Materials with Memory (C.I.M.E. Summer Schools, 74)

Dario Graffi ( E d.) Materials with Memory Lectures given at a Summer School of the Centro Internazionale Matematico E...

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Dario Graffi ( E d.)

Materials with Memory Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 2-11, 1977

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11095-5 e-ISBN: 978-3-642-11096-2 DOI:10.1007/978-3-642-11096-2 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Liguori, Napoli 1979 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(c.I.M.E.

I Ciclo

-

)

Bressanone d a l 2 a l l 1 1 1 giugno 1977

MATERIALS W I T H MEMORY

C o o r d i n a t o r e : Prof.

R. W.A.

BOUC

- GI

DAY

M.

FABRIZIO

G.

FICHERA

M.

RIBARIC

R.S.

RIVLIN

D.

GRAFFI

GEYMONAT : P e r i o d i c problems i n thermoviscoelasticity Pag. : The thermodinamics of m a t e r i a l s w i t h memory " : S u l l a nozione d i s t a t o n e l l a t e r momeccanica d e i c o n t i n u i l1 : A n a l y t i c problems o f h e r e d i t a r y phenomena : The b l a c k box approach and s y s t g ms w i t h memory " r The t h e o r y of c o n s t i t u t i v e equations l1

"

7

55 95 112

173 185

CEN TRO INTERNAZIONALE MATEMATICO ESTIVO (c.I.M.E.)

P E R I O D I C PROBLEMS I N THERMOVISCOELASTICITY

Re BOUC e G. GEYMONAT

Corso tenuto a Bressanone d a l 2 a l l 1 1 1 giugno 1977

Periodic Problems i n Therrnoviscoelastici t y Two Seminars given a t the Centro Internazionale Maternatico Estivo, Bressanone, I t a l y , june 3-11, 1977.

R. Bouc Laboratoire de Mecanique e t dlAcoustique C.N.R:S. ,!?arseille G. Geymonat Pol i tecni co di Torino

We thank Professor D. GRAFFI f o r his kind invitation t o t h f s t a l k , the aim of which is t o give a survey of some recent work done a t Laboratoire de Mecanique e t dlAcous+ique of C.N.R.S. Marseille, p a r t l y i n collaboration with M. JEAN, 5. NAYROLES and M. RAOUS, specially during the second author's year of leave from the Politecnico di Torino.

Contents Introduction. 1. Background on the mechanics of continuous media with hidden variables.

2. A lineasization result. 3, Some exsrnples.

4. Some resugts Qn the nonlinear heat equation.

5. Duality and virtual work. 6. A viscoelastic constitutive equation with periodic coefficients. 7. The periodic bilateral problem f o r a Maxwell body. 8. The periodic unilateral Signorini problem.

.

B i b ?i o ~ r a p h y

Introduction Following the fundamental work of V. Volterra 1371, I381 a 1391 ,1401 , hereditary phenomena in mechanics have been deeply studied. A great part of the work that.has been done (see 118 1,135 1,136 I,. ) treats the case called, by Volterra himself, "the case of closed cycle" (see T. VOGEL 134 I) which corresponds to the case where the relaxation function in viscoelasticity is of the type G(t--c ). However, in 1907, HATT 1191 has discovered the phenomenon of creep in concrete whiohpresents stress- independent deformations which, in addition to thermal dilatation, includes shrinkage ; the material properties of concrete change indeed as a result of internal chemical reactions and the deformation problem coupled with complicate moisture diffusion through the material, as well as heat conduction. For these reasons, in a first approximation, concrete may be regarded as an aging viscoelastic material whose creep law can be written in a rate-type form, i.e. as a system of first-order differential equations, involving hidden strains, with time-dependent coefficients 6 More recently it appears that also for other materials, especially polymers in a temperature depending situation, the relaxation function is not of type G(t i ) but following a fundamental remark of Morland and Lee 127 1, the relaxation function can be written as G(5 E') where E = E ( 0 ) i s the reduced time (fee also PIPKIN 1311). From another poirit of view the extension of phenomenological laws based on spring and dashpot models to the temperature depending case has been proposed by many experimentalists (see e.g. 1 4 /) specially for metal s. In this paper we shall recall in 51 some results on the continuum mechanics of materials with hidden coordinates (indeed hidden strains) and some consequences of the Clausius-Duhem inequality on the constitutive equations due to Coleman-Gurtin / 12 1 and Bowen 1 10 .1

..

.

-

-

(r)For a very deep review of the bas; c facts on this subject see Z.P. BATANTI 3 I.

10

Because we are interested in the study of a phenomenon w i t h high temperature variations, we develop i n 9 2 , in the case of infinitesimal strains, a linearization of the equations obtained i n 51 only w i t h respect t o strains and hidden coordinates. We conclude t h i s analysis in 53, w i t h some remarks on the possibility of uncoupling the nonlinear heat equation, similar t o those developped by Crochet-Naghdi 1 13 If o r therrnorheologicall y simple solids. In 54 we recall very briefly how the nonlinear heat equation obtained i n t h i s way can be studied in the framework of nonlinear evolution equations a s developped i n the book of Lions 1 24 1 In 95 we s t a r t the study of the equation of motion (with temperature a s a data, i . e . a given function of time and space-variables),recalling some results on duality and virtual work principle. In 56 we consider a constitutive equation of axw well-type where the "stiffness" and "viscosityn matrix are temperature-dependent and thus are time dependent. More nrecirely the temperature i s T-periodic i n time and therefore the stiffness and the viscos i t y are also T-periodic. With t h i s constitutive equation we survey, i n 97 and 58 some results concerning existence, uniqueness, asymptotic s t a b i l i t y of a T-periodic stress-field for the dynamic and quasi-static periodic b i l a teral problem (1 6 1 ,I 9 1 ,I 17 1 ) and also f o r the quasi-static Signorini unilateral periodic problem (1 7 1 , 1 8 b. We refer t o the lectures of G . Fi chera i n the present session for the corresponding Cauchy-prcblems . The applicaiions of our results on the thermal fatigue of metals due t o cycle heating and cooling will be developped i n the thesis of M. Raous 132 By lack of time we cannot develop these f i r s t results ; we can only say t h a t the numerical experiments agree wf t h the ecperimentaT results of F.K.G. ODQVIST and N.G. OHLSONl 30 f"The virginal specimens behaved i n a normal way, whereas those already cracked apparently proved t o be stronger against the formation of new cracks!'.

.

1.

1- Background on t h e mechanics o f continuous medium. with hidden v a r i a b l e s .

1.1- The mechanical and thermal behaviour i n t h e time i n t e r v a l ? ' ? C R ,o f a nonpolar body occupying t h e reference configuration Q C Ft3 a t time toE C '? i s completely determined by a vector function p(X.t) (giving t h e p o s i t i o n a t time t of a material point which has t h e p o s i t i o n X i n t h e reference configuration Q ) and by a p o s i t i v e s c a l a r function B(X.t) (giving t h e a b s o l u t e temperature a t time t o f a material p o i n t which has th. p o s i t i o n X 9n t h e reference configuration S l ). As usual we d e f i n e F(X,t) = Grad p(X,t), t h e deformation g r a d i e n t t e n s o r and we s h a l l assume t h a t p(X,t) is always smoothly i n v e r t i b l e , i .e.

(1.1)

d e t F(X,t)

>

forall

0

tee

Using Lagrange's coordinates, t h e local form o f t h e laws o f balance o f l i n e a r momentum, o f moment of momentum and o f energy a r e the following (4 (Xw) ( s e e 116 1) : (1-2)

Div FS

+

pof

Po

(1-4)

E

=

0,

=

tr SE

.-

Div q

+par

where q,=p#)is t h e mass d e n s i t y i n t h e reference position. 5 i s t h e synmet r i c second Piola-Kirchoff ( o r Lagrangean) s t r e s s tensor. E = $F'F 3 ) is t h e Lagrangean s t r a i n tensor, E = aE is t h e Lagrangean s t r a i n r a t e , Po P = Po a t 2 is t h e i n e r t i a forceat f is t h e body f o r c e f i e l d p e r u n i t

.

(x) If A (xx) t r ( . )

.

3

.

-

is an m x n matrix, A' denotes t h e transposed matrix. = t r a c e of (.).

-

mass, E i s the internal energy of the body per unite mass, q i s the heat conduction vector, r i s the heat supply field per unit mass. Let u s also recall that the law of the conservation of the mass allows us t o compute the mass density a t the time t with the formula det F =

"0 P

The local Clausius-Duhem inequality

where q i s the specific entropy per unit mass can also be written, using (1.4), i n the form

where g = Grad 8 Defining the Helmoltz free energy per unit mass by

we can also write (1.6) i n the form (1.8)

- P,

~1a

-

p0q6

+tr(~.~j-fi 2

e

o

1.2- The characteristics of material composing the body are usually stated by additional equations, the so-called cansti'tutive equatjons, which give thes t r e s s , the internal energy, the entropy and the heat conduction in terms of the Lagrangean strain tensor and the temperature field. Obviously the constitutive equations depend on the properties of the material that we are modelling, and i n the following we construct a model for solid-like materials (e.9. metals, polymers, concrete,. . ) whose response depends t o a large extent on their past history (a qualitative explanation of t h i s fact can be given in terms of various microstructural rearrangements due t o dislocatians motions, longchain molecules, phase transformations,...).

.

13

We will account for such microscopic structural rearrangements by the introduction of additional state variables called internal or hidden coordinates and denoted collectively by E which in a certain average global sense represents the internal changes. As is pointed out by S. Nemat-Nasser (129 Ip. 110) : "The representation is macroscopic in the sense that there may exist multiple (in fact, probably infinitely many) microstates corresponding to the same values of these coordinates. However, inasmuch as these coordinates are characterized by certain constitutive relations involving various parameters, which are fixed by means of suitable macroscopic experiments, they signify the most phenomenologically dominant aspects o f the microstructural changes". On? can assume that the hidden coordinates are various tensorial qutntities that transform in a suitable way under a change of frame, here we shall assume for simplicity that F is a symmetric positive definite tensor invariant by orthogonal change of frame. '? 1.3- A thermodynamic process is a set of functions of X E 0 and t a ?

that satisfy (1.2), (1.3) and (1.4). In order to be frame indifferent, the lagrangean stress, the free energy, the entropy and the heat flux are defined as functions ofthe material point and of the actual values of the state variables E, E, 0 g, E (the thermodynamic state) :

.

I n order t o f i x t h e v a r i a t i o n o f t h e hidden coordinate 5 we s h a l l assume ("1 : For a l l X e R , there e x i s t s a function h o f t h a t alonq any process durinq the time i n t e r v a l B

E,

' 6 , 8 , g,

Moreover f o r a l l t o e and a l l E0 there e x i s t s a unique ( (X,t) f y i n q (1.13) f o r a l l t E e and 5&t,) = COW.

5 such

satis-

1.4- The c o n s t i t u t i v e equations o f materials t h a t we have i n mind are based on analogies t o spring-and-dashpot models ; indeed these simple models display q u a l i t a t i v e l y retarded-elastic, creep and r e l a x a t i o n phenomena t h a t are encountered i n polymers, concrete, metals

- (Thermoelasticity)

w

Example 1

5, = h

We take

K

.

0 and S = K( 0 )E

5

elasticity. If

eo

+ A(B

)

. We have the usual thermo-

i s the reference temperature i n

the reference configuration, without stress, we must

- K(8)

eo

write A ( O ) . =

E

i s the thermal d i l a t a t i o n tensor. , ,e = X( 8 ) ( 8 - go) i s the thermal d i l a t a t i o n and thus S = K( €r )(E t eo).

,~(8)(8-0,)

where x . ( 8 )

-

Fig.1 Examp'le 2- (Maxwell body) We have t h e r e l a t i o n s

E

(1.14) 5

1

S

=

K(e )

s

.=

v(e)i

[E

+

eo

-5)

from which i t f o l l o w s

(x) As has been pointed out by G. CAPRIZ and L.M. SAHA 111 ](see a l s o F. 1 33 1) the Clausius-Duhem i n e q u a l i t y implies t h a t eg t h e r depends on the other f i e l d s o r $ i s independent o f 4 SIDOROFF

.

i

One can a l s o consider N Maxwell elements i n p a r a l l e l . This model i s very i n t e r e s t i n g f o r concrete (see Z.P. BATANT 1 3 1 , where i t i s a l s o studied a possible dependence from the temperature and t h e humidity). Example 3- (Standard Sol i d )

s

Fig. 3 Example 4- (Jeffrey's element)

Fig. 4 W i t h respect t o the choice of a good model and t h e influence o f the thermodynamics we wish only quote S. NEPAT-NASSER (1 29 1 p. 110) : "In general, the selection of the hidden coordinates represents a s i g n i f i c a n t problem. An experimentalist can only monitor c e r t a i n *inputs" and measure certain "outputs". The material then represents a black box, whose internal s t r u c t u r e i s manifested through such input-output r e l a t i o n s . The optimal selection of suitable internal variables, minimum i n number, which provides maximum information f o r a given input-output setup, is an i n t e r e s t i n g nontrivial problem outside the realm of therr,odynamics.

Thennodynamics can only provides-ageneral framework within rvhich one must operate. The detailed selection of the parameters, however, must be guided by other considerations". 1.5- Ife shall now recall here some results essentially obtained by B.D. COLEMAN and M.E. GURTIN 1 12 1 and by R.M. BOWEN 1 10 1 on the thermodynamics with hidden variables. I t is clear t h a t i n order t o specify a process f o r the body a l l we need do is t o give the motion p(X,t), the temperature f i e l d 0 (X,t) and the value Fo(X) of the hidden variables a t some instant . tocz G , f o r then a l l the other quantities can be computed. Moreover from the conservation laws one can choose the f i e l d s f (X,t) and r(X,t) t o maintain the motion and the temperature. Theorem 1 (1 12 /

,I 10 1 ).

i ) The Clausius-Duhem inequality (1.8) is verified for a l l X C Q and a l l admissible thermodynamic process A ( i .e. a therand t 6 modynamic process which i s consistent with (1.9) t o (1.13))if and only i f the constitutive .equations (1.9), ...,(1.13) satisfy : (1.18)

$ and ;i a r e independent frbm

and g

CI

i i ) If we assume a priori that h , S and are independent from E, the Clausius-Duhem inequality (1.8) i s verified i f and only i f (1.18 bis)

h

and $

are independent from

a$ and S = (1.19bis) rl=-ae AX a$ j * A 9 (1.20 bis) t r po ac e

po

, 0 ( i f meas

then the condition v, = 0 must be dropped i n the d e f i n i t i o n l e t V be equiped w i t h the h i l b e r t i a n structure induced by

H I ( s-2 )n. Let E be the space of infinitesimal tensor s t r a i n fields, i.e. o f w i t h e.. E L 2 ( n ) and l e t % be symmetric matrices e = (eij)i ,j=l,. 1J the space o f tensor stresses fields, i.e. o f symmetric matrices with s €. L2(n ). The spaces E and S form a s = ('i j)i ,j=l,. .,n i,j dual system w i t h the separating b i l i n e a r form

.

which represents, from a mechanical point ~f view, the opposite o f t h e work of the stress

E

s

i n the deformation

may be i d e n t i f i e d t o S

we shall denote by

11. 11 the

The load space

L

, and

e

. From a mathematical

then (5.1)

corresponding norm i n

and the space

V

p o i n t o f view

represents t h e scalar product ;

E or

S

.

are i n d u a l i t y w i t h respect t o the

separating b i l i n e a r form a v , 4% which represents the work o f the strenath 4 under the displacement v ; i f I$ = (f,h), where f i s a regular volume

R and h a ~ e g u l a rsurface force' on 3n \ alQ having only a tangential component on a+l ( t h i s means t h a t L: hi(x) vi(x) = 0 for i a.e. x E a 2 n), then force distributed i n

I t is easy t o see t h a t t h i s formula is t r u e when f 1. 6 L 2 ( n ) and hi E L 2 ( aa\als2) but i t s validity can be extended t o a much more general situation, a t l e a s t when both boundaries of alR and a a r e regular i n an. 2 D will denote the synunetrfc gradient operator

.

1 I t is a l i n e a r continuous operator from H ( n )" i n t o E Thanks t o Korn's inequality and t o the f a c t t h a t meas ( a n ) > 0 D is a 1 one-to-one bicontinuous mapping from v onto DW and D l f is closed i n IE (see e.g. DUVAUT-LIONS 114 Ichap. 3).

Let t~ denote the transpose of (5.4)

< Dv, s >

D

, defined

= t v r t ~ s >

dj€

.

by

v

. ds

6 3;

I t i s easy t o see t h a t t~ i s linear. continour and onto ; formally t ~ =s + means (we use the followi:~gclassical notations : n IN IfJ Sijvivj* siT 3 z sijvj sNvi and sT = (siT))

S=l

-

and the methods of LIONS-MAGENES 125 1 render t h i s interpretation rigorous. For a more detailed analysis of the duality and the virtual work princ i p l e , see MOREAU 1 26 1 , NAYROLES 1 28 1

.

6- A viscoelastic constitutive equation w i t h periodic coefficients. Let T be a positive number. Let us assume that e(X,t) i s the unique T-periodic solution of the T-periodic boundary value problem associated with the non5inear heat equation (4.1) and l e t us consider the constitutive equation of Maxwell type as i n example 2 o f 91.4, i.e.

-

where eo = x(X,8)( 8- 8,) is the thermal dilatation field which corresponds t o a non-stressed state in the reference configuration R(S, = 0). For simplicity we put

rl

(6-3)

s = S (The total stress)

(6-4)

e = E

d

+

eo (The total strain)

and then

We assume :

AI-

K(X,t) and V(X,t) are s.metrica1 fourth order tensor, measurable and bounded on R X R and such that for almost a l l (X,t)

and there exist 0 < k 4 matrices (V . .) 1J

, 0 < r s7

such t h a t far a11 symmetrical

k z v?. ( 2 Kijlmo(.t) 'J ijm

vijvlm

-i,j

(k

Z v?. i j 1J

W2- The d i f f e r e n t i a l system

atdy + v-'(x,t)

(6.9)

i s uniformely ( i n X € A(X,

= 0

K(X,t) y

Q ) exponentially stable @

.

L e t A(X,~,T ) be the fondamental resolvant o f (6.9) such t h a t T ) = j = I d e n t i t y From A2 we have f o r almost a11 X

where

.

cl, c2 are some positive constants

Furthermore from (6.5).

(6.10) we can obtain and integral correspondance

between s

and e

o f the form (6.11)

F(X,t. r )

aG

where F(X,~,T ) = provided the i n i t i a l that :

Obvious!y,

$(x)

E,(X)

=

[: e(X.t)

- e(X. t ) 1 d~

( X S ~ S) T , G(X,t.r ) = K(X,t) A(X,t,r ) r e s u l t s from a *pastu s t r a i n h i s t o r y 3, such

A

0 (

) ~r

. Ue remark t h a t

we can also j f o m a l l ~ ) w r i t e the inverse form :

( x ) In fact, if V-'K (xx) w i t h aging .

i s s p e t r i c , then A 1

.+. 1 2

w i t h c.

I

.

1 and c2

.

;.

Remarks1) From (6.5). (6.11), (6.13) it appears t h a t t o a T-periodic s t r e s s s(X,t) corresponds a strain e(X,t) = a(X)t + p(X,t) which i s a sum of a secular term a(X)t and a periodic term p(X,t) (and conversely). 2) I t will appear clearly l a t e r on, t h a t the assumption A 2 can be weakened by the following one : A2 bis : The differential system (6.9) has no non-trivial T-periodic solution. However then the integral representation (6.11) does not necesseraly holds.

7- The periodic bilateral problem f o r a Maxwell body.

7.1- We consider in t h i s section some questions concerning existence. uniqueness and asymptotic s t a b i l i t y of a T-periodic solution of the equation of motion (2.12) i n the case of a17 the data are T-periodic in time and with the constitutive equation that we have studied i n 5 6. The data are :p(X)f(X,t) a density of forces i n fi , h(X.t) a surfacic 0 density of forces on a0 la,Q with only a tangential component on a2Q(see 55). e0(x.t)(*)the thermal dilatation f i e l d and two functions ul(X,t) and u2(X,t) which f i x the displacement u(X,t) on and a2Q i .e.

ap

.

A l l the data are assumed T-periodic in time. I t is convenient t o introd x e a smooth function uo(X.t) ( a t least(")uo(t) c H'(Q)") such t h a t uo = u1 on als2 and uoN = u2 on , and put in (2.12)

ap

I t is also convenient t o introduce

where D, e , s are defined in (5.31, (6.3), (6.4) and from (5.4) we can define so as a particular solution of

From (2.12). (6.5).

