is a solution of the homogeneous problem (f)+0,0, then from the indicated formula we obtain J'(t)~-0, where t >i0,

D+

and, consequently, u --- 0. 2. Reduction to a Special Case. The functions ~((x]) , m = 0, i, 9 ., 5, and consistency conditions (1.5) allow us, without loss of generality, to reduce the problem (I)~.?,~ to a similar problem, but with zero initial data, with a special "right-hand side of the equation" and with special boundary values for the solution on S. This is achieved by introducing some vector-function C(x, t) which is defined in the following manner: Vx@~ ~, t>/0, C(x, t) is five times continuously differentiable with respect to t, satisfies the conditions

vxED+:{ o~c~ (x, t) ) ,=o=~,~)(x),

ra=O,

1. . . . .

5,

(1.8)

and vanishes for large values of t. Such a function has been applied for the first time in the theory of the Cauchy problem by S. L. Sobolev [37], Subsequently, this function has been used by O. A. Ladyzhenskaya [16] in the mixed problems of the hyperbolic equations. Under the assumptions made, the vector-function C(x, t) can be expressed in volume integrals, based on the properties cf the averaging kernel and mean functions (see [15]). Let

go(X, t)=t~(x, t ) - - C ( x ,

t),

(1.9)

where u(x, t) is the solution of the initial problem. By virtue of (I.i), (1.2), (1.3), and (1.8), uo(x, t) will be the solution of the problem .

tto--p--~-~ ~ - - - F o ( x , t),

vx~D+: t~.+o lim Uo(X , t):= tlira Otto(X, t) .+o ot =0,

(I.i0)

(l.ll)

477

(1.12)

v (y, t)es~o : . ~ (y, ~)--f0 (y, O,

c}muo(x, t)

] < C exp (%0 !

O<m 0, S' being the boundary of the domain D' Consequently, for xEO" we shall have

where

O ~ (2'1'1) (X, ~) ]

OxiOxy

C

..< .~--f-, Vx@D'.

(1.52)

Then 0qt(2'1'2) (X, ~)

=p.21 0F

OXl

D" " ~ i

=p~2

•(1)tr t*O ~ y

Ia/~0(1) COS (t~, xi) dyS--pz ~ r

S"

'

or OU(o~)de;

I-a~176 7

OxiOx7

S"

from here, by virtue of (1.32), (1.36), for

D" xCD" we obtain

(x, x) I ~ c [0 ~ (2'1'2) OxiOxj " ~" T~-P" v'c6IL,, Finally, from (1.47), (1.48), (1.50),

(!.51),

(1.52),

I 02~(x,'0 [ ~ c OxtOx] ~" T~" 0.

With the adopted notations, __

;@) (x, ~)=

dyS,

D"

Vx6D',

Yx6D'.

(1.53)

(1.53), we have V'c6B~o.

(1.54)

(1.29) takes the form

p~t

~ i c.~ (x, v) g~)(v, ~)ev+ 89v (x, ,),

~1.55)

D+

whence (1.56) and

~[IOu(~ l dx "< and according to ( 1 . 9 ) , Chap. I I ,

dx] (1.43),

(y,

dy

I

O~

(1.46),

S[ 0~2, dx~< C

(1.57)

From (1.55) we have Om(o:)= 07,

_ _ ~ ~2

S 06(i) u(2) ~.

O~

and by virtue of (1.46)

483

(1.58) In order to estimate the integral

~" =

!'? CI~

' ~(~) dy

D+

we represent it in the form of the sum 2

'~ =-~-E ,Yk,

I IO~

;~=i-?

r,(~)

~=I

whe re

Dx=D+~III(x,

D~=D+NDT,

[~i-~),

~>0.

We have

DI where

Dl

x* is some point in D~.

According to (1.44), from here we obtain 1 9 I '~+~

(1.59)

"

Then, on the basis of (1.43) 1

;, (x, v; - ~ ) ~U~(~, ~)d~ D+

=

D+ On t h e o t h e r hand, i t i s known t h a t t h e homogeneous problem c o r r e s p o n d i n g to (1.93) a d mits only the trivial solution for c~H~0. Therefore, the homogeneous equation corresponding to (1.96) does not have nonzero solutions and, consequently, the nonhomogeneous equation (1.96) is solvable for an arbitrary right-hand side u Representations (1.95) and (1.96) show that ~o(X' ~)_____~)+U~2) is an analytic function of ~6~eo vx6O +. The required estimates for

~o

are obtained from the representation

S

by the application of Theorems 8 and 31, Chap. I, while the estimates for and estimates (1.14), (i.16), Chap. II.