{w)

(7.2). (7.3), (7.4). (7.5). we have now t o solve

He assume here that the nonlinear heat equation is uncoupled from the motion motion equation.

(xx) I t i s always possible i f the boundary an i s smooth and i f the boundaries in aQ of alR and a2 n are regular.

K ( t ) and V ( t )

I n (7.6)

are l i n e a r continuous h i 1b e r t i a n operators i n E.

I n t h e f o l l o w i n g we s h a l l use the notations : Let

7.2-

h i 1bert. space. L e t T be a p o s i t i v e number. L$ (H) space o f (classes o f ) functions g defined a.e on g(t)

g(t+T),

=

H be a r e a l

denotes the h i 1b e r t

R , T-periodic,

with values i n H and such t h a t

r;r

For these functions we define

and we s h a l l use the decomposition

where

dl&ylcl

we have

11eIlf

e

E Using v a r i a t i o n a l techniques i t can be proved Theorem 2a)

Exlztenc~

Assume, e , , s o E0 L system 17.6) possess a solution a €

*;

:

Ci

L> (V1) (XI

[x) Using r e s u l t s o f

1 25 1

be satisfied.Thm the (E) ,Al. 12.rJ and (7.12) & L2# (E), be E, v & V w i t h ~ ( t =) B t + q ( t ) ,

we can improve the r e g u l a r i t y i n t.

b)

Uniqueness-

a K 0 (and then.(7.12)is satisfied) the elements o, B ,qrr are If moreover = unique. Any displacement-solution is written v ( t ) = Bt + $(t) 9 C where C 6 V is arbitrary.

7.4- The quasi-static case. can be neglected In many interesting problem, the acceleration term i n (7.6)2 (see e.g. BOLEY-WIENER [ 5 1 and M. RAOUS 132 1 ) ; i t is the socalled "quasi-static hypothesis" We want t o show that the quasi-static periodic problem reduces t o the study of an ordinary differential equation i n E. For t h i s , l e t I = DV C E be the hilbert space nith the scalar product 1 of E. We put E = I 8 J where J = I . Note t h a t ker t~ = J Recalling the "Virtual Work Principle'' (5.1), (5.2). (5.3), (5.4)

.

.

we see t h a t the equations (7.6), i n the quasi-static case, become

= 0

(7.6 bis)

and are equivalent t o

(7 -6 t e r )

I

~ ( t )L I

E

, \(V

E I,

~ ( t ) + K(t) 5

=

G

v. ~ ( t (E )

-E

)

+

~ ( t eo )

- ;s

= 0

K ( ~ ) E + K(t) eo

5 I, and our problem i s t o find ~ ( t =) a t + p(t) w i t h a .5 I, p ( t ) t 5 ( t ) = a t + q ( t ) with q ( t ) G E (the same secular term as E in order t o have a ( t ) T-periodic). We refer t o 1 9 I f 17 1 f o r d e t a i l s concerning the proof of the existence and uniqueness theorem. We have f i r s t l y :

Lemma 4-

Let the assumption A1 be satisfied and l e t eo,-s&L; (5). Then for a.e. t 15. R, there exists an operator L(t) : E + I such that L(t) = L ( t + T ) , JI L(t)ll 4 l/k and

Moreover i f we put (7.15)

x(t) = ( j

we have a.e.

- K(t) L(t))K(t)

(j = identity i n E)

t 6 R

1) g ( t ) i s a T-periodic symmetric uniformely bounded operator from E to J 2) ker $ ( t ) = I 3) dx d J , e x, X ( t ) x > II II XU + l D ( t ) ~ ( t ) x l I ~ I -5 E

> L

Prom t h i s lemma, problem (7.6 t e r ) i s equivalent t o (7-6 quart.) (7.6quat-t.)

:Find

V(t) r(t)

~ ( t ) =a t + q ( t ) w i t h a B I , q ~ L ) ( E ) s u c h t h a t

i + x(t) 5

=

r ( t ) with

= K ( t ) L(t) (so

- K(t) eo) G L i (E)

which reduces t o (7.6 quint.)

: Find a E I ,

q E L i (E) such t h a t

In order t o solve t h i s l a s t problem the following l e m a i s basic. Lemma 5If there exists a T-periodic solution z

E

the3 (7.6 quint.) has a solution a -

#

L2 (J) t o the equation #

E I , q € L2 (E) given by

.

where c I is arbitrary. Conversely, i f a and q = q Isolutions of (7.6 quint.), then z = q", s a t i s f y (7.16) O

+ q,

are

We can give now the main theorem.

Theorem 3Let A 1 ,(7.12) be s a t i s f i e d and l e t eGso (E) Then (7.16) has one and only one solution z E L$ ( J ) such t h a t i E L$n- (J). Consequently problem (7.6 t e r ) has a solution (2 , and a l l the solutions A 4 are E = E + c 5 = 5 + c where c E I is arbitrary'. Moreover the corres= K(t) ( 2 2) + K(t) so-e, is unique 8 ponding stress

gL2 .

.

2)

-

-

.

We study now the asumptotic s t a b i l i t y of $ This r e s u l t is useful f o r the numerical investigation of the problem (see 1 17 1). Because t h e operators L(t) and x ( t ) a r e T-periodic l i n e a r and continuous the d i f f e r e n t i a l equation (7.6 quint.) is an ordinary l i n e a r , T-period i c d i f f e r e n t i a l equation i n E, and the Cauchy problem

Go

has one and only one solution, C I and h L$ (E). From t h i s , it can be proved the following asymptotic s t a b i l i t y theorem : Theorem 4- (Asymptotic s t a b i l i t y of $ ) Let A 1 ,(7-12) be s a t i s f i e d and ac be the Cauchy stress-solution correspondinq t o go and eC = ~ ~ ' 0 . Then we have

39 (7.20)

$&

-1 c uc.

K

so t h a t , from (7.12)(x) (7.21)

11 uC(t)ll

(

E

(x) I t can be proved t h a t

meaning.

uC,

-

gc

+

v

1

exp

-k

( 0C '

11 oC(0)ll

E

exists and

- 7K

-

ic k L

~

K

t a-e.

i (E)

]

U

=~

0

t b 0 0

and so uc(0) has a

8- The periodic unilateral Signorini problem. 8.1- We are dealing now with a signorini(*) periodic viscoelastic problem i n the quasi s t a t i c case. In a naive formulation we impose on a part a4R of the boundary an the complementary possibilities between the unknown displacement v = u - u o (u) and the unknown s t r e s s o = s so :

-

The f i r s t corresponding t o "no-contact and no-reaction" and the second one t o "contact w i t h only normal reaction", i .e. "contact without friction". The unknown s t r e s s o balancing the reaction on a 4 Q 8.2- Example . Let u s consider a plane medium with a crack (on a4n ). This medium i s submitted t o T-periodic forces or stresses of mechanical o r thermal nature. This problem, f o r a given configuration of the crack, bears unilateral constraints since the edge of the crack can part but cannot interpenet r a t e each other. The crack i s supposed t o l i e i n a plane of symmetry for the mechanical problem, which allows us t o formulate Fig. 5 conditions of contact mathematically identical t o those of contact without f r i c t i o n , as i n (8.1) (for deta5ls see

1

8

1).

(x) Ti?? Signorini problem has been f i r s t solved by G. FICHERA~5 [ i n the elast i c ca e. ( X X )~i tag, u0 = o on a4n

.

41

8.3- In order t o give a rigourous formulation of t h e Signorini problem we must f i r s t of a1 1 precise what vN 4 0 on -a@ means. Let a3n and a4n be open d i s j o i n t subset of an w i t h (n-1)-measure 2 0 such t h a t an = aln u a2h 0 3 3 IJ and we shall suppose t h a t on a2n U a3n a regular surface force, with only a tangential component on a2n a is given (the tangential force i s zero on a2R i n case of example of Fiq.5). , Let ~ ~ ' ~ ( a $ U a4n ) =($ & H ' / ~ ( $ Q) ; supp$sagLJa4n ;it i s a closed subspace of HI/'( an ) and s o i t i s a h i l b e r t space f o r t h e induced norm ;

ap,

1

B = HX1/2 (a4* ) = [ $ € L2(a4n) ; t h e r e e x i s t s * e ~ : ' ~ ( a ~ua4n n ) with

i t is a h i l b e r t space f c r the natural quotient topology. ; 4J E Ho1/2(a3Q c)a4n ) and q = $ r;l

a4n

3.

I f the boundary of a451 i n an i s regular a more manageable characterization of H:/' (a4n ) i s given i n 1 25 1. Let L be t h e usual positive cone i n L2(an ), i.e. L = { $ € L ~ ( ~)R ; f o r every f representative of $ , f ( x ) 2 0 a.e. x e an 1; then L2 (28 ) is an ordered topological vector space, moreover L2(an ) is a l a t t i c e , i.e. f o r every f i n i t e family[ml one has sup{$ @ n l c L 2 ( a n). Using tne t r a c e theorem and t h e order properties of H ( R ) established by LEWY-STAMPACCHIA 1 23 1 i t i s possible t o prove (see 1 20 1 ) t h a t t11/2(a~ ) i s an ordered topological vector space s u b l a t t i c e of L 2 ( aR ). The same r e s u l t s a r e valid f o r H ; ' ' ( ~ ~ u a4n), obviously. Let now d ii~'2(a4Q) then we define

-

-

,..., mnl

....,

supx ( 41 ,$ 2 ) = r e s t r i c t i o n to' a4n of sup($ 1 2) where JI i I a i = 1, 2 and . qi 6 Ho1/2 (a3n Ua4n)

,+

p=

$i*

I t i s e a s i l y seen t h a t does not depend on the choice of t h e representatives ql and q2 and a.lso t h a t t h i s d e f i n i t i o n coincides with i n L2(a4n). In the same way t h e positive the definition of 6 of H:'*(~~G) i s the r e s t r i c t i o n t o a4D of t h e positive cone cone --

-

42

H:"(a3Qua4n)

of

and we have

B = H : ' ~ ( ~ ~ Qi )s an ordered topological vector space sublattice of

-

L~ (a451) Let now L :

V-

~ : ' ~ ( a ~ nbe ) defined by

L v = restriction t o

a4n

of vM

Obviously L is linear and continuous ; moreover L i s onto as composition of the trace application u ~ ( y ~ = u ~ ) and~ the projection of a vector of IR" onto the linear subspace generated by v(x). We can now formulate rigorously the condition vN < 0 on a@ as

6 , the dual space of IB , is a hilbert space of d i s t r i h u t i ~ n sdefined in aR that can easily be characterized by using the r e s u l t s of LIONS-MAGENES125 The duality between IB and C will be denoted by a point and b g is the virtual worK of the normal contact force g f o r the virtual displacement b normal t o the boundary ; when g e L2 (a4Q) then

.

KO

Let C 6 be the polar cone t o can be written as :

5

.

then the condition uN ( 0 on a4n

The transpose t~ : 6 +.lL of L for the duality bc-tween B , 6 a n d v , IL i s linear continuous, one-to-one and has a closed image t~ 6 in IL Thanks t o t h i s f a c t K and have mutually polar images 5. L-'(KJcv = t~ KO c r h . Moreover thanks t o the i n j e c t i v i t y of D and the and closure of D V in IE , and also have mutual 1.y polar images t l o C = D LCS IE and CO = D) 1CS The situation i s sumiired up i n the fol lot~ingdiagram :

.

GO

-

-

-

-

.

KO

GO IG-

.

It is now easy to see that the unilateral constraints (8.1) can be written as the complementarity system :

(8.4)

L v E

-J

,

aN

E

-KO,

Lr.uN = 0

or in the two others equivalent ways :

An other formulation of (8.6) is

8.4- We give now a mathematical formulation of the periodic Signorini problem. Because we seek v(t) of the form v(t) = $t + q(t) with 8 e V, q 6 L$ (V), it is impossible to satisfy vN(X,t) $0 on aqQ for a1 1 t € I? However we are interested by the asymptotic character-of the periodic solution and so we shzll only ask that the unilateral conditions b2 satisfied for all t to. More precisely, let us consider the following linear continuous mapping

.

z

which can be deduced f r o m (6.11) provided Al, A2 be satisfied. The periodic Signorini problem for a Maxwell body can be stated as follows, recalling (8.7) :

44

(8.10) : Find ( t o , a, p, a ) E 1RxExL2 (E)xL2 (E) such that # # i ) a.e. i i ) a.e. VV

t 6 If?

o ( t ) + sO(t) = % ( a ) ( t ) + $(eo+

t h

[ .tO,+a,

-6 - C

,

>

0

With the method of convex analysis i t can be proved that this problem can be reduced t o (see 1 7 1) : (8.10 bis) : Find ( a i ) a.e.

t

ER

.

p, a) L E x L> (E) x L$ (E) such that a(t)

+

so(t) =

&O ( a ) ( t ) +gl(eO+ p ) ( t )

8.5- A mechanical comment. Connection between the formulations (8.10)

and (8.10 bis) can be more easily understood i f they are expressed in terms of the duality between IB, the space of the restriction on a451 o f the normal t o the boundary displacements, and 6 the space of associated normal forces. In these terms (8.10) becomes (8.11)

f o r a.e.

t > to : b ( t ) 6-5

,

g(t)

62,b(t).g(t)

where b(t) should be of the form b(t) = B t + q ( t ) w i t h B EIB q E L2 ( IB). I n the same terms (8.10 bis) becomes

= 0

and

#

(8.13)

f o r a.e.

t 6 R

q ( t ) E-L,

g(t)

,

q(t)-g(t) = 0

The cjnsti tutive law is implicltely determined as the general solution o f a boundary value problem on i2 , that we shall write :

where E 2 ( q )

depends only on

z,

i.e.

=

3f2(q).

Equation ( 8.12 ) implies, a t l e a s t formally, t h a t f o r a.e.

x

€a@

and furthermore, since

%(x)

B(x)

= 0

g i s non-positive,

i s zero (together w i t h a l l Hence we see t h a t the normal reaction g(x,t) the other reactions, thanks t o T si = 0) a t a.e. p o i n t x where the secular term B(x)

i s negative ; t h i s i s a mechanically obvious r e s u l t : f o r time t

large enough there w i l l be no contact a t t h i s point. Moreover i t i s easy t o understand why q may be choosen as a negative function. Indeed l e t (b,gj be a solution of (8.11) and (8.12) and l e t bo be a constant f i e l d belonging t o bo. then (b

+

5

B

such t h a t

= 0 and f o r a.e.

t EIR

q(t)

+

bO, g) i s another solution of (8.11) and (8.14)

boE-s as i s seen d i r e c t l y .

I f we choose, as i s possible

then (b + bo, g) i s a solution o f (8.11) and (8.14) and so we may ask q t o be a negative function a t l e a s t i f ( 8.:5 ) defines. an element o f IB , Hence i t appears natural ask f o r the existence of such a supremum which w i l l be ensured using the r e s u l t s o f

8.6-

17

/and

1 20 1 .

I n order t o solve problem (8.10 bis) o r equivalently problem (8.10) we

must introduce the following assumption on K and V A3- &,t)

i s o f the form V(X,t)

= v ( t ) Vo(X)

.

where

twv(t)

is a

T-periodic scalar function, continuously d i f f e r e n t i a b l e and w i t h p o s i t i v e values and there exists a positive constant

JJ K(X,tl)

- K(X,t2)J

kItl-

t such t h a t f o r a l l (X,

t

t ),

1' 2

t21

b!ith A 3 the periodic Signorini problem (8.10 b i s ) can be stated i n terms o f E ar,d a* = a/v

which s a t i s f y the system :

Assuming A t , A2 defined i n (8.81,

we denote by J(:

(8.9).

A;

and

It can be proved (see

the corresponding mappings

Lemma 6Assurni ng Al, A2 and A 3, -the 1inear mappings (8.8).

(8.9) and corresponding t o

K(t) vel(t)

.

1 7 1)

and

:Yo

and 9(; defined i n are continuous.

Furthermore

ii)' d p e ~ ; p E)~

;(p)

=

X;(*p)

and $;(P)

= 0

From t h i s lemma we can prove Theorem 5Assuming A l . A2 and A.3,

eo,-6,

s ,? & i L2 (T ;E) then : -0-0-#

The problem (8.10 bis) has a solution

Let ( ~l,-pl~ul) -

( a, p a ),

be a solution ; then (-9Lp2,-v2)

i s an another solu-

t i o n i f and only i f : a1 = a2

,

N

p1 =

4

al = u2

p2,

Furthermore we can choose pl

-

t h a t every solution

-

,

p2(t)

e-g

such t h a t

p2 must be w r i t t e n

sup

pp =-pI

=

and 0 c ' P un intorno 0 di a2 X ~ che:

S

Infine ricordiamo il teorema di [ C, 01 : Teorema 3.3. Se c'2 uno 6neR y u d e uvuz a z b n e ha .La pphophietZ di conhmvazbne ( Ceaub&], &0ha L'azbne ha un potenzia.Le 16opka-potenzia.Le) che E continuo (hemi-conCinuo 6 u p ~ m e n t e l . E' possibile cost formulare il primo e il second0 principio $ella termodinarnica nella forma ([ C, 0 1 , n. 5) (*) ; Phima P h i n c i p 2 . L'azione E(P, u) ha la proprieta di conservazione in oo. Secondo Phirzcip&. L7azione/h(P, o) ha la proprieta di Clausius inloo. Come conseguenza della prima e della seconda legge della termodinamica abbiamo per il Teorema 3.1: EhLte un inhieme EE Claoo di a w nei quati E(P, u) ha la pnophim di Conhmvazione e un imieme r.6 di b u n& 6 (P, o) ha pptr~p~P.#. di C L m i u . I n o h e c' E una 6unzLone colztinua E di n m o ckiamata dunziolzc e n m g h che 2 un poffenzia.Le p a trE, .ie dominio 9(E) 2 demo i n in e e e el, u 2 E E 9(E), &ow pwc ogni E > 0, c ' 2 un intophno 0 d i a2 a%& che:

7

p a ogni p m c e s ~ oP M e che Pol E 0 (*a) l . ~ d i n e ,bempptre p a /r Teolrema 3.1, e s h t e una dunzi.one di 6Z.uXo s, chinmata dunzionc entropia, che 2 un ~opm-po;tenzia.Lepwcla, eemicontinua nupehiomente neL dotninio 9( s ) demo i n in c en el, u2 E 9(s), &om peh ogru E > 0, c'E un in.tohn0 0 di o2 M e che:

pen ogtu pmcesno P M e che Pol E 0. Coleman e Owen nel loro lavoro, formulano anche un criterio per costruire iL potenziale o il sopra-potenziale una volta assegnata l'azione. I1 metodo 5 importante, ma non ne faremo cenno rimandando direttamente a ((C, 01, Teorema 3.3 e Teore~a4.4).

(*) La nostra formulazione differisce in parte da quella citata di Coleman e Owen, in quanto ora si specifica lo stato a nel quale deve valere il 0 Prino Principio. ("'3)

possono finizione. E

Inoltre ([C, 01, Teorema 4.6) dimostrano che tutti i potenziali per differire solo per una costante nel dominio oomune di de-

lim P1 =a2, e s e o f 9(P1), t"lpl

ii) Continuits.

Peh

ogni P E n, .Pa ~ u n z b n eap : 9 (P)

-+

R cobL d

c

~

E continua. Definizione 3.2. Sia .a E X c h m o 0 di oo Raee che pm ogni:

&om ao1

k3

una a z i o n e . Se p m ogni

E

> 0

c l E un &-

dioidnw che a ha la p~~ophiea% di Conservazione i n uo (di Clausius in

Inoltre 2 possibile sintetizzare alcuni risultati di [C, 01 cbl teorema:

Teorema 3.1. Se e.b.ibte uno &ahto oo n d q u d e uvllr az.ione !a .Pa p m p h i & i d i Clausius (Conservazione), atbm L 1 d i e m e d q L i b W XO(*) neL q u d e m e pt3j3hi&i B ~ P I L i d i r a t aB d m o C. 7n p a ; t t i c o h e n oo c zO. Indichiamo con & il gradiente di velocits, cio? !, = F ~ - l , con B = 2 = (1/~a) grad 8, allora risulta: Teorema 3.2. P a un &enrevL;to matehiaee, ne.R1.iporJatcui di v f i d . i z 3 d&e Condizioni di Invariabilit2 e di Limitatezza, Le dunzioni E , s de&inite nu I: 0 n n d modo deguente: t P ,1

=

0

,

1

L(t)

+

c(o, PT)l dr

t +

%(a, P,)

'

E(?)1

dt

3(7)

hono a z b n i .

(*) Per una p r e c i s a def i n i z i o n e d i ZO vedi

I C,

.