~2) from (1.96)

We turn now to the fact that the kernels and the right-hand sides of Eqs. (1.29) and (1.96) depend equally on the parameter T. Due to this, the transformations and the proofs from Subsections 5, 6, Sec. i, remain valid and just as in the first problem, in this case we also arrive at the asymptotic inequalities ^ iu~

: ) i ~ ] - ~:~7 '

!t a~o(X, c ax~ ~)i~< ivlet~,

[ O2~Zo(X,oxiOxjT)]~ [T,!I'~1/9,

u .-+ ,

VxED'cD*,

2 v-~IL0, 0. 0, being an arbitrary number, and we shall not go into the details which can be found in [33, 34]. We assume that the following conditions hold:

1. VIE[0, T] : F c C 4 (/~+), VX6f) + :F6C 5 (0 ~< t < T), _-~+ /OmF ( x , t)\

Vxet) :(

~t-~

^

)t=o=U,

r n = 0 , 1,2, 3.

2. vt(~[0, T] :/tiC 2'" (S), ~ ;>O, vg6S :/~C 7 (0 4 t 4 T),

0~(~ GL2(D+), nl+n.2+rta-----4, o~'04"04'

3. ~(~247

(1.102)

vV(~S: [~(~ (g)l+ = [ A ~,oy{/'1re(o)(g)]+= 0, 0%(I)

. . D +,

vyeS : [~(')(v)I+ = [A~('>(y)]+ =0, 5. SEJI~(~), ~ > O.

(1.103)

We represent the solution u(x, t) in the form

u(x, O==u(~)(x, O§

(1.lO4')

t),

where u (*) and u (a) are the classical solutions of the following problems:

~ v

d2u (1)

0,

t)cQ :A

vx~D+: lira t~(I)(x, t ) ~ lira au(1)(x' 0 ==0, t~+o

t~-t-o

v (y, t)ESr:[u(1)(g, t)] §

Ot

(i. 104)

(g, t),

and

vx6O+:lirnu(2)(x, 0=?(O)(x), lira -au(2)(x, at t)=?(~)(x), t~+0

(1.105)

t-~+0

v (g, t)~Sri[U(2)(V, t)]+=0. The existence of u(X)(x, t), under the assumptions made, follows from the results of Subsections 3, 4, 5, 6, Sec. i; in this case, by virtue of the fact that in problem (1.104)

490

the initial conditions are homogeneous, it is sufficient to require from S that it should belong to the class ~]~(::),~ > 0 . In addition, the vector functions F(x, t) and f(y, t) must be extended for t > T so that the extensions should have a growth with respect to t not faster than c.exp(oot), oo > 0. We solve problem (1.105) by the Fourier method.

Applying the scheme formally, we have

%")

sinVp---~Ont,

(1. 106)

n=l

where

{~~

oo {?),,(x)}.=1

are the eigenvalues and the eigenvector-functions

,

v (x) + ~ov (x)

of the homogeneous problem =

O, vy6S: v+(v) = 0 ,

(1.107)

and

v(o) = (n)

~ v(O)(x) v.(x) dx, ,o") --~ ~,('(x) v.(x) d x r(n) --~

,I D+

(1.108)

D+

are the Fourier coefficients. According to (1.21), Chap. II, problem (1.107) is equivalent to the integral equation (i .109)

(x) - 2 i O(D(x, y) v (v) ay D+

From (I.i09), on the basis of estimates (1.7), (1.9), Chap. II, one can see that

v.(x)eC~(D +) n o(o+), ~. > o. Applying Green's formula (6.7) ' Chap. I to the vectors account that [p(o)]+~v+~O, we obtain

(A,~(o)~.. T ]~n)-where

(A,p(o))(n)=I

A (~x)?(~

~

~(0) T(X) and Vn(X) and .taking into

,on ,|( n ) ,

(i.iio)

are the Fourier coefficients for AT(~

D+

Similarly, as a consequence of (AT(~ taking into account (i. Ii0), we obtain

(A2Tf~176176

and

~(o)__ (A'q~(~ (n) ~

(i. Iii)

Proceeding in the same manner, for ?(t) we shall have (1.112)

(A~(1))(n)

•(I)(n)

o)n

By virtue of (i.iii) and (i.i12), formula (1.106) will take the form

u(2)(x, ~~I--~ - - ~ v " (..-7X)(A2?(~ n=l

LEMMA i.

If

,)cos V o - ~ . t

t'vtt

x ) (A~(1))(,)sinVp-~% t. - - V. rP- ~~v . (~---~-_/2 n=l

(i.i13)

n

~(x) satisfies the conditions i) ~C0(D+),

2) o~iEL2(D+), 3) ,~+=0 , then

r p

,,=!

~,~,,;,>,,.~: \ E (v, ~) ax, /5'+

(i. i13' )

where

~''> = i ~ (x) v.(x) dx, E @, ~,) D+

491

is the bilinear form defined in (6.4), Chap. I. The proof follows from the formula

i E (~, v,,)clx- %~?(,,}.

(1.114)

D+

In turn, from (1.114) we have, in particular,

~De

-

-

-

{

rn ~/z, + n,

(1.115)

+

LEMMA 2.

If

~(X) satisfies the conditions

I)~6CI(D+), 2) ~ eo~L 2 ( D +),

3)

v+=O,

then we

have the equality

n=l

(1.116)

D+

For the proof we apply to ~ and Vn Green's formula ( 6 . 4 ) , Chap. I; we o b t a i n I (~A~.--v.A~) dx---O; D+

consequently,

f A~'vn=--o)n~r

),

D+

or

( A ~ ) ( n ) -= --COn~(n ) . P

The Parseval equality for A~? ~__~(A~)~n)-----~. ]AcpI2dx n=l

yields (1.116).

With the aid of the

D+

proved lemmas and the representation (i.i09), based on estimates (1.7), (1.9), Chap. II, we prove the following statement: the series (1.113), as well as the series obtained by a single term-by-term differentiation of (1.113), converge uniformly in the closed cylinder QT, while the series obtained by the double differentiation of the same series converges uniformly in any strictly interior subdomain of QT; thus, (1.113) is indeed the classical solution of the problem (1.105). For details and extension to other problems see [33, 34]. 2.

Initial--Boundary-Value Problems of Thermoelasticity

i. The First Problem. Formulation and Reduction to a Special Case. The method of solution of the nonstationary problems of the theory of elasticity presented in Subsections i, 2, 3, 4, 5, 6, 7, Sec. i, are extended to the solving of the fundamental problems of thermoelasticity. We show this on the example of the first problem and with respect to other problems we shall give short explanations and we shall indicate references. Problem (I)+,H,w. Determine a vector U = (u, u~), regular in the cylinder Q~, from the conditions :

0 ) u ( x , t ) - - Tgradu4(x , t)--p O~u(x, V(x, t)EQoo:A {lfi--iOt2 t) AU4 (X, t)

10u,(x, t)• Ot

h(x, t),

(2.1)

v O div U (X, t)=--114 (X, 1~),

vx6D+: lira u (x, t ) = ~ (~ (x), t--,+0

lim ---~---=0u(x't) '~(~ t~+0

lim u4(x, t)=~l~

v (v, t)~s~, u + (v, t) = F (y, t),

492

(2.2)

i--*+0

(2.3)

for large values of t O'nU (x, t)

otmlOXr~zOx~n~Ox~n, [ ~< c- exp (;or), 4

m=~

m~, rot,>0, m----0, 1,2, % > 0 .

k=l

Here H ( x , t ) : ( h , h~), v(x, t)GQr .~(o) ?(1), ?~o), vxE-D+; F(y, t ) = ( f , f4), v(y, t)fiS~ a r e g i v e n f u n c tions. The problem of the smoothness of the given data, basically coinciding with those which were assumed in Sec. i, will not be considered here; the necessary corrections will be made in the course of the presentation. For the existence of the classical solution "natural consistency conditions"

U6CI(Q~) NC2(Qoo)

it is necessary that the

v y 6 S : ~(o) ( v ) = f (v, 0), ~ (~) (V)= /o/ (~, t)~, ,

,

~---07--

h=o'

?(~) (Y)=f4 (Y, 0).

(2.4)

be satisfied. From the same considerations as in the elasticity problems, we also require the validity of the consistency conditions of "higher order"

(~], t))t=0 , vy6S: ?(~) (y) = (O \ mfor"

?~,~-1) (y) = (

Om-'/g(q, t)) Ot~-' ]t=o' m = 2 , 3 . . . . . 7,

(2.5)

where

( O~-'h (x, t)~

@m) (x)=p-lAb(m-2)(x)- ~-lTgrad~m-2) (x) + p-a \

(

otm-2--h=o'

%~

(2. 6)

These conditions are obtained by the successive differentiation with respect to t of Eq. (2.1) and the substitution of the given data for t = 0. The uniqueness of the classical solution follows directly from formula (6.16), Chap. I. If U = (u, u4) is a solution of the homogeneous problem (I)0~0.0, then from the indicated formula we have

j , (t) = -- ~- I Igrad u412dx ~ 0 , q D+

where

J

[O" 2 '

1 E ( l l , U,)Af- 2_~i_]lg4!2]dx>O;

D+ from here, by virtue of 2(0)-0,

it follows that O - 0.