O] formula ( 3 -24)

~

:

4. In questo numero verificheremo che, assegnato il punto materiale X e comunque presi due strati a', a" E 9 ( E ) che differiscono solo nella coniiyrazione attuale per i valori circostanti il punto X, risulta:

Un discorso analogo vale per s(o). Fonnuliamb pertanto il seguente teorema:

Teorema 4.1. Petr ogni &emento (LFX,IV, n, E, S, 8) i l d o d n i o g(E) deLLa dunzhne enmgia E e qu&o 53 (s) d W dunzhne entrtoph s E .taec. chc? be a' E 9 (E) , o p p a e o ' E 9 ( s ) , LLUQJUX hiape&Xvam&e u i a p p W e n e ancke ogpL

* a;

u" peh cLL~

6

I:

cot(XI = CL7,,(X).

Ino&e:

'Jimostrazione. Dato o1 E 9 (E) , considerimo un generic0 o" = (Call, or), con Cu,,tX) = Cul(X) Per lVAssiomaVI, per ogni E < 0, esiste I I ~ \ intorno 0" di a" e un processo Pt di durata t < E/Q dove Q = sup Ih(o; PT) 1, tale che O
.

i) opt, 0') E 0" ii) PtCX) a CoI (XI mentre l'azione: (4.11

t ~(p,, a') Ijgf(ol;PT)

-L(T) d

6'

~ + fi(ol;PT) d~

verifica la disuguaglianza (4.3) . Per la condizione ii) risulta in particolare che P(X) = -FF-' - = 0 , si ha cost: (4-2)

=

0, quindi I, =

16(01;PT)I d r s t Q < ~

o' E

Per la Definizione 3.3 di [ C , 01 , a" 5 E-approssimabile da a ' , ma poi&$ 9 (E) , allora;per il Teorema 4-4 di [ C, 0 1 , a" E O (E) Quindi

.

Da cui per la limitazione (4.2) e l'arbitrarietg di E, abbiamo:

La dimostrazione relativa alla fimzione s(o) 2 naturalmente del tutto simile.

111 E. T. Gnat: The nofin 04 and i & impficcatians & inhenmodcj&cn &-$kc b o u , Proc. Iutam Symposium, 1966, pp. 292 - 314. (21 W. Noll: A new mathematicae theohg Anal. 48 (1972), pp. 1 50.

05

06 bhp.&2matcmki.b, Arch. Rational Mech.

-

t31

B. D. Coleman - D. R. Owen: A mathematical 6oundcLtion doh inhehmody&cb, Arch. Rational Mech. 54, (1974), pp. 1 - 104.

[-I1 B , D. Coleman: Thmodgnanticd od mat&

Lclith memohy; Arch. Rational Mech.

Anal., 17 (1964), pp. 1.- 46. Noll: Matehiaeec~unidohm b.impLe bodieb uLith inhomogeneitiu, Arch. Rational Mech. Anal. , 27 (1967) , pp. 1 32.

[S] W.

161

-

S. Lang: Intmductian to di~$ehenzi.abLemani6oLd4, New York Sydney, Interscience, 1962.

- London -

Nagata: Modehn genend Zopohgy, North Holland Publishing Company, Amsterdam, 1968.

[71 J.

CEN TRO INTERNAZIONAL E MATEMATICO ESTIVO (c.I.M.E)

ANALYTIC PROBLEMS OF HEREDITARY PHENOMENA

G.

FICHERA

Conso tenuto a Bressannone daf 2 allrIl giugno 1977

ANALYTIC

PROBLEMS

HEREDITARY

OF

PHENOMENA

Gaetano F i c h e r a U n i v e r s i t # t d i Roma

1. The V o l t e r r a i n t e ~ r o - d i f f e r e n t i a l e q u a t i o n s i n t h e mathe

-

m a t i c a l t h e o r y of e l a s t i c i t y . I n t h e c l a s s i c a l t h e o r y of e l a s t i c i t y when one t a k e s i n t o a c c o u n t t h e h e r e d i t y of t h e body l ~ n d e rc o n s i d e r a t i o ~ 1 , i . e . t h e t o r y of -

e-

t h e d e f o r m a t i o n s u n d e r which t h e body was s u b j e c t e d i n

t h e p a s t , one i s f a c e d w i t h t h e a n a l y t i c problems a r i s i n g from t h e i n v e s t i g a t i o n of a c l a s s of i n t e g r o - d i f f e r e n t i a l e q u a t i o n s . T h i s k i n d of e q u a t i o n s f i r s t a p p e a r e d , i n c o n n e c t i o n w i t h e l a s t i c (4)

p r o b l e m s , i n t h e work of L. Boltzmann f 1 1 , 1 2 1 . About t h i r t y y e a r s l a t e r V.Volterra s u p p l i e d t h e f i r s t approach t o an a n a l y t i c theo r y f o r t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n s of h e r e d i t a r y l i n e a r e l a s t i c i t y 133, C43, E51,161, h71. L e t A be a t h r e e d i m e n s i o n a l bounded domain of t h e c a r t e s i a n s p a c e X'

of t h e r e a l v a r i a b l e s x ,

orthogonal c a r t e s i a n coordinates

, x,, M s . L e t x be t h e p o i n t w i t h x, , x, , x* . Suppose t h a t A is t h e

r e f e r e n c e c o n f i g u r a t i o n of a n e l a s t i c body i n t h e fzame-work of

t 4 ) Numbers i n b r a c k e t s r e f e r t o t h e b i b l i o g r a p h y a t t h e e n d of tbese lectures.

t h e l i n e a r t h e o r y of e l a s t i c i t y . Denote by 1 , 2 , 3 ) t h e e l a s t i c i t i e s of A

j k ( x ) ( i , a, j , k

Q

=

, s a t i s f y i n g t h e w e l l known symmetry

conditions

C o n s i d e r i n g t h e e l a s t ' i c h e r e d i t y of A

, t h e c o n s t i t u t i v e equa-

t i o n s a r e w r i t t e n a s follows:

where t is a r e a l v a r i a b l e ( t i m e ) ,

6.

.

i s t h e s t r e s s t e n s o r and

E., is t h e s t r a i n t e n s o r which, i n t h e l i n e a r i z e d t h e o r y lk

l a t e d t o t h e displacement v e c t o r

The f u n c t i o n s q.

ck jk

u S{U,

,Uz.U3

3

,

is re-

a s follows:

c a n be d e n o t e d , f o l l o w i n g V o l t e r r a ,

(x,t,s)

the heredity coefficients. From t h e e q u i l i b r t u m e q u a t i o n s 6;C,L = f;cx,+),

where f

5

If,, fa, f,f

is t h e body-f o r c e a c t i n g upon

A , one g e t s

,

r e c a l l i n g (1.11, (1.2), (1.3) t

i

(1.4)

j

x

u

'

-0

Yirjr/r ( x , ~ , T ) u . Ik (

~ 1 d~r

7

f t

where 4

Y ; R j k ( ~ , t =, ~5 ) { 9 ; k j L ( x . t . s )

+

'f'.Lkj(~.t.t)\.

I t is t a c i t l y assumed t h a t t h e f u n c t i o n s

qLjks a t i s f y

suitable

hypotheses such t h a t t h e i n t e g r a l s extended t o i n f i n i t e i n t e r v a l s be c o n v e r g e n t and t h e d e r i v a t i o n u n d e r t h e i n t e g r a l s i g n

(l)

, with

We assume t h r o u g h o u t t h e s e l e c t u r e s t h e summation c o n v e n t i o n ,

i . e . a summation must be u n d e r s t o o d when a n i n d e x is repeated twice. (')

By /k w e mean t h e p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o t h e -s-p a c e '3= x k . By /LA we mean -. 9xk>xr

v ariable -

r e s p e c t t o t h e s p a c e v a r i a b l e s , be f e a s i b l e . Follouring V o l t e r r a we assume t h a t t h e " p a s t h i s t o r y W o fthe body

is known, i . e .

t h a t the functions 0

-OD

a r e known f o r x

€

A

.

and f o r any t (')

Set

Fi ( x , t )

= f;

(x.t)-

-

J;

(x,t).

Equations (1.4) a r e now w r i t t e n ri

I f we suppose t h a t t h e body i s clamped a l o n g i t s boundary, we must a s s o c i a t e t o (1.5) t h e boundary c o n d i t i o n s (1.6)

u; ( x , t ) = o

for

(x,t

1

t '3A x

Rr.

Hence t h e a n a l y t i c a l problem t o be i n v e s t i g a t e d c o n s i s t s i n m l v i n g f o r any t

0

t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n s (1.5) w i t h t h e

boundary c o n d i t i o n s (1.6)

;

A s an example 1e.t u s c o n s i d e r t h e s i m p l e s t c a s e of a one-dimen=

s i o n a l problem. I n t h i s c a s e (1.5)

becomes

and we have t h e f o l l o w i n g boundary c o n d i t i o n s (1.8)

u ( o , t ) = u ( . r , t )= o

t ro.

(4)Actually t o assume t h a t t h e body can be compared t o a good family g i r l of t h e p a s t y e a r s , such t h a t %wrything of her past is knowd'is a hit too much. En f ortunately the probl em becomes more complicat_ed i f this hyphothesis is not assumed (see Section 7) . V o l t e r r a a s u m e s f l x , t ) 5 0 , which,of c o u r s e , f r o m t h e p o i n t of view of a n a l y s i s is e q u i v a l e n t t o t h e hypothesis i n the t e x t .

We assume t h a t acx).o a(x)

e 4co, 11; y + ( x , t , t )

belongs t o

io,+=)

x CO,

+-)j

Set A(x)=

;

~ ( x , t )

lX3, a(?)

a

and moreover t h a t t h e f u n c t i o n

( 0 6 X s 1)

and ?v,(x,t,o)

belongs t o

,

R e c a l l i n g ( 1 . 8 ) we have uix,t,=

e 4 {[o,.I I x

a

v ( * , t )=

belong t o

c'[ COA3

CO,+-)

f

'3u

a x [ Q ( x ) - ] a. x

/ ~ ( x , ~ ) - - ( r , t ) d ~

where

The problem (I-. 71,(1.8) is e q u i v a l e n t t o t h e f o l l o w i n g one:

We have a x

where

J

I

/

I

~ ( x , j ) v ( ~ , i ) =d J G ( < , J ) n ( F , ' ) d T

.

0

a

*

S i n c e t h e f u n c t i o n G,(*,J)def i n e d i n t h e s q u a r e [o,il jump when t h e p o i n t

( ~ ~ c7r o) s s e s

x

C0,il

has a

t h e d i a g o n a l x = y d e r i v a t i o n under

t h e i n t e g r a l s i g n i s n o t p e r m i t t e d f o r computing u x x ( * , = ) . Since a'(x)

au

ox +

9 2 u a(x) -;:=

ax

Y

From ( 1 . 9 ) w e have 1

u(x.t

1 + /tdr j ' H

( r . . i , t , ~ ) v ,( y~) d y

where

Equation (1.10)

is a n i n t e g r a l

A n a l y s i s i s known a s t h e

e q u a t i o n , which i n

Volterra-Fredholm

classical

i n t e g r a l equation

( s e e f 8 1 , E91 1 . T h i s e q u a t i o n has one and only one

continuous

s o l u t i o n . Hence problem ( 1 . 7 ) , (1.8) h a s one and only one s o l u t i o n belonging t o

e2t0,11.

The a n a l y s i s of t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n s (1.5)

, when

t h e dimension of t h e s p a c e is g r e a t e r t h a n one, is much more complicated. V o l t e r r a , a s an i n t r o d u c t i o n t o t h e g e n e r a l system (1.5), proposes t h e p r e l i m i n a r y s t u d y of t h e s c a l a r i n t e g r o - d i f f e r e n t i a l e q u a t i o n

w i t h t h e boundary c o n d i t i o n (1.12)

r*ix,t)=o

for (x,t)6aAxR7.

Equation (1.11) i s nowadays knou7n a s

the

Volterra integro-

d i f f e r e n t i a l equation. I t i s i n t e r e s t i n g t o p u t i n t o evidence t h e t e c h n i c a l d i f f i c u l t i e s which one has t o face,when one t r i e s t o apply t o (1.11)(1.12) t h e same procedure used f o r s o l v i n g t h e problem (1.7) I f we s e t v (x,+ ) = A 2 u

we have, because of (1.12),

(5)

A2 d e n o t e s t h e Laplace d i f f e r e n t i a l o p e r a t o r A,:

'3= ' 3 -q+3 ~ :

+

3:

ax:

.

,(1.8).

where G ( x , F ) i s t h e Green f u n c t i o n f o r t h e D i r i c h l e t problem f o r t h e Laplace o p e r a t o r i n t h e domain A

, which is .assumed t o s a t i s f y

s u i t a b l e r e g u l a r i t y hypotheses. I n t h i s c a s e f o r e x p r e s s i n g t h e second s p a t i a l d e r i v a t i v e s of u we need t o u s e s i n g u l a r i n t e g r a l s i n t h e Cauchy s e n s e ( s e e [ l o ] ,

Hence problem (1.11)

, (1.12) l e a d s t o t h e f o l l o w i n g

integral

equation

The c l a s s i c a l procedures f o r handling t h e Volterra-Fredholm e q u a t i o n s do n o t apply i n t h i s c a s e , because of t h e presence of t h e Cauchy s i n g u l a r i n t e g r a l s . I t would be of some i n t e r e s t

to

develop a t h e o r y of Volterra-Fredholm i n t e g r a l e q u a t i o n s , e x t e n d i n g t h e c l a s s i c a l t h e o r y , when Cauchy s i n g u l a r i n t e g r a l s

occur

w i t h r e s p e c t t o t h e space v a r i a b l e s . T h i s , i n o u r o p i n i o n , s h o u l d

2

be an i n t e r e s t i n g s u b j e c t f o r a good T h e s i s , e s p e c i a l l y i f '2A p e r m i t t e d t o have s i n g u l a r i t i e s . However i n t h e s e l e c t u r e s

we

s h a l l u s e a d i f f e r e n t approach. Volterra

r 71

proved f o r t h e problem (1.11)

theorem under q u i t e g e n e r a l hypotheses on e v e r y compact s e t ]

.

Pi

, (1.12) a

uniqueness

(t,z) [boundedness i n

We s h a l l o b t a i n t h e V o l t e r r a r e s u l t

as

a

p a r t i c u l a r c a s e of t h e t h e o r y we a r e going t o develop i n t h e next lectures.

2. The a b s t r a c t V o l t e r r a i n t e g r a l e q u a t i o n . L e t 5 be a complex Banach s p a c e , where t h e norm i s denoted '1

11

.

by

L e t u ( t ) be a f u n c t i o n of t h e rea.1 v a r i a b l e t with values

in S

(5)

.

Suppose u ( t ) is d e f i n e d i n t h e i n t e r v a l 1 of t h e r e a l axis

and is c o n t i n u o u s i n 1 , i . e . f o r e v e r y t o €I

W e shall write that u(t) r

e"(1,s ) .

L e t u s now suppose t h a t I : [ d > f . l i s bounded and c l o s e d ( i . e . c o m p a c t ) and l e t u s c o n s i d e r a decomposition of C d , p l i n t o p a r t i a l i n tervals

Lt

k ,tk+,3 d r

t,

f

tZ _L

. - - . 5- t,, 2

t m +=, ( 3 .

By u s i n g t h e e l e m e n t a r y arguments f o r d e f i n i n g t h e i n t e g r a l of a r e a l v a l u e d c o n t i n u o u s f u n c t i o n e x t e n d e d t o C d , p l , one p r o v e s that

, if

u ( t ) E C o ( I , ~ ) , t h ef o l l o w i n g l i m i t e x i s t s i n t h e s t r o n g

convergence of S :

n

$+*

where

Zlr

k.4

i s a r b i t r a r i l y chosen i n [tyl,ik+,fand

By d e f i n i t i o n

The i n t e g r a l i s a d d i t i v e , i .e . i f a

The i n t e g r a l is l i n e a r , i .e. if

_c

y

_z (3,

u f t ) = a, U, it) t &+u2(t)( 4,

$4

complex c o n s t a n t s , u , ( t ) , u l ( * ) c o n t i n u o u s on [ d , p J )

Moreover

When we c o n s i d e r an i n t e r v a l 1 of t h e r e a l a x i s w i t h o u t any f u r t h e r s p e c i f i c a t i o n . we mean t h a t I may be bounded o r unbounded, c l o s e d o r open.

(5)

Although t h e r e s u l t we a r e going t o c n n s i d e r i n t h e f i r s t p a r t of t h e s e l e c t u r e s hold i n t h e more g e n e r a l c o n t e x t of Lebesgue i n t e g r a t i o n t h e o r y , we s h a l l r e s t r i c t o u r s e l v e s t o c o n s i d e r i n t e g r a l s of g e n e r a l l y continuous f u n c t i o n s which can be e a s i l y i n t r o duced and have enough g e n e r a l i t y t o cover most of t h e applications t o Mechanics. I f I is an i n t e r v a l of t h e r e a l a x i s (bounded o r not),the function

it 1, d e f i n e d i n I and w i t h v a l u e s i n S ,i s g e n e r a l l y continuous in I -

i f any bounded i n t e r v a l

i n t o i n t e r v a l s J, ,--.,1 , t e r v a l Jk

c o n t a i n e d i n I can be decomposed

such t h a t

( C = 4 , ...,m ), considering

k(t)

i s continuous i n any i n -

1% a s an open i n t e r v a l . I t i s

e v i d e n t t h a t sums of g e n e r a l l y continuous f u n c t i o n s i n J

are

g e n e r a l l y continuous f u n c t i o n s . I f a ( + ) i s c o n t i n u o u s i n t h e open and bounded i n t e r v a l (

) , we s a y t h a t

4

u ( t ) is summable i n

(d,rJ)

(d,p) whenever t h e func-

is bounded.

I f u ( t ) is summable i n P

1u c t , d t d

p

(4, )

t h e n we d e f i n e

= Aim jY;t)dt. &--)O

d+E

I t is v e r y e a s y t o prove t h e e x i s t e n c e and t h e f i n i t e n e s s of the l i m i t . If

I i s bounded and

u ( t ) i s g e n e r a l l y continuous i n

t h a t u ( t ) is summable i n 1 i f u ( t ) open i n t e r v a l c o n t a i n e d i n

I

,we s a y

is summable i n every bounded

I where a c t )

i s continuous. I f we

decompose 1 i n t o a f i n i t e s e t of such i n t e r v a l s : J, , ... , 5,

,

we put

The i n t e g r a l on t h e l e f t hand s i d e does n o t depend on t h e part i c u l a r decomposition of I i n t o t h e open i n t e r v a l s J,,... , J, I f I i s unbounded and u(t) is g e n e r a l l y continuous i n I ,we say

t h a t u ( t ) i s summable i n I i f 1) u ( t ) i s summable i n e v e r y bounded i n t e r v a l J c o n t a i n e d i n I ;

2 ) a p o s i t i v e c o n s t a n t L e x i s t s s u c h t h a t f o r e v e r y bounded J c I

jl I u ( t , l l d t

5

L

.

J

I f o n l y c o n d i t i o n 1) i s s a t i s f i e d , t h e f u n c t i o n

u (f)

is

said

l o c a l l y summable i n 1. If

I

( t ) is summable i n t h e unbounded i n t e r v a l

u

n o t e by and by

, p t h e e x t r e m a o f I ( o n e o r b o t h o f them b e i n g i n f i n i t e )

ot G

a n d i f we de-

t h e e x t r e m a o f t h e bounded i n t e r v a l J c I

,b

, we s e t

The e x i s t e n c e a n d t h e f i n i t e n e s s o f t h e l i m i t c a n be p r o v e d v e y easily. The i n t e g r a l o f a g e n e r a l l y c o n t i n u o u s f u n c t i o n ~ ( t j e x t e n d e dt o an i n t e r v a l

I

f i n i t e o r n o t , e n j o y s a l l t h e u s u a l p r o p e r t i e s of

t h e i n t e g r a l s , f o r i n s t a n c e t h e p r o p e r t i e s e x p r e s s e d by (2.1).(2.2). (2.3). L e t u s now d e n o t e by A a n i n t e r v a l o f t h e r e a l a x i s and by L' a n open s e t o f t h e Banach s p a c e d e f i n e d on

A x A x

V

.

5

with values i n

L e t f ( t , r , v ) be a f u n c t i o n

S

a n d s a t i s f y i n g t h e f o l l m-r

ing conditions:

1) a r e a l v a l u e d n o n - n e g a t i v e f u n c t i o n L summable i n A

,

(T)

s u c h t h a t f o r a n y p a i r u ,v

exists, locally

o f v e c t o r s of

V

and f o r a n y t t A

2) f o r a n y v ( s ) 6 e G ( i , V ) [ i - e . c o n t i n u o u s i n 1 a n d w i t h v a l u e s

, w h e r e I is a n y c o m p a c t s u b i n t e r v a l o f A

i n V] fixed

t

€

A

t h e f u n c t i o n of f :

u o u s a n d ) summable i n 3 ) f i x e d t, i n A

,

1 ; the function &

,

and f o r e v e r y

[ t , ~v,( t ) J is ( g e n e r a l l y c o n t i n -

belongs t o

e

.

where 1 i s any compact s u b i n t e r v a l of A

O ( I , S ) ,

L e t W ( t ) be a g i v e n f u n c t i o n b e l o n g i n g t o ~ ' ( A , v ) . ws~h a l l c o n s i d e r t h e following a b s t r a c t Volterra non-linear

i n t e g r a l equations

The f o l l o w i n g e x i s t e n c e and u n i q u e n e s s theorem h o l d s . 2 . I . Under t h e s t a t e d h y p o t h e s e s f o r f ( t , r , w )

EXISTENCE: a compact s u b i n t e r v a l

t,

I

and

of A

*(t) we

have

e x i s t s containing

s u c h t h a t t h e r e e x i s t s v ( t ) e0(1,v) ~ which is a s o l u t i o n of (2.4)

f o r every t c l . UNIQUENESS: If

(2.5)

U C ~ E)

,t

eacr,v)

,

~ c t =,

q ( t ,+ j - f [ t , r , u ( r ) ] d t

(~cI),

0

t h e n uCt) 5 w ( t ) . L e t I be a compact s u b i n t e r v a l of A c o n t a i n i n g t, and b a p o s i t i v e r e a l number s u c h t h a t i ) t h e s e t of 6? x R X

is contained i n

d e f i n e d by t h e c o n d i t i o n s

A x A xV ;

ii) f o r every t 6

I

I t is e v i d e n t t h a t by t a k i n g t h e l e n g t h of

I

and b s m a l l enough

c o n d i t i o n s i ) ,i i ) a r e s a t i s f i e d . Let

2 be t h e s e t o f t h e f u n c t i o n s u ( t ) o f e0(I,S)d e f i n e d by t h e

condition

Iluit)-9(t)il 6 b

Yte.1.

Let us consider f o r every u ( t ) ~ e o ( I, v ) and f o r t € 1 t h e mapping

T u = 'P(t) + \ L f ' [ t , ~ , u ( ~ ) ] d i .

(2.6)

t.

I f we s e t z v = T u , and s u p p o s e

~

€ , 2we h a v e

t

I I ~ ( t ) - 4 ( t )Ll l )I/ f [ t , ~ . ~ ( i - ) ] d l - / ~ $ [ t , Z , q ( l ) (1] d ~ t.

to

1,

+

[+,T,QCTI]~T to

4

1/ t.

1

t ~ C ~ I I I u ~ ~ j - ~ ~ ~ ~ I l d ~

Hence (2.7)

L e t u s now c o n s i d e r t h e f u n c t i o n

p(t)

E

~-'(I,R+ )

.t

= e x p ~ L) ( ~ ) d ' t

t L t c ,

+ " ,

exp 2 J t 0 ~ ( = ) d ~ t ? t o

=

eO(I,S )

and c o n s i d e r

IJIu(t)jll : m a x ( p ( f ) l l u ( f ) l l ] .

(2.8)

I

Since

L i s , w i t h r e s p e c t t o t h e norm

eo( I , 5 ) , For

t

a s a Banach s p a c e through t h e norm

we c a n c o n s i d e r

u(t) E

a s a complete m e t r i c s p a c e .

eocr,s), v ( t ) ce3(1,s)

/l

p ( t ) ~ I T u =- ~p (~t )~ ~ ~f t.

Hence

(2.8), a c l o s e d s u b s e t of

we have t

[ t , ~ , u ( t ) ] a -J r It

t . i . ~ ~ ~ ) ~ c ~ r ] l

t o

+

l l l ~ ~ - 5~ ~ I\\~ u l- w\ \ \ \ . I f T is r e s t r i c t e d t o 2 we have t h a t T i s a c o n t r a c t i o n mappi n g of C i n t o i t s e l f . By t h e c o n t r a c t i o n p r i n c i p l e of T e x i s t s i n

: v =T v

, i . e . a s o l u t i o n of

I f u s a t i s f i e s c o n d i t i o n (2.5)

hence

I*

we have

r v.

-

(6)

See, f o r i n s t a n c e ,

1127

,

p.394-399.

(6)

(2.4)

a f i x e d point

The t h e o r e m , which h a s b e e n p r o v e d , is a t h e o r e m " i n t h e s m a l l " s i n c e i n g e n e r a l t h e i n t e r v a l I where t h e s o l u t i o n e x i s t s w i l l n o t coincide with A

. However

i f a p a r t i c u l a r h y p o t h e s i s is made, we

have a t h e o r e m " i n t h e l a r g e " .

P r e c i s e l y t h e f o l l o w i n g theorem

holds. 2 - 1 1 . I f i n a l l t h e h y p o t h e s e s c o n c e r n i n g $ ( t , ~ , vwe ) may assume V

5

S , t h e n t h e s o l u t i o n ~ ( tof) ( 2 . 4 ) is d e f i n e d i n 4he wholeofA.

In fact i f V

I

5 ,we

of A c o n t a i n i n g

ii)

,

c a n c h o o s e a r b i t r a r i l y t h e compact subinterval

t o , w i t h o u t i m p o s i n g t h e two c o n d i t i o n s i) and

c o n s i d e r e d i n t h e p r o o f of t h e o r . 2 . 1 .

T u is g i v e n by ( 2 . 6 ) , w i l l b e l o n g t o ea(?,S).Thus ( 2 . 7 ) h o l d s a s s u m i n g 2 = t h e f u n c t i o n w ( t ) = T u , where

F o r a n y m(t) t e ' ( I , S )

e a ( i , S ) . The r e m a i n i n g p a r t of t h e p r o o f of t h e o r . 2 . 1 r e m a i n s unchanged. T h i s p r o v e s , b e c a u s e of t h e a r b i t r a r i n e s s of

is defined f o r every t

t h e s o l u t i o n v(t) o f ( 2 . 4 ) 1st Remark. Suppose f

o n l y d e p e n d i n g on

?

t

and

I

,

that

A.

V

and s a t i s f y -

i n g t h e above s t a t e d c o n d i t i o n s . 1) and 2 ) . Assume q ( t ) r vj , where L ;

i s a v e c t o r of V

.

Then t h e i n t e g r a l e q u a t i o n t

.i(tI= w . ,+

f[i,v(~)]d~ o

i s e q u i v a l e n t t o t h e Cauchy problem where t h e s o l u t i o n ~ ( ti s ) s o u g h t i n t h e c l a s s of t h e f u n c t i o n s which, i n a s u b i n t e r v a l

I

of A c o n t a i n i n g t, , c a n be r e p r e s e n t e d

a s follows:

I v ( t ) = u(t,l+ j w ( r

dr

,

to

where

w(T)

i s g e n e r a l l y c o n t i n u o u s a n d summable i n 1

~ ( t c)o i n c i d e s

.

The function

w i t h t h e d e r i v a t i v e w l ( t ) o f ' d t ) i n a n y p o i n t of 1

where ~ ( t i)s c o n t i n u o u s , i . e. e x c e p t i n a f i n i t e s e t of p o i n t s , where t h e d e r i v a t i v e of w C ~ ) c o u l d f a i l t o e x i s t .

(')

When we s a y t h a t v'(t) i s t h e d e r i v a t i v e of

(3)

V(t)

i n the point t

The d i f f e r e n t i a l e q u a t i o n ( 2 . 9 ) must b e s a t i s f i e d i n e v e r y p o i n t

t where

2r1[f)

is c o n t i n u o u s .

Theorems 2 . 1 and 2 - 1 1 p r o v i d e e x i s t e n c e and u n i q u e n e s s t h e o r e m s f o r t h e Cauchy problem ( 2 . 9 ) , ( 2 . 1 0 ) 2nd Remark.

.

The r e a s o n why we have n o t assumed p l t ) a I i n d e f i n ( 2 . 8 ) i s b e c a u s e , o t h e r w i s e , f o r p r o v i n g t h e con-

i n g t h e norm

t r a c t i v e c h a r a c t e r of t h e mapping T we had t o impose a f u r t h e r r e s t r i c t i o n on

I

.

T h i s would h a v e p r e v e n t e d u s from g e t t i n g a n

immediate proof of t h e theorem " i n t h e l a r g e " 2 . 1 1 and a l e s s e l e g a n t p r o o f f o r t h e e x i s t e n c e i n t h e l a r g e of v ( t ) h a d t o b e p r w i d e d ( s e e f o r i n s t a n c e 1131

,

p.292-291,theor.10.6.1).

By % (5)we d e n o t e t h e s p a c e of t h e l i n e a r c o n t i n u o u s mappings

into S

of 5

. &(S) i s

K

a complex Banach s p a c e when endowed w i t h the

norm

L e t K(+,r) be a f u n c t i o n d e f i n e d i n A x A

with values i n % ( S ) .

Suppose t h a t t h e f o l l o w i n g h y p o t h e s e s a r e s a t i s f i e d .

, k(t;t)

I ) For every t e A

c'

is a g e n e r a l l y c o n t i n u o u s f u n c t i o n d

i n A ;

1 1 ) II K (t,t)llL

t

L (7) w i t h L ( T )2 o

and l o c a l l y summable i n A

cA

;

t

and v(+)€eo(I,5) imply w ( t ) = k ( t , r ) u ( r ) d r ~ e ' ( I , S ) . t I t i s e a s i l y s e e n t h a t h y p o t h e s e s I , I I , I I I on k ( t , t ) i m p l y t h a t I

)

.

t h e h y p o t h e s e s 1 ) , 2 ) ,3) above s t a t e d f o r E ( t , ~ , v )a r e s a t i s f i e d when we assume F ( t , ~ , v ) :k

(t,r)v

, Vs S .

Hence from t h e o r e m 2.11 we deduce 2.111.

Given q(t] e

C'(A,S)?

u n d e r t h e h y p o t h e s e s I ) , 11) ,111)for

K ( t , 2 ) t h e r e e x i s t s one and o n l y one s o l u t i o n v(t)6

we mean t h a t

e0( A ,

S)

of t h e V o l t e r r a l i n e a r i n t e g r a l e q u a t i o n

~ ( t= )q ( t ) +

(2.11)

1;

K(~,T).L~(T d)

~

.

I t is e a s y t o s e e t h a t H y p o t h e s e s I), 11) ,111) f o r

2. IV.

compact i n t e r v a l A' c A

K (t,T) a r e

s a t i s f i e ? i n any

if K ( t , z ) ~~ ' [ A x A ,&,(s)] ,

hence t h e con-

c l u s i o n s of t h e o r e m 2.111 h o l d t r u e u n d e r t h i s h y p o t h e s i s on K(t,7)

3. The Peano-Gronwall lemma.

Let u ( t ) c e o ( l , ~and +)

3.1.

able function i n

If f o r

toh

I

I

l e t L ( t ) b e a n o n - n e g a t i v e and summ-

.

and f o r any t e

I

1

t

u(t,

(3.1) where

c

(3.2)

c r

5

I L ( ~ ) ~ (, ~ ) ~ ~ I

is a non-negative c o n s t a n t , then

~ ( ti) c exp

1 JT~(T)dil

( t ~ )1.

+*

A l t h o u g h t h e p r o o f of t h i s lemma, which c a n be s t a t e d i n a s l i g h t l y more g e n e r a l f o r m , c a n b e f o u n d i n many t e x t - b o o k s ,

we

s u p p l y h e r e , f o r t h e c o n v e n i e n c e of t h e r e a d e r , a s h o r t p r o o f . L e t u s f i r s t assume t

>t, . From (3.11, by m u l t i p l y i n g b o t h s i d e s t

for

we g e t

2- [ e r p (- / ci t

t

/

t

~ ( ~ ) d i L) ( ? ) a ( x , d r t o t*

]

t

t c l(t)ex,(- J~(.r)cir) t,

and, a f t e r i n t e g r a t i n g both s i d e s i n t h e i n t e r v a l

(3.3)

L

L -etcexp

S u b s t i t u t i n g t h e r i g h t hand s i d e of s i d e of

J+:

It.,

t3

l.(~)di.

(3.3) i n t o t h e r i g h t hand

(3.1) ,we g e t (3.2).

The p r o c e d u r e i s a n a l o g o u s f o r

t

cto.

Under t h e h y p o t h e s e s I), 2) ,3) on f ( t , t , v ), assumed i n S e c t . 2 ,

and f o r c Q ( t )e~o ( ~ , S ,) y . ( t ) € e o ( ~ , s ) , l e t u s s u p p o s e t h a t f o r t ~ I

(to E I

C A

)

v c t ) = q t t ) +f l i t , t , v ( i ) l d T

,

/

;

to

Hence

t o

t

I j ' L(7)I I W ) - u ( ~ ) I I ~. T / P

i t - t i L

I I L P ( O - Y ( ~t) I I

t

F o r t h e Peano-Gronwall

.

lemma, we g e t

I I V ( ~ ) - U ( + )-1I I mar I I ? ( ~ ) - Y ( + ) exp II

(3.4)

t

~ ( tt ) p[t,r, g ( T ) l d ~.

u(t)

I

t

1 j~ ( r ) d r l to

which e x p r e s s e s t h e c o n t i n u o u s dependence of t h e s o l u t i o n v ( t ) of

.

( 2 . 4 ) on t h e datum 4'Ct)

I f # ( t , ~ , v ) = ~ ( t , f ) v( l i n e a r c a s e ) and

K(~,~)E~~[AXA,%(S)I,

we have ,Iv(t)ll 5

w a x llQ(t)ll

I

exyilt-t.i

max ~ l ~ ( t , r ) l l ]

IxI which i s a n "a p r i o r i " e s t i m a t e of t h e s o l u t i o n vlt) of t h e l i n e a r i n t e g r a l e q u a t i o n (2.11) i n t e r m s of t h e d a t a

q ( t ) and

K

(t,f ) .

I t must be remarked t h a t from ( 3 . 4 ) one g e t s a g a i n t h e p r o o f of t h e u n i q u e n e s s s i n c e CP(t)r Y ( t ) i m p l i e s VCt) f u ( t )

.

However we have

s e e n t h a t t h e u n i q u e n e s s i s , i n o u r a p p r o a c h , a n i m m e d i a t e by- p r o d u c t of t h e same p r o c e d u r e which g i v e s t h e e x i s t e n c e of a solution.

4. Some a p p l i c a t i o n s of t h e a b s t r a c t t h e o r y . L e t A b e a bounded domain of t h e C a r t e s i a n s p a c e re21 variables defined i n

A

, xl

A,,...

(

5 is

.

t h a t f o r any p a i r x

h

We s a y t h a t t h e r e a l v a l u e d f u n c t i o n v(x)

t h e c l o s u r e of t h e domain A ) is d -H8lder-

,?

of p o i n t s of

Ivt*)- v ( J ) ! G H

A

Ix-3 I d

where d i s a g i v e n r e a l c o n s t a n t s u c h t h a t o c

d

L i

d-~glder-continuous functions defined A

s h a l l d e n o t e by

of t h e

whenever a n o n - n e g a t i v e c o n s t a n t H e x i s t s s u c h

-continuous i n A

of t h e r e a l

xt

e

a

(6) ,

. The ,

space

which we

i s a Banach s p a c e i f we i n t r o d u c e t h e

f o l l o w i n g norm:

Iv(x)-

l l v l l = wax l ~ ( x ) +l bup d A x,JtA

~ ( 7I )

Ix-ytd

x+7

This p a r t i c u l a r s p a c e w i l l be c a l l e d a Schauder s p a c e . L e t u s suppose t h a t A

e3

a r y of c l a s s

(8)

h a s a smooth boundary 3 A

, s a y a bound-

and c o n s i d e r t h e f u n c t i o n u which is a so-

l u t i o n of t h e D i r i c h l e t problem

A,u =

(4.1)

Suppose t h a t v t (4.3) where

in A

.IY

ed( A )

I1 u/;L ll,

(4.2)

, ( 0

4

d 4 i )

.

.

2A

The f o l l o w i n g e s t i m a t e s hold

c t l Itd~

5

on

u =G

( i , L = 4 , ..., t )

c i s a c o n s t a n t only depending on A and

d

. These

are the

c e l e b r a t e d Schauder e s t i m a t e s f o r t h e s o l u t i o n u of t h e problem (4. I), (4.2)

( s e e C14 1 , p . 3 3 5 , CIOI, S e c t . 3 4 ) .

We s h a l l denote by class to

in

u s denote by G t h e l i n e a r o p e r a t o r

a function wc

Hence

ril

s p a c e of f u n c t i o n s which a r e of

/? and s u c h t h a t e a c h n - t h p a r t i a l d e r i v a t i v e k l m g s

ed(aI L e t

( 4 . 1 ) , (4.2).

entd(E)t h e

C * (3) i n t o t h e s o l u t i o n

The range of

G

which t r a n s f o m

u of t h e D i r i c h l e t problem

belongs t o

e

.3.+6

(i).

is a bounded l i n e a r o p e r a t o r from

Moreover i f we put

eb(i) into

L e t u s now c o n s i d e r t h e V o z t e r r a i n t e g r o - d i f f e r e n t i a l

itself equation

which we s h a l l w r i t e i n t h i s more g e n e r a l form (4.5)

b 2 u = $ ( r , t 1-

jtZ,(Xp*.T)

ulLk( x J T ) 4%

( t lo)

0

We assume t h a t

, fLL a r e r e a l v a l u e d f u n c t i o n s such t h a t

1) + ( k , t ) c e O ( r \ x R + ) and, moreover, f o r e v e r y t > o : + ( x , t ) € C a ( b } ; 2)

f.

( x , ~ , )z E LL

eO(ax R + X R'

) and, moreover, f o r e v e r y ( t , )~E R+x RT :

e"

When we s a y t h a t t h e boundary Q A of A i s of c l a s s we inean t h a t i n some neighborhood of any p o i n t of t h e boundary, QA can be p a r a m e t r i c a l l y r e p r e s e n t e d by f u n c t i o n s of c l a s s e n .

("

E;R(x,t,t )

r Cd(i).

L e t u s a s s o c i a t e t o (4.5) t h e boundary c o n d i t i o n u (x,t

(4.6)

: o

( x , t ) ~2 A x

R'.

W e look f o r a s o l u t i o n of ( 4 . 5 ) , (4.6) which

i ) is c o n t i n u o u s i n A i i ) f o r e v e r y t r R+

,

x

,

'R

u(x>+)E

Set Azu = V(x,t

(4.7)

1.

W e may i n t e r p r e t V ( x,t)a s a f u n c t i o n d e f i n e d i n RT and w i t h valug

in

ed(A). W e

s h a l l d e n o t e by v(t) t h i s f u n c t i o n . Analogously

d e n o t e by q ( t ) t h e

ed(A)v a l u e d

function t

4

we

H(x,t).

Set

-!;c,(~.t,f)rLCI = k(t,t).

(4.8)

Because of (4.4) and of t h e assumed h y p o t h e s e s on f t h e o p e r a t o r K ( t , ? ) i s a bounded l i n e a r o p e r a t o r of

K ( t , r ) E C*{R'X RT, & [ ~ * ( A J I

i t s e l f . Moreover

F o r (4.7) and (4.8)

(y.t,C

ea(h1

)

,

into

1.

we deduce t h a t t h e problem ( 4 . 5 ) , ( 4 . 6 ) is

equivalent t o t h e Volterra l i n e a r i n t e g r a l equation

v ( t ) = q ( t ) +(

t

J

K(~,T)W{.;)~T

c o n s i d e r e d f o r f u n c t i o n s va:ued

i n the

C d( A )

space.

From theorem 2. I V we deduce 4.1.

Under t h e assumed h y p o t h e s e s on A

, 4 , PiL , t h e problem

( 4 . 5 1 , (4.6) h a s one and o n l y one s o l u t i o n u ( r , t ) s a t i s f y i n g t h e above s t a t e d c o n d i t i o n s ij ,i i ) . I t must be remarked t h a t , a l t h o u g h t h e o r . 4 . 1 p r o v i d e s an e x i s t e n c e and u n i q u e n e s s theorem, an u n i q u e n e s s theorem u n d e r less r e s t r i c t i v e h y p o t h e s e s on h and on t h e f u n c t i o n E a p ( x , t , t )c a n be 1 F.

o b t a i n e d by employing t h e o r i g i n a l p r o c e d u r e of V o l t e r r a , s i m p l i f i e d by t h e u s e of t h e Peano-Gronwall lemma. To t h i s end l e t u s assume t h a t A

h a s a piece-wise smooth boundary and i s s u c h t h a t

t h e c l a s s i c a l Gauss-Green f o r m u l a s h o l d i n r e g u l a r domain)

.