In the same way as in Sec. I, the formulated problem can be reduced, without loss of generality, to a similar problem with null initial data, with a special right-hand side of the equation and with special boundary values on S. The solution of this "reduced" problem, which we call Uo(x, t) = (uo, uo~), is connected with the solution of the initial problem by the relation

U0 (x, t) ----U (x, t) -- C (x, t),

(2.7)

where C(x, t ) = ( c , c4) i s an e a s i l y c o n s t r u c t e d v e c t o r - f u n c t i o n w i t h f o u r components ( s e e [ 6 ] ) , s i m i l a r t o t h e one g i v e n in S u b s e c t i o n 2, Sec. 1, Chap. I I . For Uo(x, t ) we have t h e p r o b l e m

fo:f

.+

(l)Fomo,0, where Fo = ( f o ,

f o ~ ) , Ho = (ho, ho~) and

(Y, t)--c(y, t), fo~ (Y, t ) = f 4 ( g , t)--c4(Y, t),

493

! 0 k O~c(x,l) t)+A'--l~Ox / e (x, t) -- 7gradc4 (x, t)--9 az=

h~

! Oc,(x, t)

ho~ =h4 (.x:, t)=z-~c4(x, t) From ( 2 . 8 ) , ( 2 . 5 ) , we have

by v i r t u e

of the d e f i n i t i o n

of

'5

(2.8)

divc (x, t).

~ ..(m), ~.,) and C(x, t) , on the b a s i s of (2.4) ,

vxGD* : ( Om/~~ (x, t))t=0= 0, O~m m=0,

1.....

5, (\-Omho.(X, ~,~ O)/t=o = 0 ,

(2.9)

re=O, 1 . . . . . 4,

vvES : (, 0~/o(,,t)) o~'n j~=0=C, m = 0 , 1 . . . . . 7,

( 09/~Otto !-!)t=o=0, m = 0 , 1,. 9 6. 2.

The Laplace Transform.

The Solution of the Elliptic Problem.

complex variable in the half-plane

H~o:~>~o>%,

where

Let T = o + i~ be a

X+2~t ~ e = ~ +• 2 ~ {a8,%},%= P~ ~1 --'i,

%=max

(see Subsection 7, Sec. 2, Chap. II), ~o is the index of the exponential growth with respect to time for the given problem. Let V~EII%, co

/-/o(X' ~)----I e x p ( - - z t ) H o ( x '

t)dt,

0 oa

Po(Y, ~)= I e x p ( ' ~ t ) F o ( g '

(2.10)

t) dr.

0

We consider the boundary-value problem

Vx6D+ : _ A (oO.)u o (x, , ) - ~gradtto~- ~-~-^uo (x, , ) = -- h~ (x, ~), (2.11)

AUo4(x, -.) -- -~- t~o4(x, ,)-- ~divu o -- --)Zo4(x, ,), vv~s: Oo+ (v, ~1= Fo (v, ~1, which is obtained from the "reduced" problem place transform

+ (I)~0,H0,0 by the formal application of the La-

co

/Jo (x' ~)=S exp ( - - ~ t ) U o ('x, t) dr.,

(2.12)

0

Integrating by parts, from (2.10), by virtue of (2.9) we obtain

h~

~ ) = ~- \

=

Or"

~

exp(--~t)

(2.13)

or'

0

/~o4(X, ~)= -~- ~ Or5 it=o + I . exp(--~t) 06ho4(x'ot~ t) dt'

(2.14)

0

- j ~ dt, ]o(Y, :}---- ~- \:3V'lt=o + ~- j exp (--,t) O'/o

(2.15)

0 co

]o4 (y, ,)= ~ (0'1~

+~TIexp(--,t)O'/~

(2.16)

0

and, consequently, we have uniformly with respect to xCZ)~, yGS

e

~<e__

c

[ho(X, ~)i~< i-~, I~o~(X, ,)i~ i~io, ]]o(V, ~)1-.< ~, 494

the estimates

c

l]o~(v, ~)1-. 0 ,

~CII,.

"< 3'

Vx~-D +,

-, ~ zOLo, vxED ~-~D.

(2.40

(2.41)

Combining the estimates (2.22), (2.23), (2.39), (2.40), (2.41), we have on the basis of (2.18)

:'8o (x,

o, v~q%o, vx~D ~,

/O~D~ t ) / J e---i-- ~>0, V:EI1~, Vx6/ff'cD +.

(2.42

These estimates allow us to verify in a straightforward manner that the vector cr+ioa

1 I exp(~t)gT~ Uo(x' t )='T~7

,)d~

(2.43)

a--ieo

I + satisfies all the conditions of the "reduced problem" ()F0.~0,0. Then, from formula (2.7) one finds the vector U(x, t), the solution of the initial problem (I)+H,~, and then, in the same way as in Sec. i, one eliminatas the vector C(x, t) by reduction to the nonhomogeneous elliptic problem (2.11) with right-hand side equal to the vector

(x, . ) _

(x).

This problem is solved explicitly in generalized Fourier series (see Subsection 8, Sec. 2, Chap. II). W e have considered the first problem. The extension to other problems and the construction of the theory of the nonstationary problems of thermoelasticity are contained in [6]. LITERATURE CITED i.

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