A

(briefly, A

is a

L e t u s assume t h a t f i t (0,T)

(T 7

0

pi R 1; ( x , t . z ) is

f o r e v e r y t , i s continuous

ir,

A

x

, a r b i t r a r i l y f i x e d ) and t h e s p a t i a l d e r i v a t i v e piece-wise continuous i n A x ( O , T )

Let a c o n s t a n t C, e x i s t (4.9)

( x , t r t 1,

If'i,(x.t,7 ' I

+

(3)

.

such t h a t

lfiR,i(x,t,r)l

5

CT

(x,t,~)

4.I I. Under t h e assumed hypotheses on A

c A x (b.T)x(O,T).

and Pik t h e problem e4(A) and

(4.6) h a s a t most one s o l u t i o n , w h i c h belongs t o

(4.5), has

piece-wise c o n t i n u o u s second d e r i v a t i v e s i n A . We have t o prove t h a t i f we assume i n (4.5)

( ~ , t ) 0r , Eq. (4.5)

,

(4.6) imply u ( x . t ) s 0 . Let u s multiply

b o t h s i d e s of (4.5) f o r u ( x , t )and i n t e g r a t e

over A . A f t e r an i n t e g r a t i o n by p a r t s and a f t e r applying Cauchy i n e q u a l i t y , we g e t

Hence,by u s i n g Schwarz and P o i n c a r 6 i n e q u a l i t i e s ,

C(T,A)

depends only on T and A

and from (4.6) we deduce

. From

u(x,t)i0

for

t h e Peano-Gronwall

(x,t)gAx

lemma

[O,Tl.

The e x t e n s i o n of theorems 4.1, 4.11 t o t h e problem (1.5), (1.6) p r e s e n t s only formal d i f f i c u l t i e s .

Of c o u r s e one has t o assume

s u i t a b l e smoothness hypotheses on t h e e l a s t i c i t i e s aibjk( x )

and

i n a d d i t i o n some " e l l i p t i c i t y h y p o t h e s i s " , f o r i n s t a n c e t h a t t h e

(9) A f u n c t i o n $ d e f i n e d i n a r e g u l a r domain 3 is s a i d t o be piecewise c o n t i n u o u s i n f5 i f

B = B f u 8 , u ... ~ 4 , . , . . . , B-... a r e mutually d i s j o i n t r e g u l a r domains; i i ) $ c o i n c i d e s i n Bk w i t h a f u n c t i o n continuous i n & .

i)

where B,

q u a d r a t i c form i n 6 v a r i a b l e s a;Rjkfx)&i;l, & .

( ELL : EL; )

Jk is p o s i t i v e d e f i n i t e f o r every x .

We s h a l l n o t d e v e l o p h e r e t h i s s u b j e c t and r e f e r t h e r e a d e r t o p a p e r s [ 151, i161, where t h e a n a l o g o u s of theorems 4 . I , 4. I I have been e x t a b l i s h e d , a l t h o u g h by somewhat more c o m p l i c a t e d p r o o f s , i n t h e c a s e of a homogeneous i s o t r o p i c body. We p r e f e r t o . c o n s i d e r i n t h e n e x t S e c t i o n t h e more i n t e r e s t i n g c a s e when on t h e boundary of t h e body, i n s t e a d of t h e d i s p l a c e ments, t h e s u r f a c e . f o r c e s a r e g i v e n . We wish now t o show, a s a u s e f u l example, how t h e a b s t r a c t t h e o r y of S e c t . 3 a p p l i e s t o a n o n - l i n e a r i n t e g r o - d i f f e r e n t i a l Let F(x,u,w,z ) for

XE

be a r e a l f u n c t i o n o f f o u r r e a l v a r i a b l e s defined

, u L V , w cW , P E Z ,

[o,.i]

problem.

where V

,W , Z

a r e closed in-

t e r v a l s of t h e r e a l a x i s . W e assume t h e f o l l o w i n g h y p o t h e s e s :

2) f o r

(X,U,W,Z)E

have 3) f o r

[o,ii

F(x,u,w,2 )

X E

XU x

x

Z

,

and f o r e v e r y

3

Z

, we

('O)

+ ~

[0,-i] and f o r u,;

W

€ ;2 E

U

;

w,GEW;

2,s E

Z

~ F ( x , u , w , z ) -F ( x , G , G , ~ 1 )G c , ( ~ u - ~ I + I w - ~ ~ + I z - ~ I ) ,

where c, is a c o n s t a n t i n d e p e n d e n t of x L e t H (x,t,'t, u , W , 2 )

.

be a r e a l f u n c t i o n of s i x r e a l v a r i a b l e s

s a t i s f y i n g t h e following hypotheses: 4 ) H(x,t,z,u,w,

5) f o r

f ) 6

, o [ ~ ~ , x4 R ~

( x , t , r ) t [0,-13xRXR

x R x ux W x Z

and f o r

3

u , ~ E U , w , ~ z t, $~E, Z

)I

I ~ ( x , t , t , ~ p ~ , 2 ) - H ( X , t , f , ; , ~ , ~

C4O)

;

H y p o t h e s i s 21 is e q u i v a l e n t t o t h e f o l l o w i n g ones: i) F r 0 i f Z i s bounded, ii) Ffo if inPZ =-C- , sup2 r +-, i i i ) F z o i f inPZ > - m , s u p z - + - .

,

where

C,

i s a c o n s t a n t i n d e p e n d e n t of

( x , t , ~

.

L e t L, and L,be t w o l i n e a r r e a l v a l u e d f u n c t i o n a l s d e f i n e d f o r every r e a l valued f u n c t i o n belonging. t o every p a i r of r e a l c o n s t a n t s a , b

e 4[ O , j j

and s u c h t h a t f o r

the following hypothesis

is

satisfied:

imply a r b : 0 . H y p o t h e s i s 6) a s s u r e s t h a t t h e s o l u t i o n of t h e problem

I,"'= .v.

L, ( u )

= L ~ ( = ~o ,)

~

r

~

~, lV C ~~ ' [ ,O , I~I

l

c a n be r e p r e s e n t e d a s f o l l o w s ~ ( x =,

(4.10)

I'G(X.~)Y(~)~-( 0

and moreover 1

Gx(x,)

u1(x) =

(4.11)

0

L e t u s d e n o t e by

~ () d1j

,

and by G d u t h e o p e r a t o r s g i v e n by (4.10)

Gu

and (4.11), r e s p e c t i v e l y . W e s h a l l c o n s i d e r t h e s e o p e r a t o r s a s o p e r a t o r s from t h e Banach s p a c e

e O [ o , i li n t o i t s e l f and assume

these f u r t h e r hypotheses

7)

c , ( l l ~11 t 114, 11 t i ) = 6 c2

8) F o r e v e r y

4

i

[(IIGIJ + I I G ~ I I ) ( . ( - s ) - ' i+ I

v ( x )

4

4.I 1 1 Given q ( x , t )

E

3) , 4 ) . 5 ) , 6 ) ,7),8), a a

i.

e* 1 [ 0 , i J , 2 ]

e'{[ o , { ] ~ R , Z-321, u n d e r )

o

c o n d i t i o n s 1) , 2 ) ,

e x i s t s s u c h t h a t one and o n l y

s o l u t i o n u ( x , t ) of t h e i n t e g r o - d i f f e r e n t i a l problem

one

exists for

-a

_L

t

5 a

large ( i . e . f o r g

6

.

[ - a , o.]

,

the solution e x i s t s i n the

R 1. The s o l u t i o n

e0{ [o, i]x

of f u n c t i o n s belonging t o every t

= R

If

,u ( x , t )

E

u (x,t)

[-a, a

i s sought i n t h e class

I] and such t h a t , f o r

C2[0,41.

Let u s f i r s t c o n s i d e r t h e a u x i l i a r y problem

where

u

E

e210,AJ

and (P(*) i s a given f u n c t i o n belonging t o

.

eo(Co,dl,Z),Set uH=2

If (4.15) a r e s a t i s f i e d we have u = G 1 .

Problem (4.14), (4.15) is e q u i v a l e n t t o t h e f o l l o w i n g one Z=Tz+q

(4.16)

2 = eo([o,l],z), which, being a c l o s e d s p a c e eo[0,4 ] , can be c o n s i d e r e d a s a

considered i n the space s u b s e t of t h e Banach

complete m e t r i c space. L e t II II

denote t h e norm i n

e0r 0 , d 1 i . e .

IIPII = * n a x l z ( x ) l . l0,il

Because of t h e assumed hypotheses we have T Z + q

E

z

for 1 6 C

and, moreover,

Hence t h e mapping $: T z + ~is a c o n t r a c t i o n of L e t u s denote by

2.a q

2

into itself.

t h e unique s o l u t i o n of (4.16). L e t

9 :a;.

We have Hence

II r (4-3 I-' I I ~ - $11

1 1 6 1- ~ L e t u s now s e t ~ ( x , t )=

ux J

a

( x , t ). Problem (4.12), (4.13) is equiv-

a l e n t t o t h e f o l l o w i n g one (4.11)

I-Te

= L P ( x , t ) + l t H [ x , ~ , T ~ G . z ,~z I d i .

I f we s e t 2-TZ =ir(x,t),we have t h a t (4.17) i s e q u i v a l e n t t o

Assuming

$ ( t , r , ~= )H [ x , ~ , T GGv,G,GLY,v] , ,

where $ ( x , t , v ) m u s t be viewed a s a f u n c t i o n v a l u e d i n t h e s p a c e

e0[o, { ] , a n d

g i v i n g t o ~ ( t and ) Q ( t ) a n o b v i o u s meaning, we may

w r i t e (4.18) a s follows:

v(+)=q ( t ) + [

f [ t , r , v ( T ) ] d ~ .

From theorems 2 . 1 , 2 . 1 1 we deduce t h e a c t u a l one. A l t h o u g h t h i s theorem must be c o n s i d e r e d n o t h i n g more t h a n a mere e x e r c i s e , i t is p e r h a p s u s e f u l p o i n t i n g o u t t h a t s e v e r a l p a p e r s a r e nowadays produced i n mathematics by t h e same procedure, which c o n s i s t s i n c o n s t r u c t i n g a r t i f i c i a l problems where t h e hyp o t h e s e s a r e a d a p t e d i n s u c h a way t o match t h e r e q u i r e m e n t s of some p r e - e x i s t i n g

general theory.

5. E l a s t i c body w i t h memory s u b j e c t e d t o g i v e n body- and surface-forces. L e t u s a s s o c i a t e t o t h e i n t e g r a l - d i f f e r e n t i a 1 s y s t e m (1.5)

the

f o l l o w i n g boundary c o n d i t i o n s on 9 A

v(*)t

I\),(*)

point

x

,\?L(x),

Q ~ ( X ) {

i n the

i s t h e u n i t inward normal t o Q A

.

The i n t e g r o - d i f f e r e n t i a l boundary v a l u e - p r o b l e m (1.5)

(5.1)

c o n c e r n s a n e l a s t i c body, which h a s a memory i n t h e s e n s e of V o l t e r r a and which i s s u b j e c t e d t o g i v e n body-forces t o given surface-forces

2(x,t)

F(x,t )

and

on '2A.

Although t h e r e s u l t s we a r e g o i n g t o p r o v e h o l d i n s l i g h t l y more g e n e r a l h y p o t h e s e s , we s h a l l assume, f o r s i m p l i c i t y , t h a t

e"'(x3)and v

c+A i s of c l a s s Coo , a L L j k ( ~ '

~ ~ ~ ~ ( ~ , ~ , T ) L

B e s i d e s ( 1 . 1 ) , we s h a l l suppose t h a t t h e q u a d r a t i c form

a ; L j t (E~ L L)E jk i n t h e s i x r e a l v a r i a b l e s &;A ( l 5 i f o r any x E

(Eik=

&Li )

3 ) i s positive definite

6

X3.

From ( 1 . 5 ) , (5.1) we deduce t h a t t h e g i v e n f o r c e s must s a t i s f y f o r any t 2 o

t h e "equilibrium conditions"

+

x A F(r,t)dx

IQAxAq(x~t)dG=O.

Hence we s h a l l assume t h a t

F(x,t)

, q ( x , t ) a r e smooth v e c t o r valued

functions, f o r i n s t a n c e belonging t o (5.2)

em(xLR t , x 3 )

and s a t i s f y i n g

.

L e t u s c o n s i d e r t h e problem (5.3) (5.4) Assume

(x>

4 ,y

c

4(x)

=

u. ) J l k I,

( G ; ~ , ~ ( X )

~ ~ ( Ux )

'lc

r(

=

in

A,

on ? A

XI

em(x3,X3)

and s u c h t h a t

J . A P ~ X

+

r ~ y d a = ~ . L A

A

T h i s problem h a s one and o n l y one s o l u t i o n

ue(x)

E

eoo(A )

such t h a t

L e t u s d e n o t e by L*O(X)

t h i s solution;

G

= G*+

HY

and H a r e l i n e a r o p e r a t o r s .

The most g e n e r a l s o l u t i o n of (5.31, (5.4) is g i v e n by u(x)=

q?'+H

r t a t b

~

x

where a and b a r e a r b i t r a r y c o n s t a n t 3 - v e c t o r s Let o c a

1

1

.

(see [ 171).

With t h e same meaning g i v e n t o t h e norm II

II,

t h e f o l l o w i n g Schauder e s t i m a t e h o l d s f o r u o ( s e e 1183,

i n Sect.4, p. 74) :

where 1 I

means t h a t t h e Schauder norm of t h e r e l e v a n t func-

I QA,d

t i o n is taken over 9 A

i s t h e g r a d i e n t of y o v e r a A ; c is

;

a c o n s t a n t o n l y depending on A and on t h e e l a s t i c i t i e s . I f we a s sume

4

6

ed(A)and

vf E ed(aA),

y such t h a t

(5.4) h a s a u n i q u e s o l u t i o n

LL'

t h e n t h e problem ( 5 . 3 )

,

s a t i s f y i n g ( 5 . 5 ) and ( 5 . 6 ) .

Set

e

L e t us d e n o t e by

I td

( 3) ~ t h e s p a c e of t h e ( v e c t o r v a l u e d )

f u n c t i o n s w(x) d e f i n e d on '2A C o n s i d e r t h e Banach s p a c e

and s u c h t h a t Ow

e b ( A ) x e'td(3,4),where

6

e d (aA ) .

t h e norm II Lr I1 of

t h e v e c t o r ZTi{y(~),~(x is) ! llv 11 = 11 'J;

(Id+ Il v-(x) ll

/3A ,d

t It V

and c o n s i d e r ?r(t)ijv,(x,t),v 2 ( x , t ) fa s a f u n c t i o n , v a l u e d i n t h i s s p a c e L e t us i n t r o d u c e t h e o p e r a t o r s K i j ( t , ~by ) setting

K

K

( t , ~V,) t 24

Denote by

22

it.r 1%

=

uif.,&

t h e s u b s p a c e of

S

(x,~,T)

%(*) [ 4 5 t HV-1jlh.

ed(i) x eitd(2A)

v e c t o r s U ; v,(xl,v;(x){such t h a t

I ,Av,(rldx

+

A

lxhq('0d6=

0.

aA

I f we c o n s i d e r t h e m a t r i x o p e r a t o r K,, ( t , z ) k,, (t,z) KCt,t)

r

k t

kaz f t , ~ )

formed by t h e

we see t h a t i t maps, f o r any t ? o ,2. > v e c t o r of S

.

Denote by

$ ( x , t ) ] . Because of

G

,

a v e c t o r of

S into

t h e v e c t o r valued function

ip(t)

( 5 . 2 ) we have 9 (*) 6

.

5

a

1F[x,:

The problem (1.5),(5.1).

(5.5) is e q u i v a l e n t t o t h e l i n e a r V o l t e r r a i n t e g r a l equation functions valued i n the space

vci,

(5.7)

+ jtk

(t,r, V

.,

for

5 ( T )

nr

=

it).

0

(5.6) we have

Because of

and o n l y one s o l u t i o n

Zr(t)i

k ( t , z ) ~ e O [ ~ + x ~ + , ~ ( ~ ) ]one . ~ e n c e

Iy

3

(x,t),~~(~,t)

C'(R~,S) of Eq. (5.7)e*ts

( t h e o r . 2 . IV) .

is t h e

The most g e n e r a l s o l u t i o n of problem ( 1 . 5 1 , ( 5 . 1 )

follow-

ing: u(x,t)r

~ v ; ( j , t ) i +i u 2 ( i f , t ) + a ( + ) +b i t ) / \ h ,

where a(*) and b ( t ) a r e a r b i t r a r y 3 - v e c t o r v a l u e d f u n c t i o n s of t

6 . I n v a r i a n c e of t h e s p e c t r u m w i t h r e s p e c t t o p e r t u r b a t i o n s

due t o V o l t e r r a ' s l i n e a r i n t e g r a l t r a n s f o r m s . Let

L

be a l i n e a r t r a n s f o r m a t i o n ( o p e r a t o r ) w i t h domain

t h e complex Banach s p a c e s u b v a r i e t y of

DL . We

SL

in

5 and r a n g e i n 5 . L e t U be a l i n e a r

s h a l l c o n s i d e r t h e r e s t r i c t i o n of L t o L'.

We s h a l l s a y t h a t t h e complex v a l u e 1 is a r e g u l a r v a l u e f o r L (restricted t o into

u

U

) i f a bounded o p e r a t o r

and i s s u c h t h a t

(L-1 ) G

z

C 1 e x i s t s , which maps 5

-

)

i

:

I

z

identity

operator). The s e t of a l l t h e s p e c t r u m of

L.

1 ' s which a r e n o t r e g u l a r f o r L form

the

(restricted t o U ).

L e t k l ( t , ~ )be a f u n c t i o n d e f i n e d i n A x A of t h e r e a l a x i s ) and w i t h v a l u e s i n Z ( S )

(

,

A compact i n t e r v a l s a t i s f y i n g t h e con-

d i t i o n s of theorem 2 . 1 1 1 . L e t ~ t ( t ) b ea c o n t i n u o u s f u n c t i o n d e f i n ed i n A with values i n the operator

U

.

Let us f i x

tu i n A and l e t u s c o n s i d e r

Lu(t)-l u(t)

+

1,H ( t , c ) , ( t ) d s t

t

5

t i n t h e v a r i e t y CO(A,V ) of t h e Banach s p a c e

Ou-lu(t)

C o(A,S)

where t h e f o l -

l o w i n g norm h a s b e e n assumed:

Ill vlll = max llw(t)ll The v a l u e 1 i s r e g u l a r f o r

.

O t i f a bounded o p e r a t o r

r,t e x i s t s

which maps e ' l ( ~ , S i n)t o e " ( A , U ) and i s s u c h t h a t f o r t~ A

.

( o t - 1 l )Xr t rAt ( Q ~ - X I 1 - 1 . se t

The

of a l l t h e

t h e s p e c t r u m of

t

@

3

' s which a r e n o t r e g u l a r f o r

C O ( ~ , u1.)

[restricted t o

From now on w e s h a l l c o n s i d e r L to

?' ( A , u ) L

6. I.

Let

where

h

Qt form

and

ot

restricted t o

and

U

respectively.

a n d O t have t h e same s p e c t r u m . be a r e g u l a r v a l u e f o r

L

.

Consider t h e equation

t.

qi*)tea(A.S).

Set

L u ( t ) - Xu(+) = u(t) W e have

u jt) =

is e q u i v a l e n t t o

G l ~ ( t ) Eq. . (6.1) r t

where

KA ( t , ~= )-!-I ( t , ~Gi ) Since f o r any

Kx

s a t i s f i e s t h e h y p o t h e s e s of t h e o r . 2 . I11 we have

,

c p ( t ) E C 3 ( ~ , ~ )one , and o n l y one s o l u t i o n w ( t ) 6 e 3 ( ~ , s )

Let us represent

vit) a s

~ ( t = ,v1 '9 i

.

By a c l a s s i c a l t h e o r e m of Banach ( s e e 1191 p . 1 0 2 ) bounded l i n e a r o p e r a t o r f r o m Assuming

r Xt = G XvXt

1 is regul.ar f o r

1 is r e g u l a r f o r L

.

is

a

itself.

we s e e t h a t 1 is a r e g u l a r v a l u e f o r

L e t u s now s u p p o s e t h a t we s e e t h a t

e ' ( ~S ,) i n t o

V :

@t

t

0.

. Assuming(;l= rt' a

7. The V o l t e r r a i n t e g r a l e q u a t i o n i n a i n f i n i t e i n t e r v a l .

The t h e o r y of t h e i n t e g r a l e q u a t i o n ( 2 . 4 ) becomes much more

-

d i f f i c u l t when we assume t, =

.

00

A l t h o u g h t h i s c a s e i s of l a r g e r

p h y s i c a l i n t e r e s t , t h e a n a l y t i c a l r e s u l t s which c a n be g i v e n

are

much more r e s t r i c t e d w i t h r e s p e c t t o t h e c a s e when, f o l l o w i n g Vol-

t,

t e r r a , w e assume t h e " c o n v e n i e n c e h y p o t h e s i s " L e t f ( t , r , v ) be a f u n c t i o n d e f i n e d i n i n t h e Banach s p a c e

R

x

R

> - a .

and w i t h v a l u e s

x S

S , s a t i s f y i n g t h e following hypotheses

1) a n o n - n e g a t i v e f u n c t i o n L ( t , r ) d e f i n e d i n Rx 2 e x i s t s t h a t , f i x e d t h e r e a l number c

i s a f u n c t i o n of for

(t,t)

E

t

E

, f o r e v e r y t ~ 1 , :(-=.z) , L

summable i n t h e i n t e r v a l

I Z x I Z,

Moreover, f o r where

7

1%

U,u

I,.L(t,s)is

::,-!

such t h a t

€ 5

,t

-

Lit,t)d.c

c, ,

5

0

c,, i s a p o s i t i v e c o n s t a n t o n l y d e p e n d i n g on z ; -,

2) l e t C 0 ( I , , S ) be t h e s p a c e of t h e f u n c t i o n s of

-t h a t

such

lj tr(t ) 11 is a bounded f u n c t i o n of t i n

e(I,.S)and

f o r e v e r y t c I t t h e f u n c t i o n of

t:

f

1,

.

ko(I,,5

!

such

For every v i z )

[ t , ~v (, r ) ] i s

6

summable ini;

3) t h e f u n c t i o n

(7.1) belongs t o

(L, S) .

7. I . Under t h e assumed h y p o t h e s e s on f

0

,

where

~

6

) t h e, i n t e g r a l e q u a t i o n

1 i s a complex p a r a m e t e r , h a s one and o n l y one s o l u t i e n

the space Set

( t , ' ; , ~, )g i v e n

E O ( I ~ ,if~ )I X I < T v = .P(t)+

-4 C

-00

Z

f

,

[ ~ , T , u ( T ~ ]d i .

T is a mapping from E a ( ~ , , ~ ) i n t iot s e l f . L e t u s assume i n ( ? " ( I , , S ) t h e norm

-

1%

which makes e m ( I , , S ) t o become a Banach s p a c e . We have

hence t h e proof f o l l o w s from t h e c o n t r a c t i o n p r i n c i p l e . L e t u s now assume t h a t f t

uniformly with respect t o t r 1

Z

.

I n t h e a p p l i c a t i o n s i t i s n a t u r a l t o assume t h a t t h e " p a s t h i s t o r y " of t h e m a t e r i a l s y s t e m u n d e r c o n s i d e r a t i o n h a s n o i n f l u e n c e

eJ(Ic

w h a t s o e v e r f o r v e r y remote p a s t t i m e , i n o t h e r words t h a t a e x i s t s such t h a t f o r every v ( T )E the integral

/*-LF

,S )

k>0

t h e c o n t r i b u t i o n of

( t 1%1

[ t , r , w ( r ) ld i

-m

c a n be n e g l e c t e d . We e x p r e s s t h i s f a c t by s a y i n g t h a t t h e l e n g t h of t h e memory i s

!I . I n h y p p t h e s i s 3) we k u s t r e p l a c e t h e i n t e g r a l

on t h e r i g h t hand s i d e of

(7.1) by

t

(t -C $ [ t , t , v ( z ) ~ d r .

7 . 1 1 Under t h e assumed h y p o t h e s e s , i f t h e l e n g t h of t h e memory is s h o r t , w e have e x i s t e n c e and u n i q u e n e s s .

We have now t o c o n s i d e r t h e i n t e g r a l e q u a t i o n t

u(t) = * ( t ) + I f w e set

jt-t, f l t , t ,

W ( ~ ) ] A T .

A

W e see that T

maps

- (It,S )

e

IIITv-T2111

O

,L

i n t o i t s e l f and moreover

Illw-2lll

I*

L(t,t)dr.

t-R

I f R is s u c h t h a t

l t-R L ( t , r ) d ~

d i ,

t

we have t h e proof of t h e theorem.

I t must be remarked t h a t i f

i s n o t s u i t a b l y s m a l l , theorem

k

7 . 1 1 c o u l d f a i l t o be t r u e . I f we c o n s i d e r , f o r i n s t a n c e , t h e v e r y simple l i n e a r i n t e g r a l equation. t

(7.3)

vlt): 9(t)t

We s e e t h a t t h e c o r r e s p o n d i n g tion for

0 4

c 1

,

v(7)dt.

L-t.

T

,

g i v e n by ( 7 . 2 ) , i s a c o n t r a c -

a n d , i n . thiS c a s e , w e have e x i s t e n c e and unique-

n e s s . However f o r d = 4

u n i q u e n e s s f a i l s t o h o l d s i n c e t h e homo-

geneous e q u a t i o n a s s o c i a t e d t o (7.3) i s s a t i s f y e d assuming arbitrary constant

%-!+)?

(I+)

.

L e t u s now c o n s i d e r t h e l i n e a r c a s e , i . e f(t,t,w)= K ( t , ~ ) u

where K ( t , ~i )s a f u n c t i o n d e f i n e d i n

.

1,

x

1 % and w i t h v a l u e s i n

L e t u s assume t h a t

I ) f o r every

I, and

+E

1%t h e f u n c t i o n of

J~ li K ( t , t ) u -

d r 6 c,

Z : Illm

~

t ] -, I ~

~ i f* a 5 mix l i m / I A k l L-7-

flit

r t w,

i f max l i i IA ll'lk 0 , k

9 is t h e convergence r a d i u s of t h e power series (7.5) ,

t h e s e r i e s ( 7 . 5 ) is t o t a l l y c o n v e r g e n t i n t h e d i s c

Pz0 I >t I 6

i .e . i f

f o r any

9'

9's

p

.

If

D Y' .-

t h e s e r i e s ( 7 . 5 ) i s n o t con-

Y L too

v e r g i n g i f I l l > ? . The convergence must be u n d e r s t o o d i n t h e s t r o n g t o p o l o g y of

If T

6

L(S)

( s e e [ 9 ] p.115-116).

c ( S ) a n d i f we c o n s i d e r t h e e q u a t i o n

(7.6)

V =X T u

+

rP

i t is w e l l known t h a t i f t h e power s e r i e s (7.5), where A h = 7 1 c o n v e r g e s , f o r some

# 0

, t h e n (7.6) h a s t h e u n i q u e s o l u t i o n &

L

v = z X T q For any XJ7.6)

k-o h a s one and o n l y one s o l u t i o n i f and o n l y i f t h e QO

convergence r a d i u s of k - o 1 ' ~ '

is t

=

( s e e E91 p. 117-1251.

On t h e o t h e r hand i t c a n be shown t h a t t h e f o l l o w i n g l i m i t 1, 'Ik

R;m I, T h-7-

e x i s t s and is f i n i t e ( s e e C91 p. 1 2 8 ) . T h i s l i m i t is known a s t h e s p e c t r a l r a d i u s of t h e o p e r a t o r T 7.111.

.

Hence

h a s e x a c t l y one s o l u t i o n , f o r a n y f i x e d 3.

Eq. (7.6)

,when

and o n l y when t h e s p e c t r a l r a d i u s of T v a n i s h e s . L e t u s c o n s i d e r a s a n example t h e i n t e g r a l e q u a t i o n (7.7)

q.r(t):

1 \fe-(t-r)P(-c)dl

+ LP(t)

-0

where

v ( t ) , cp (t l c

E" ( I,,

@ ) i. e . zr and

cP

a r e complex v a l u e d

f u n c t i o n s bounded and c o n t i n u o u s i n 1,. Set V ( t ) = etw(t)

, &(t)=et'f'(t)

Set

w ( * )=

/-

.

Eq.(7.7)

becomes

t l f ( ~d)i QI

From ( 7 . 8 ) we deduce

Hence

w(+,=ye1'

+

I

t

e

A ( t -t

*

(f )dZ

0

where y is a c o n s t a n t . W e h a v e

0

v l t ) i s a s o l u t i o n of ( 7 . 7 ) b e l o n g i n g t o

We have t h a t

i f and o n l y i f t h e r i g h t hand s i d e of

(7.9)

-

e3(Ic,&?)

r e p r e s e n t s a function

of t h i s s p a c e . I t i s e a s y t o s e e t h a t : ' 1) I f @& 1 2 .i , %he f u n c t i o n zrCi) = i v e n by ( 7 . 9 ) b e l o n g s t o -eU(Ic , 6?), no m a t t e r how t h e complex c o n s t a n t i s chosen.Hence C

E q . ( 7 . 7 ) h a s i n f i n i t e l y many s o l u t i o n s . 2 ) I f 6?e

1=

1

, t h e f u n c t i o n v(t) b e l o n g s t o

o n l y i f t h e "datum"

cQ(*)

E Y ( ~ @)t , i f

i s s u c h t h a t t h e f u n c t i o n of t

and

is bounded i n

I,

.

I f t h i s c o n d i t i o n is s a t i s f i e d t h e problem has

i n f i n i t e l y many s o l u t i o n s s i n c e c can be chosen a r b i t r a r i l y . 3) I f

Re 1

1

v ( t )belongs t o

,

E D( I,,

6? ) i f and o n l y i f t h e

c o n s t a n t e is g i v e n by

hence

and Eq. (7.7) h a s one and only one s o l u t i o n .

-

From t h i s a n a l y s i s we deduce t h a t t h e spectrum connected w i t h t h e i n t e g r a l e q u a t i o n (7.71, c o n s i d e r e d i n

e'(I,

,$1,

fills

the

whole h a l f -plane (ii.e 1 2 4.. Before e n d i n g t h i s s e c t i o n we propose t h e f o l l o w i n g two exerc i s e s t o the reader: 4 .

1) Consider t h e l i n e a r o p e r a t o r from

eo(j,,5 ) i n t o

i t s e l f given

by t h e V o l t e r r a l i n e a r i n t e g r a l t r a n s f o r m Tw = /k(t,r)vh)dr,

t

0

where k (t,t) s a t i s f i e s t h e hypotheses of theor.2.111.

Prove,

by

computation, t h a t t h e s p e c t r a l r a d i u s of T is z e r o . 2) Let

& ( 5) for

3 (5 )

be a f u n c t i o n d e f i n e d i n

and summable i n

LO,+

00).

LO,

+w

) with values i n

Consider i n t h e s p a c e

3 ( t ) c E"(I, ,S), t h e i n t e g r a l e q u a t i o n

Eo(1,, S )

t

w(t)=

Prove t h a t i f (7.10)

1

I,

% ( t - z ) v ( t ) d +~ 9 ( t ) .

111 4 ( ( l ~ ~ ( 5 l l l d ~ ) ~

t h i s e q u a t i o n h a s one and o n l y one s o l u t i o n . The example g i v e n by Eq. (7.7) shows t h a t , i f c o n d i t i o n ( 7 . 1 0 ) i s

143 v i o l a t e d , e i t h e r e x i s t e n c e o r u n i q u e n e s s c o u l d f a i l t o be t r u e .

8. Summary on Sobolev s p a c e s . Let A {v$(x)

be a domain (open s e t ) of t h e c a r t e s i a n s p a c e be a s e q u e n c e of f u n c t i o n s be'longing t o

them w i t h a compact s u p p o r t c o n t a i n e d i n A s p a c e of

('2)

(

X?

Let

er(A,en)e a c h

e nis

of

the vector

n - v e c t o r s w i t h complex components).

The f o l l o w i n g theorem due t o H.Weyl is of f u n d a m e n t a l importance i n t h e t h e o r y of f u n c t i o n s of r e a l v a r i a b l e s .

8.1. Suppose t h a t f o r e v e r y &

7 0

a

and f o r e v e r y k

3 &

e x i s t s such t h a t f o r

s>+

i

! ~ ~ + ~ ( x 5) - r ( ~ ) i ) . I n f a c t i t is e a s y t c ' b r o v e t h a t I= the function a , t x ) (-r 4 a r g i X ) b e l o n g s t o ?"(A) b u t d o e s n o t b e l o n g t o H _ , ~(A ) . (IL) These s p a c e s , a t l e a s t i n t h e p a r t i c u l a r c a s e p - 2 ,were known s i n c e t h e v e r y b e g i n n i n g of t h i s c e n t u r y , t o t h e I t a l i a n mathemat i c i a n s Beppo L e v i r201 and Guido . F u b i n i r217 who i n v e s t i g a t e d t h e D i r i c h l e t minimum p r i n c i p l e f o r e l l i p t i c e q u a t i o n s . L a t e r on many m a t h e m a t i c i a n s have used t h e s e s p a c e s i n t h e i r work. Some French m a t h e m a t i c i a n s , a t t h e begi-nning of t h e f i f t i e s , d e c i d e d t o i n v e n t a name f o r s u c h s p a c e s a s , v e r y o f t e n , F r e n c h m a t h e m a t i c i a n s l i k e t o do. They proposed t h e name Beppo L e v i s p a c e s . Although t h i s name is n o t v e r y e x c i t i n g i n t h e I t a l i a n language a c d i-?sc,xnds because of t h e name "Beppo",somewhat p e a s a n t , t h e outcome i n F r e n c h must be gorgeous s i n c e t h e s p e c i a l French p r o n u n c i a t i o n of t h e names makes i t t o sound v e r y i m p r e s s i v e . U n f o r t u n a t e l y t h i s c h ~ i c e

W e list h e r e some of t h e most i m p o r t a n t r e s u l t s

on

Sobolev

s p a c e s . We r e s t r i c t o u r s e l v e s t o t h e p a r t i c u l a r c a s e p = Z . t h i s c a s e we s h a l l u s e t h e s h o r t h a n d n o t a t i o n Hm,2

(A )

.

Suppose from now on t h a t

mapping which transform: restriction )

'tv

to

A

a function(")

3 A

H,

is bounded. W X )

In

i n s t e a d of L e t r be t h e

i n t o its t r a c e

( or

. The f o l l o w i n g i n e q u a l i t y h o l d s ( s e e

1173, p.353) :

and

IIU !Im

i s t h e norm i n

be e x t e n d e d t o t h e s p a c e of

DPi*

for

H, .

Through ( 8 . 2 ) t h e o p e r a t o r

t

can

and we c a n d e f i n e t h e b o u n d a r y v a l u e s

H,

D r 1 p 1 4.n1-1.

I t must be remarked t h a t ( 8 . 2 ) h o l d s u n d e r much more h y p o t h e s e s on

( s e e r223p.52-64)

QA

general

and t h a t , u n d e r t h e assumed

s m o o t h n e s s h y p o t h e s e s on 3 A , an e s t i m a t e s h a r p e r t h a n ( 8 . 2 ) c a n be p r o v e d . The f o l l o w i n g t h e o r e m s h o l d ( s e e [23]p. 1 1 1 , C171 p . 3 5 2 , 3 5 4 ) : 8.I.

I f we assume i n t h e r a n g e 7 ( H m ) of t h e o p e r a t o r r t h e norm

( 8 . 3 ) , t i s a compact o p e r a t o r , i . e . gent i n t h e space

H ,,

maps a s e q u e n c e weakly c o n v e r -

i n t o a sequence s t r o n g l y convergent i n t h e

s p a c e T (/-I,,).

was d e e p l y d i s l i k e d by Beppo Levi,who a t t h a t t i m e was still a l i v e , and as many e l d e r l y people w a s s t r o n g l y a g a i n s t t h e modern way of v i e w i n g m a t h e m a t i c s . I n a r e v i e w of a p a p e r of a n I t a l i a n mathematician,who,imitating t h e Frenchmen,had w r i t t e n s o m e t h i n g on "Beppo L e v i spaces',' he p r a c t i c a l l y s a i d t h a t h e d i d n o t want t o l e a v e h i s name mixed u p w i t h t h i s k i n d of t h i n g s . Thus t h e name had t o be changed. A good c h o i c e was t o name t h e s p a c e s a f t e r S. L - S o b o l e v . S o b o l e v d i d n o t o b j e c t and t h e name S o b o l e v s p a c e s is nowadays u n i v e r s a l l y a c c e p t e d . (47) The t e r m f u n c t i o n must b e u n d e r s t o o d a s f u n c t i o n w i t h v a l u e s in and b e l o n g i n g t o e-(A).

-

en

-

8.11. L e t u s c o n s i d e r t h e mapping which makes t o correspond t o t h e same f u n c t i o n viewed as a f u n c t i o n of

H,,,

U ( * ) E

(embedding of

into HI 1.

H,

T h i s mapping is compact

H

withbcm 1. (Rellich

s e l e c t i o n p r i n c i p l e 1. 8.111. I f

err,

-,t , u ( x ) e H , i m p l i e s lul L

-_ax A

C

u(x)eeO(A)

and

IIuII~

)

where c is a p o s i t i v e c o n s t a n t only depending on A

and on m

(Sobolev theorem).

9. E l l i p t i c and s t r o n g l y e l l i p t i c d i f f e r e n t i a l o p e r a t o r s .

=

If

a;;1 ( i = 4 ,..., rn

; j = 4 ,...,n

m a t r i x w i t h complex

is a m x n

)

e n t r i e s and u t {u,, ...,unfa

n - v e c t o r w i t h complex components, by au

w e denote t h e m - v e c t o r

la,, u; ,.-- .-,a m j u j

n

{ dj; 1 where

matrix

x In

d-. r

Eij

. We

1 % ,...., ~ ~ 1 , i . ue ~. = i . i ~ ~ %u~is. I af

- v e c t o r , we s e t a u w = W e have

L e t a,

auv (x)

1

(OU)V

=

- . .; ~ Q . -Vu &J

1 and

m-vector and v a m vau

L.

:t

r ( a u ) = q &..&. . Lj

1

.

eO"(Xe).Set

x ( i n t h e s e n s e of petrowski)

L

J

be a n x n (complex) m a t r i x which, f o r s i m p l i c i t y ,we

The m a t r i x d i f f e r e n t i a l o p e r a t o r

drt

u, - - . . -,U,

u z

uhv.

assume belonging t o

point

J

We denote by c? t h e

have a l r e a d y used t h e symbol

u w f o r i n d i c a t i n g t h e s c a l a r p r o d u c t of '4:

3'

a,

15I:V

(x)

'#

0

L ( x , D ) is

e l l i p t i c i n the

if

f o r every r e a l

i-vector J # o .

(48)

The o p e r a t o r

('8)

F o r a more g e n e r a l d e f i n i t i o n of e l l i p t i c i t y s e e E I g l y - l . B - S S , L ? ~ s l l p . 2 i o

i s c a l l e d t h e ( f o r m a l ) a d j o i n t of t h e o p e r a t o r L(x,D).This o p e r a t o r is (formally) s e l f - a d j o i n t i f

operator

L

(x,D) = a

c a n n o t be e l l i p t i c i f t h e a

1,

2.2

L (x,

D ) = L * ( ~ , ~ ) . Af i r s t o r d e r

scalar

a

L

(x)

'3Xb

are real or i f

(x)

.

r>2

I n t h e case

t h e most c e l e b r a t e d f i r s t o r d e r e l l i p t i c s c a l a r o p e r a t o r i s

t h e Cauchy-Riemann o p e r a t o r

a -

.

F o r t h e t h e o r y of

o

+b----

ax,

0 XI

-

e l a s t i c i t y a r e p a r t i c u l a r l y important

the

2nd o r d e r m a t r i x o p e r a t o r s which we w r i t e a s f o l l o w s : (9.2)

QeL.k (*)ujlrk + b .

L;u

'

1

L

(x) J

ujlk+ c . . ( x ) u '1

~

( i , j = f ,..., m

) .

T h i s m a t r i x o p e r a t o r is e l l i p t i c i f f o r any r e a l

7 ' q,,... ,Fa 3 #

6

S & ' F ~ #] O .

det { a i r i a ~ x )

F o r a n i s o t r o p i c homogeneous body we have t h e 2nd o r d e r m a t r i x o p e r a t o r of c l a s s i c a l e l a s t i c i t y (9.3)

L.u

-.

Ui/aL

(t = w

=

3)

,

+

where p and X a r e r e a l c o n s t a n t s (Lam6 c o n s t a n t s ) . I t i s e a s y t o s e e t h a t t h e o p e r a t o r is e l l i p t i c i f b o t h t h e c o n d i t i o n s

p # O

,

1+2p # o

are satisfied. The o p e r a t o r (9.1) i s s t r o n g l y e l l i p t i c ( i n t h e p o i n t x ) i f

f o r any r e a l

t-vector J # o

and any r e a l n - v e c t o r

2

#

0

.

I t is

e v i d e n t t h a t s t r o n g e l l i p t i c i t y i m p l i e s e l l i p t i c i t y . The c o n v e r s e however, is f a l s e a s t h e example of t h e Cauchy-Riemann proves. I n f a c t

operator

vanishes f o r

7,

=

0

,

y1

7

and

arbitrary.

In t h e c a s e of t h e o p e r a t o r (9.2) t h e s t r o n g e l l i p t i c i t y condit i o n is

Qe

L'x))~

TL T ;7j + 0

f o r e v e r y ~ ~ { ~ , , . - . , ~ ~ e~v + e r yo ra ena ld Suppose

7 : { ?,,-...,r n j

and t h a t - t h e c o e f f i c i e n t s a i L j

.n=z

(x)

+O.

s a t i s f y condi-

t i o n s (1.1). Moreover suppose t h a t t h e q u a d r a t i c form W ( x , E ) = ( EiL = EL; ) i n t i L€ j ktive definite We have

a iLj

(x)

ir ( c + * )

r e a l v a r i a b l e s be p o s i -

.

,

x

i

t ( ~ ~ ? ~ + ) ;(jk.rj ~ ~ 'Ti) l [rk)l ! 'O .

7;

~;t.~tt(~'[

Hence p o s i t i v e n e s s of W ( r , E ) i m p l i e s s t r o n g e l l i p t i c i t y .

In the

c a s e of the o p e r a t o r (9.3) w e have P o s i t i v e n e s s of W : f o r Strong E l l i p t i c i t y : f o r Ellipticity

: for

p 7 O

31 t2p > 0 ;

t~ ( A + 2r+) 7 0 ; $ 0 ,X t l p $ : O .

~

10. Boundary v a l u e problems f o r s t r o n g l y e l l i p t i c o p e r a t o r s and

r e l a t e d integro-dif f e r e n t i a l equations. L e t a,,?(*) be a n x n m a t r i x belonging t o 0 5

Iq 1

m )

.

em(X')( 0 s Ipl 5 m ,

Consider t h e matrix d i f f e r e n t i a l o p e r a t o r of o r d e r

Rnt :

(10.1)

L(X.D)U

~

~

o

~D ~~ .(U x

)

I f A is connected, t h e s t r o n g e l l i p t i c i t y h y p o t h e s i s i n A can be

written

We a s s o c i a t e t o t h e o p e r a t o r (10.1) t h e b i l i n e a r form

I t i s o b v i o u s t h e i m p o r t a n c e of t h e q u a d r a t i c form Q ( u , u ) i n p h y s i c a l problems, s i n c e t h i s q u a d r a t i c form is a n e n e r g y integral.

6 ( u , v ) i s d e f i n e d and c o n t i n u o u s i n H, x H, . L e t u s d e n o t e by H , t h e s u b s p a c e of H, formed by a l l t h e f u n c t i o n s u of H , v a n i s h i n g on Q A w i t h t h e i r d e r i v a t i v e s up 0

t o the order

m-4

,

i.e. such t h a t

?U 5 0 .

I n t h i s s e c t i o n and i n t h e n e x t we s h a l l s u p p o s e t h a t A bounded, c o n n e c t e d and s a t i s f i e s t h e smoothness h y p o t h e s e s

. However,

sumed i n Sect8

is

as-

i t must be remarked t h a t some of t h e

r e s u l t s , we a r e g o i n g t o c o n s i d e r , h o l d u n d e r more g e n e r a l

hy-

.

p o t h e s e s on A

The f o l l o w i n g theorem due t o ~ & r d i [251 n ~ r e l a t e s the algekaic c o n d i t i o n (10.2) t o a n "a p r i o r i " e s t i m a t e s and h a s been

the

s t a r t i n g p o i n t of t h e modern approach t o boundary v a l u e problems f o r e l l i p t i c operators. 10. I . The o p e r a t o r (10.1) i s s t r o n g l y e l l i p t i c f o r e v e r y x c

and lo *, o

i f and o n l y i f (-4 )m

f o r every u r

@c

5

e x i s t such t h a t

B ( k , ~ )2 yo llv 11' m - 3ollvllG2

H,,, .

('0)

L e t u s now i n t r o d u c e t h e f o l l o w i n g n o t a t i o n s : i)

V i s a l i n e a r s u b s p a c e of i,cVc

H ,

such t h a t

H,;

(49) A c t u a l l y B ( u , v ) i s l i n e a r w i t h r e s p e c t t o u and p s e u d o - l i n e a r r e s p e c t t o w , i . e . 0 ( L L , Q V +b w ) = E B(u,.v) + I 3 (u,Iu). Some a u t h o r s , i n s t e a d of b i l i n e a r , s a y s e s q u i l i n e a r . ('0)

F o r t h e proof s e e 117Jp.366-367.

is a b i l i n e a r f u n c t i o n a l d e f i n e d i n

ii) D ( w , z )

which is c o n t i n u o u s w i t h r e s p e c t t o t h e norm i n

T(H,)xC

~ ( H r n ) ,

(H,)

i.e.

the

norm (8.3) ; iii) f

is a l i n e a r bounded o p e r a t o r which maps into

H, i v ) w e set f o r

PROBLEM

)

;

H,

u,v a

. Given

z(H,

f c Ha ( s L'(A) ) , find u c V

(u,v) = ( f , w ) ,

(10.3)

such t h a t

f o r every

VEV.

We s h a l l make t h e f o l l o w i n g : MAIN ASSUMPTION

.

For every v c V

5

e x i s t s such t h a t

&>O

(ye7 0 ) . Znder this h y p o t h e s i s t h e f o l l o w i n g e x i s t e n c e and u n i q u e n e s s theorem h o l d s f o r problem (10.3) : 1 0 . 1 . -One and o n l y one s o l u t i o n a e x i s t s of problem ( 1 0 . 3 I . l f we d e n o t e by G t h e t r a n s f o r m a t i o n which maps f~ H a solution u e

1) G 2)

i n t o the

V of t h e problem (10.3) we have

is a bounded l i n e a r t r a n s f o r m a t i o n of

G viewed a s a mapping from

H~

into

V ;

W, i s compact.

F o r t h e proof of t h i s theorem we r e f e r t o C171p.368-371.

We only

o b s e r v e t h a t t h e p r o p e r t y 2) of G is a n o b v i o u s consequence of 1 ) and of t h e o r . 8. I I . I n a d d i t i o n t o e x i s t e n c e and u n i q u e n e s s w e have t h e f o l l o w i n g regularization results. 10.11. If

such t h g by V (10.4)

f c H, c A.

then

b e l o n g s t o HVt2,(4)

f o r any 8

Moreover, i f s u i t a b l e h y p o t h e s e s a r e s a t i s f i e d

D , u c HgtZ, 11

u:Gf

11

r+zm

(A)

and

6 C

\ I $11.

where c o n l y depends on v

,L ,A ,D ,& $ ' V.

F o r t h e proof of t h i s theorem and f o r t h e h y p o t h e s e s t o be s a t i s f i e d by V and p.355-365

D

i n o r d e r ( 1 0 . 4 ) t o h o l d , we r e f e r t o El71 ( s e e

and p . 3 7 1 ) . We o n l y mention t h a t (10.4) h o l d s i f we a s -

sume e i t h e r V r

; ,

o r VI H,,,

, DE0

,DPO.

From t h e o r . 8 . I11 we deduce t h a t , i f ( 1 0 . 4 ) h o l d s , t h e n $ E implies u E

eaD(A)

1.

Assume t h a t h y p o t h e s e s f o r r e g u l a r i z a t i o n h o l d , t h e n integration by p a r t s is f e a s i b l e and we have

B(U.Y) =

I

( L L I ) V ~ +X

1

M(u,v)~~,

A 9A where M ( u , v ) i s a b i l i n e a r d i f f e r e n t i a l o p e r a t o r of o r d e r 2m-1 i n u and of o r d e r m-1 Hence (10.3)

i n U.

can be writtten

+ If

D(h+u,~~)=

0

C"(A) we have

which i m p l i e s

L u Let WE

UV

=#'.

be t h e v e c t o r s p a c e of a l l t h e u E V

V: M ( u , ~dcr )

s u c h t h a t f o r any

D (pu,'iu)= 0 .

t

The problem ( 1 0 . 3 ) i s e q u i v a l e n t t o t h e f o l l o w i n g o n e : Lurf

,

ueUv

I n s e v e r a l p a r t i c u l a r cases the condition ucuV is expressed by a s e t of s t a n d a r d boundary c o n d i t i o n s on > A . L e t u s now c o n s i d e r t h e i n t e g r o - d i f f e r e n t i a l problem (10.5)

L U ( X , ~ )J=( x , + ) +

(10.6)

u

(x,

t

e u,

We assume t h a t

f,

and, f o r f i x e d x € A

t d ( x , t , r ) $ u ( x , T ) ~ T in A x [ t o , + - ) ( o r 1oc1 L 2 m ) for

(x,t,T )

to& t L+OO.

i s d e f i n e d i n A x [to,+-)x [to ,+

)

, i s a c o n t i n u o u s f u n c t i o n of ( t , ' t ) a n d , f o r

any f i x e d ( t , z ) , b e l o n g s t o H , ( A )

(9L 0 ) .

f u n c t i o n of t is continuous and, f o r

The f u n c t i o n cP(x,t)as a

fixed

t

,

belongs t o H,(A).

We have from (10.4) 5 C

Ilp li* .

Hence t h e o p e r a t o r

: D

fd(x.t.r)

f o r fixed

H,

.

t and

7

G w,

, is a l i n e a r bounded o p e r a t o r from

If we s e t La(*& ) = ~ ( x , t

i.e. u(x,t) =

G

H,

into

v(x,t),problem ( 1 0 . 5 ) ,

(10.6) is e q u i v a l e n t t o t h e V o l t e r r a e q u a t i o n i n t h e space

and t h e theory of S e c t . 2

-

i n p a r t i c u l a r theor.2.N

-

a p p l i e s . Thus

we have e x i s t e n c e and uniqueness f o r problem (10.51, ( 1 0 . 6 ) .

The

r e s u l t is a l s o a consequence of t h e o r . 6 . 1 . Before ending t h i s s e c t i o n we would l i k e t o propose t o t h e reade r , a s a n e x e r c i s e , t o apply t h e t h e o r y developed i n t h i s section

t o t h e p a r t i c u l a r c a s e of t h e c l a s s i c a l s c a l a r Laplace o p e r a t o r .

assuming

1)

Dso

( a,A

,

0

V z H , ;

s u b s e t of 9 A

;

I n each c a s e t h e main h y p o t h e s i s is a consequence of t h e known P o i n c a r e i n e q u a l i t i e s ( s e e t17J p.350-351,p.379).

well

11. G e n e r a l e l l i p t i c boundary v a l u e problems and r e l a t e d

integro-differential equations. The r e s u l t s on t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n s c o n s i d e r e d i n S e c t . 1 0 , a l t h o u g h c o n c e r n i n g a l a r g e c l a s s of e l l i p t i c o p e r a t o r s , a r e of r e s t r i c t e d i n t e r e s t f o r h e r e d i t a r y problems, s i n c e i n these problems t h e i n t e g r o - d i f f e r e n t i a l i n t h e domain A

,

operators a r e only considered

b u t n o t on t h e boundary. On t h e o t h e r hand, i n

a p p l i c a t i o n s we have problems where t h e i n t e g r o - d i f f e r e n t i a l

.

e r a t o r s must be c o n s i d e r e d b o t h i n A and i n Q A

op-

See, f o r instance,

t h e problem c o n s i d e r e d i n S e c t . 5. L e t u s expound h e r e a more g e n e r a l approach t o e l l i p t i c boundary v a l u e problems which p e r m i t s t o h a n d l e a b r o a d e r c l a s s of integro- d i f f e r e n t i a l problems of h e r e d i t a r y phenomena. We suppose t h a t t h e bounded domain A s a t i s f i e s t h e smoothness h y p o t h e s e s assumed i n t h e l a s t s e c t i o n s . C o n s i d e r i n A t h e l i n e a r d i f f e r e n t i a l system

where

L (x,D) is

a

nw n

matrix d i f f e r e n t i a l operator, e l l i p t i c i n

X ' i n t h e s e n s e of P e t r o w s k i ( s e e S e c t . 9) . $ is a g i v e n n-vector valued function.

Jfe a s s o c i a t e t o (11.1) t h e boundary c o n d i t i o n

where

B(x,D)

Bij (x,b)

(

is a

m x n m a t r i x d i f f e r e n t i a l o p e r a t o r whose e n t r i e s

i =4,---,m ; j = 4,--,n

)

are linear differential operators d

9;.(~,9)= ~ . . ( x ) D ~ d

whose c o e f f i c i e n t s b.. ( x ) Y

a r e d e f i n e d on Q A

p e r m i t t e d t o be of o r d e r p i suppose t h a t

4(x)

'1

J

.

(06ldll

pi)

These o p e r a t o r s am

with

pi a n a r b i t r a r y i n t e g e r . We c e*(xk) and t h a t b; ( x ) 6 e m ( 3 A ) ~ . ~~j,,..,j,,,]

i s a g i v e n m - v e c t o r v a l u e d f u n c t i o n d e f i n e d on Q A

. We

shall

d e n o t e by B; ( * . D ) ( i

=

)

4

the

--vector

,al,(,,~)].

operator{~~,(x,i)),..

I n c o n n e c t i o n w i t h t h e boundary v a l u e problem (11. I ) , ( 1 1 . 2 ) w e s h a l l consider the i n t e g r o - d i f f e r e n t i a l system

L(x,3)uscp(x,t)+

(11.3)

(~_LI~ILP,,,)

m a t r i c e s and

where f (x,t,?) a r e smooth n *lt A

- v e c t o r s , c p ( x , t ) and 9( x . + 1: ?I

[~,(v,t), ...,%

Pi

P

(x,t,=)

smooth n-

b,t){given f u n c t i o n s , r e s p e c t i v e l y

m

- v e c t o r v a l u e d and m - v e c t o r v a l u e d , and e i t h e r

o r , even,

= too r

f ~-ao :

-

= t A (where 4 is "the l e n g t h of t h e memory").

The a p ~ r o a c his t h e s t a n d a r d one w e have f o l l o w e d t h r o u g h o u t t h e s e l e c t u r e s which c a n be o u t l i n e d as f o l l o w s : 1 ) To assume t h a t t h e r e e x i s t s one and o n l y one s o l u t i o n .u problem ( 1 1 . 1 ) , ( 1 1 . 2 ) when

S and

q ~ {,,..., t gm]

2 ) Represented u

t o prove t h a t

f

is g i v e n i n a s u i t a b l e Banach s p a c e

i n a s u i t a b l e Banach s p a c e z : Z , x - - - x Z m . a s follows

d

(x,t,9)

o p e r a t o r from S x Z

:D ( G f'

In

C

t

tg; > (05

~ d j ~ _ z m )ai sbounded m

i . 4

into

5

and i i p ( r , t . = ) ! D

is a bounded o p e r a t o r from S x z t i n u o u s l y on

of

i n t o Zi

,

(G f + Z i:r

(Oi / : + P S I

b o t h d e p e n d i n g con-

(t,r).

When 1 ) and 2) have been a c h i e v e d , w e a r e i n p o s i t i o n t o r e d u c e

-

t h e problem ( 1 1 . 3 ) , ( 1 1 . 4 ) t o a n e q u i v a l e n t V o l t e r r a i n t e g r a l equat i o n , e i t h e r i n the space

$x

2 1. T h i s

(?'[I,,

Sr 2 ]

o r i n the space

e0 [: I,

,

i n t e g r a l e q u a t i o n i s one o f t h e types which have been

c o n s i d e r e d i n S e c t i o n s 2 and 7. The c o n d i t i o n s t o be s a t i s f i e d by t h e o p e r a t o r s L ( w , B ) a n d B ( x , D j f o r g e t t i n g 2 ) a r e w e l l known and c o n s i s t i n some a l g e b r a i c hypo t h e s e s t o be assumed on t h e s e o p e r a t o r s ( s e e ( 1 8 1 S e c t i o n s l,2). F o r c o n v e n i e n c e of t h e r e a d e r w e list h e r e t h e s e h y p o t h e s e s

a r e c a l l e d Sup-

which, a c c o r d i n g t o Agmon, D o u g l i s and N i r e n b e r g , p l e m e n t a r y c o n d i t i o n and Complementing c o n d i t i o n . Suwwlementarv c o n d i t i o n .

3

Set

, of e v e n d e g r e e Zm . F o r e v e r y p a i r of l i n e a r l y i n d e p e n d e n t r e a l v e c t o r s 7 ' and 7 , t h e polynomial L ( x ,'f + r 5 ' ) i n t h e complex v a r i a b l e r h a s e x a c t l y n x T h i s polynomial i n

i s , f o r every x

r o o t s with p o s i t i v e imaginary p a r t . T h i s c o n d i t i o n is s a t i s f i e d by any e l l i p t i c m a t r i x o p e r a t o r f o r 2

_*

3

,

b u t i t n e e d s t o be p o s t u l a t e d f o r 2 = 2

.

Complementing c o n d i t i o n . F o r any X E ~ and A any r e a l vector

f

tangent t o 9 A a t x

L,(x,'f where

+TV) =

set

Lt(x,t,~)L,(x,t,r),

is t h e e x t e r i o r u n i t normal t o 'JA

p o l y n o m i a l of d e g r e e

m

, non-zero

at

having a s z e r o e s t h e

x

m

and

L'

(w,),z)

a

z e r o e s of L o ( x , y + t * )

w i t h p o s i t i v e imaginary p a r t . S e t

a;. and d e n o t e by

LoL lt ( x,p

=

(x.5 J

z I ~ ! Z

p;

b:j

(.)

J~

1 t h e c o - f a c t o r of t h e e n t r y of i n d i c e s h , b

i n the matrix

L a,(x)'$. I5i=V

4

I t is assumed (complementing c o n d i t i o n ) t h a t t h e rows of t h e matrix

jk ((~'.(x.ftT~)L~(K,~ttr.))) &J

c o n s i d e r e d as p o l y n o m i a l s i n

~ f ( ,* , y , t ) , with c,

't

,

(L:,,--.,m

;

k:, ,..., +I)

a r e l i n e a r l y i n d e p e n d e n t modulo

i.e. that

,- -- ,c,,,

i m p l i e s c, : ...

complex c o n s t a n t s

.

c,,,,

c

and

Rk

polynomialsin

r,

0.

The c h o i c e of t h e s p a c e s

S

ways.We c o u l d assume as S and

and 2

C

c a n be done i n s e v e r a l

some Schauder s p a c e s a s wedidin

S e c t . 5 . However w e p r e f e r t o u s e now d i f f e r e n t s p a c e s i n o r d e r t o show t h e r e a d e r s e v e r a l ways of employing t h e t h e o r y of e l l i p t i c systems i n i n v e s t i g a t i n g t h e r e l e v a n t i n t e g r o - d i f f e r e n t i a l

equa-

tions. I n t h e spacer(ki,,,)which

we i n t r o d u c e d i n S e c t . 8 we now d e f i n e

a new norm d i f f e r e n t from (8.3). W e d e n o t e t h i s norm by II

Ilm-F L

and d e f i n e i t a s

I\

,=

Ii

m-Z where

is a f u n c t i o n of

u

by H_-

t h e s p a c e Z (H,.

in9 1 I u\Im

,

UEH,,,

such t h a t

H,

) w i t h t h i s new norm.

l e t u s now p r o v e

~ c c o r d i nt o ~ t h e theory developed i n Sect.6, t h a t , when t h e problem ( 1 1 . 1 )

, (11.2)

c a n be s o l v e d , a l s o t h e prob-

lem ( 1 1 . 3 ) , ( 1 1 . 4 ) c a n be s o l v e d ( a t l e a s t f o r Hence w e assume t h a t , s e t t i n g p = max given

, ji

$'r H,-t,

on ? A . We d e n o t e

ru=W

c 14

-r;-

(2nb

r = t , >1.- ~

, p,+i , ... ,vlllti), we have:

.

, t h e problem (11.1) (11.2) has

+

one and o n l y one s o l u t i o n b e l o n g i n g t o H p . The f o l l o w i n g t h e o r e m h o l d s ( s e e C171 ,p.78) :

11. I . Under t h e assumed h y p o t h e s e s , i f f E H+-2m j r ;r ) , t h e

s o l u t i o n of

a c (11

1I

(11.5)

( 1 1 . 1 ) . (11.2) belongs t o

where c d e p e n d s o n l y on If and

H ,-2,

3; c

9

,L ,

A

and

, t h e above i n t r o d u c e d o p e r a t o r s 2,

.

H,

Z

L u ( x , +=) v ( r , + )

a n d w e have

4.

4 f +; :L r.gi b e l o n g s t o 4

H,-:.

9; c HJ - t t ; -

F 11 ,-,,

H,-pi-r

a r e such t h a t

I ~ e l o n g st o Set

,

H,

and

0.a(x,t)

= w. (s,t ) .

Let us c o n s i d e r t h e V o l t e r r a i n t e g r a l system

and T ( G $ + ! , T ~ ; )

Fixed v z that

p , assume 5 = H ,-,, , Z ;

:

H9-p.-

where i t i s e v i d e n t

+

?

t-i+-3mmust be u n d e r s t o o d l i k e a s p a c e of

- ;l i k e

f u n c t i o n s and e a c h H ,-p,

n-vector

valued

a s p a c e of s c a l a r (complex)

val-

ued f u n c t i o n s . We s u p p o s e t h a t { q ( r , t ) ,y,( x , t ) , ... ,?v,(x,t )]can be viewed a s a t i o n $ ( t ) of CG[ I t , S x Z J i f

5

x Z ] i n other cases.

function V ( t ) r

y = t d - c - o r a s a f u n c t i o n of

where

L

jt,t)

2'1

I,,

The same must be assumed on t h e "unknown" w,(x,+)J.

(~(x,t),w,(x,t),.--,

Suppose f o r s i m p l i c i t y t h a t f d ( x , t , t ) , and m o r e o v e r , i f

func-

y=-rn ,

that for

0 5

e.

(~,t,t t

Ipl

5

e- ( x

%,

R

R )

p,

is t h e f u n c t i o n c o n s i d e r e d i n c o n d i t i o n 1) of Sect.7.

Set

\

K

t

mtr z

-.. K

(t,~)

mt4

,

( t ~ I)

mi4

W e can w r i t e t h e system (11.6) a s a s i n g l e V o l t e r r a equation r

*

which is e q u i v a l e n t t o t h e problem ( 1 1 . 3 ) If

5x2

y = t, > - -

3.

integral

, (11.4)

.

, ( 1 1 . 8 ) h a s one and o n l y one s o l u t i o n i n e'{[tb,+-j,

Hence ( f 1 . 3 ) , ( 1 1 . 4 ) h a s one and o n l y one s o l u t i o n U ( t ) s

u ( r , t ) b e l o n g i n g t o @ " { [ t, +o DO), W

,

1 . We

do n o t need t o s a y

more

on t h i s c a s e . Much more i n t e r e s t i n g i s t h e c a s e when i n the integro-

- d i f f e r e n t i a l s y s t e m we assume e i t h e r

y-- o r -00

F

t -L

.

(24)

L e t u s c o n s i d e r t h e l a t t e r c a s e , s i n c e t h e arguments developed when

y-= t - R

c a n be e a s i l y a d a p t e d t o t h e c a s e when

o t h e r hand w e b e l i e v e t h a t t h e c a s e y = t - k

g = --.On

the

i s t h e one t h a t , i n a

l i n e a r theory, has a r e a l physical significance. W e w r i t e 6q.s

(11.3), (11.4) i n t r o d u c i n g a complex p a r a m e t e r 1

The e q u i v a l e n t e q u a t i o n (11.8) is w r i t t e n t (11.9) V(t) = ( t ) + ~(t,-t)V(r)dr.

4

t -f,

L e t u s assume, i n s t e a d of (11.7), t h a t f o r any 0 i Iri L p

where

x

, t , r and f o r

L i s a p o s i t i v e c o n s t a n t . A f t e r somewhat t e d i o u s b u t com-

1:ietely e l e m e n t a r y c o m p u t a t i o n s , by a p p l y i n g t h e p r o c e d u r e

of

theorem 7 . 1 1 , w e s e e t h a t Eq. (11.9) h a s one and o n l y one s o l u t i o n if

(11.11) where N

13.1

L

(LNLC)-'

is a n u m e r i c a l c o n s t a n t , e x p l i c i t l y computable, and L

and c t h e c o n s t a n t s c o n s i d e r e d i n t h e estimates (11.10), (11.5).

I n o t h e r words t h e s p e c t r u m of (11.9) h a s n o p o i n t s i n t h e d i s c d e f i n e d by (11.11). I n p a r t i c u l a r f o r X = i we see t h a t

we

have

[^I) Leitman and F i s h e r i n t h e i r r e c e n t Monograph l26J w r i t e "E t h e v i s c o e l a s t i c i t y problem ( i .e . l i n e a r i n t e g r o - d i f f e r e n t i a l problem) i s n o t of i n i t i a l p a s t - h i s t o r y t y p e ( b r i e f l y i f y + t , ) t h e problem of e x i s t e n c e i s somewhat d i f f e r e n t " ( s e e p . 6 4 ) . And t h e y v e r y w i s e l y add " W e w i l l n o t p u r s u e t h i s problem f u r t h e r s i n c e cozside r a b l e t e c h n i c a l d e t a i l must be p r o v i d e d .

e x a c t l y one s o l u t i o n of t h e h e r e d i t a r y problem i f t h e bound f o r t h e " h e r e d i t y c o e f f i c i e n t s " and t h e l e n g t h

are s u c h t h a t

L

k of t h e memory

i

(11.12)

RLL

,c3

I n t h e e l a s t i c i t y problems t h e r i g h t hand s i d e of (11.12) must be u n d e r s t o o d a s a c o n s t a n t depending on t h e e l a s t i c n a t u r e

and

on t h e geometry of t h e body u n d e r c o n s i d e r a t i o n . I f (11.12)

is v i o l a t e d ,

c o u l d b e l o n g t o t h e s p e c t r u m of

I =l

( 1 1 . 9 ) . What is t h e p h y s i c a l s i g n i f i c a n c e of t h i s o c c u r r e n c e ?

1 2 . Z e g r o - d i f f e r e n t i a l problems i n domains w i t h s i n g u l a r boundaries. I n the preceeding S e c t i o n s , i n d e a l i n g with integro-dif ferential problems, we have supposed t h a t t h e r e l e v a n t domain ary

QA

A

h a s a bound-

which i s smooth. Although some of o u r r e s u l t s h o l d u n d e r

l e s s r e s t r i c t i v e h y p o t h e s e s on sumed by u s ( 9 A

€

QA

w i t h r e s p e c t t o t h e ones

as-

C w ) , t h e main r e s u l t s of t h e t h e o r y of e l l i p -

t i c o p e r a t o r s which we have u s e d , f a i l t o be t r u e i f 2 A

contains

s i n g u l a r i t i e s l i k e edges, v e r t i c e s , etc. I n f a c t i n t h e s e c a s e s t h e b a s i c i n e q u a l i t i e s (10.4)

, (11.5) a r e , i n g e n e r a l , f a l s e .

R e s e a r c h work i n t h i s a r e a of t h e t h e o r y of e l l i p t i c e q u a t i o n s is nowadays i n p r o g r e s s . However t h e r e s u l t s a v a i l a b l e a t p r e s e n t a r e n o t a b l e t o p r o v i d e a l l t h e t o o l s needed f o r e x t e n d i n g

the

t h e o r y of i n t e g r o - d i f f e r e n t i a l p r o b l e m s , which we have expounded i n t h e s e l e c t u r e s , t o a domain w i t h a s i n g u l a r boundary. The o n l y e q u a t i o n which h a s been i n v e s t i g a t e d t o a s a t i s f a c t o r y extent

is t h e c l a s s i c a l . V o l t e r r a

(1.11)

w i t h t h e boundary c o n d i t i o n

been c a r r i e d o u t a method

which

i n a paper c a n be

,

by

i n t e g r o - d i f f e r e n t i a l equation 1.12).

L.De V i t o

very l i k e l y

,

This analysis

has

1271 o f 1 9 6 1 , by

extended

t o more g e n e r a l

(22)

situations. The r e s u l t of De V i t o is founded on an a b s t r a c t e x i s t e n c e p r i n c i p l e due t o t h e w r i t e r (:')

This p r i n c i p l e has shown t o be a v e r y

u s e f u l t o o l i n many a p p l i c a t i o n s and must

be c o n s i d e r e d

like a

s y n t h e s i s and a g e n e r a l i z a t i o n of a l l t h e e x i s t e n c e methods founded on Banach s p a c e s , which s t a r t i n g from Caccioppoli C311 have been used i n t h e l i n e a r t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s . L e t V be a complex v e c t o r s p a c e and M; (i=r,z)a l i n e a r mapping of V i n t o t h e complex Banach s p a c e 9; . L e t 0; d u a l of

B;

.

Denote by

be t h e topological

t h e d u a l i t y between a Banach

space

and i t s t o p o l o g i c a l dual. L e t cp be a given co-vector of B:. us consider f o r -every v r V

Let

t h e system of e q u a t i o n s

where " t h e unknown" is t h e co-vector

I\y

c

8;.

Denote by V, t h e k e r n e l of M, i .e.

V2 E {w, , V , L V , M , ? = O ] . L e t Q be t h e Banach f a c t o r - s p a c e

-

Q=

6 4

M,(V,)

and denote by M, t h e l i n e a r mapping which maps t h e v e c t o r v e v i n t o the equivalence class[M,v] t h e norm 11 Cw3jl g i v e n by

Q

. We

recall that

t h e subspace of

v.

of

Q

is

.

in? i w t M,v2Il ?a€

Yo be

Q

of an element (equivalence c l a s s ) [ w ]

Iliwlllg = Let

of t h e s p a c e

9,

9: d e f i n e d by t h e c o n d i t i o n

(29 The p a p e r 1271 by De V i t o was c r i t i c i z e d by E.Magenes (Mathe m a t i c a l Reviews,vo1.25,1963,p.452-453).Futility of Magenes sriti c i s m has been shown by De V i t o i n h i s paper C281. (23)

See [29] p. 174-178 and [30] p.11-16.

and d e n o t e by 3 t h e f a c t o r s p a c e

-

0

The above mentioned e x i s t e n c e p r i n c i p l e i s e x p r e s s e d by

the

f o l l o w i n g theorem: 1 3 . 1 . N e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e of a .U s o l u t i o n of

( 1 3 . 1 ) f o r any a r b i t r a r i l y g i v e n rp

t,~

0

.

Then we can verify that

where

is the time--evolution operator, forming a semi-group, i-e.,

equals

and having the following infinitesimal generator, say

cf. RibariE 1973 [A.XI] for general properties of such semigroups. Relation (11.27) verifies that stT UT as defined by (11.241, (11.25) and (11.26) is indeed a state of VUuTu. U u It makes sense to interpret stTU(r,t) as a state of Pu (cf. RibariE 1973, XX.a.1); and we note that at the time instant T the state of Tu equals the output of Tu when its input iui" = 0 V t > 7, i.e., the state s\(T) of TU equals the observable memory of Tu at the time instant P. In an analogous way, it is possible. to define the state of a general causal system with memory, which is neither linear nor time-invariant, cf. Ribaric* 1976, § 10.

Collective memory. Relations (11.9). (11.10) and (11.11) tell us that qu 9 qI1, unless qll = 1 so that the memory of TII about i t s past t < 0 changes when TI1 is part of the system TII UTI , i.e. the collective memory qu of TII UTI at t = 0 differs from the memories ql = 0 and qll of its individual members TI and TII due to mutual interaction. This difference can be quite striking, as witnessed by the followi~gexample. When qll = qI1(0) e-ht, then qu= 2 qI1( ~ ) e - ~ " - ? ~by ~ '(11.9). ~ , (11.10), (11.11); and taking h > 0 Pnd f9 = 1 we have qu(t)=$q,l(0) b't > 0. So due to interaction between part TI having no memory, and part TII with a fading memory and whose memory qll(t) about i t s past t E (---,0) is fading too (qI1 (t -. -) + O), the composed system TI UTll is unable to forget. When f 9 < 0, this example demonstrates that it is also possible for TI to help TII to forget its past by 'negative reinforcement. The example considered again brings forth the most fascinating and useful fact in systems theory, that putting together parts of the same kind, we may succeed in creating a completely new kind of system.

References

RibariE, M.: Functional-Analytic Concepts and Structures of Neutron Transport Theory, Vof. I and II. Slovene Academy of Sciences and Arts, Ljubljana, 1973. RibariE, M.: Basic equation of input-output models and some related topics,in Control Theory andfopics in Functional Analysis, Vol. II, pp. 257-279, Int. Atomic Energy Agency, Vienna, 1976. RibariE, M. & I. Vidav: Analytic properties of the inverse A(z)-' of an analytic linear operator valued function. Arch. Rational Mech. Anal. 32, pp. 298-310 (1969). Scott, M.R.: A bibliography on invariant imbedding and related topics. SLA-744284, Sandia Laboratories Albuquerque, N.M. 1974.

CENTRO INTERNAZIONALE MATFMATICO ESTIVO (c.I~M~E~)

NOTES ON THE THEORY OF CONSTITUTIVE EQUATIONS

R o n a l d S. RIVLIN

C o r s o tenuto a B r e s s a n o n e d a l 2 a l l ' l l giugno 1977

NOTES ON THE THEORY OF CONSTITUTIVE EQUATIONS Ronald S. Rivlin Lehigh University, Bethlehem, Pa., U.S.A.

Contents Chapter 1.

One-Dimensional Constitutive Equations

Introduction Types of viscoelastic behavior A

representation for the stress.

An alternative procedure Additive functionals Hereditary materials Implicit constitutive equations Alternative constitutive assumptions Strain expressed as Taylor series 10.

Non-linear integral representations

11.

Polynomial approximation

Chapter 2.

The Effect of Superposed Rotation

1.

Conservative materials

2.

Cauchy elastic materials

3.

Stress dependent on the deformation gradient matrix and its time derivatives

4.

Functional constitutive equations

5.

Fluids

Chapter 3 .

Restrictions on Constitutive Equations Due to Material Symmetry

1. Description of material symmetry 2.

Restrictions on the constitutive equation imposed by material symmetry in implicit form Some invariant-theoretical concepts Canonical forms for constitutive equations Function basis Constitutive equations of the function type Constitutive equations of the functional Type Invariant functionals Peano's theorem Tables of typical invariants Canonical form for invariant functionals Canonical form for tensor functionals

Chapter 4.

Integrity Bases for Finite Groups

1.

The basic theorems

2.

Determination of the integrity basis in a simple case

3.

Some results from group representation theory

4.

Transformation of the carrier space

5.

An example

6.

Irreducibility

7.

Determination of linearly-independent invariants

8.

Historical note

Chapter 5.

Integrity Bases for the Full and Proper Orthogonal Groups

1.

Introduction

2.

Isotropic tensors

3.

Isotropic tensor polynomials

4.

The integrity basis for N second-order symmetric tensors

5.

The integrity basis for six or fewer second-order symmetric tensors

6.

Irreducibility

Chapter 1 One-Dimensional Constitutive Equations 1.

Introduction The object of these lectures is to present the manner in

which constitutive equations in mechanics and other branches of continuum physics can be formulated in a systematic manner on the basis of clearly stated concepts of the type of physical behavior that it is intended to model.

In order to do this, we have first

to decide on the variables which it is appropriate to relate to each other, in view of the physical situations to which it is intended to apply the constitutive equations.

Secondly, we have

also to consider such questions as the smoothness of the relations between these variables, since different assumptions regarding continuity or differentiability of the constitutive relations can radically alter the physical behavior which they model. Both of these matters can be conveniently illustrated by studying one-dimensional constitutive equations, without introducing the complexities which arise in the three-dimensional case. In this chapter, these aspects of the problem of formulating constitutive equations are discussed in the context of the mechanics of viscoelastic materials, in which a tensile force is assumed to depend on a tensile strain, or rate-of-strain, or on the history of these.

It will be evident that similar situations apply in a

wide variety of other physical contexts. Once these one-dimensional aspects of formulating constitutive equations have been understood, there remain two further

aspects which arise only in the three-dimensional case*.

The

first of these involves the effect of a superposed rigid motion in restricting the manner in which the dependent variable in a constitutive equation can depend on the independent variables. These aspects of the subject are discussed in Chapter 2.

The re-

maining chapters are devoted to the further restrictions which may be imposed on a constitutive equation in view of any specified

**

symmetry that the material which is modelled by it may possess 2.

.

Types of viscoelastic behavior Consider that a thin rod of perfectly elastic material of

uniform cross-section is loaded by a time-dependent load a(t), per unit area measured in the undeformed state. We suppose that, as a result of this loading, the rod undergoes a simple extension E(Z]

per unit initial length. We shall call ~ ( t )and ~ ( tthe )

stress and strain respectively at.time t.

Since the material of

the rod is perfectly elastic, the value of the stress at time t depends only on the strain at time t and not on the strain at any other time.

We shall assume that the material is such that the

relation between ~ ( t )and ~ ( t ) is linear, thus; rlt) = EeCt),

(2.1)

where E is a constant, which is called the tensile modulus for the material of the rod.

*

**

One might also be concerned with the formulation of twodimensional constitutive equations. The problems which then arise are essentially the same as those which arise in the three-dimensional case, but are usually much more easily dealt with from a mathematical standpoint.

I am grateful to Dr. G.F. Smith for many valuable discussions in connection with this nart of the work.

There are many materials for which the assumption of perfect elasticity is not valid and for which the stress at time t depends not only on the strain at time t, but on the strain at all previous times.

Such materials are said to have memory.

In mathematical

terms we may write

where

f

is a functional of the strain hZstory

the range (-m,t].

E(T)

defined over

We will see later that there may exist mater-

ials with memory for which the stress at time t is not strictly expressible as a functional of the strain-history, i.e. by the constitutive relation (2.2).

However, for the moment, we will

exclude such materials from our discussion. We consider only strain histories for which

E(T)

= 0

up to

and including some specified time to. Then we may replace the relation (2.2) by

We shall now describe two typical kinds of material behavior. These do not exhaust all the possible kinds of material behavior which are observed, but are characteristic of many materials. (i)

At time ?, say, greater than to, a very rapid - idealized as

instantaneous

-

increase in the strain takes place from zero to

some finite value and the strain is subsequently held constant, as shown in Fig. l(a).

Correspondingly, the stress increases

instantaneously by a finite amount and then decays with increase of time, either to zero as shown in curve I of Fig. l(b) or to some finite non-zero value as shown in curve I1 of Fig. l(b). The material is said to exhibit inszantaneous elasticity and

Fig. 1 ( a )

Fig. l ( b )

Fig. 2 ( a )

Fig. Z(b)

stress rezaxation.

Again, if we increase the stress instan-

taneously at time t, say, by a finite amount and then hold it constant, as shown in Fig. 2(a),

the strain increases instan-

taneously by a finite amount and then continues to increase more slowly with time either to a constant finite level, as shown in curve I of Fig. 2(b),

or indefinitely (i.e. until the onset of

rupture), as shown in curve I1 of Fig. 2(b).

The material is

said to exhibit instantaneous elasticity and creep. (ii)

In the second type of material which we consider here, an

instantaneous change in the stress at time

t does not result in a

corresponding instantaneous change in the strain.

If the stress

is increased instantaneously from zero to a finite value and then held constant, as shown in Fig. 3(a),

the strain increases con-

tinously from zero, either to a finite value, as shown in curve I of Fig. 3(b),

or indefinitely, as shown ih curve I1 of Fig. 3(b).

In such a material, it is not possible to increase the strain instantaneously by applying a finite stress. The essential difference between the two types of material we have considered is that, while both of them are materials with memory for which the stress may be regarded as a functional of the strain history, the first material exhibits instantaneous elasticity and the second does not. 3.

A representation for the stress There are many materials of the type described for which

at any rate for small enough values of

E(T)

-

-

the relation bet-

ween the stress and strain history is linear; i.e. if al(t)

and

-n2(t) are the stresses at time t corresponding to strain his-

tories

E~(T)

and

E~(T)

respectively, then the stress at time t

Fig. 3 ( a )

Fig. 3 ( b )

Fig. 4 ( a )

Fig. 4C'b)

corresponding to the strain history

.

n2 (t)

linear

*

is nl(t)

E~(T) + E*(T)

+

For such materials, the functional f in (2.3) is a functional of the strain history.

We shall first consider materials for which'the stress is a linear functional of the strain history.

To emphasize this we

write (2.3) as

*(%I

t

L

=

[E(T)~,

T'=t

where

Our first objective wiil be to obtain expressions for ~ ( t ) in useful analytical form.

We shall first do so in the case when

the material exhibits instantaneous elasticity. Consider the strain history E(T), funct.ion of

T,

space C[tO,t]),

as shown in Fig. 4(a), or

E(T)

where (i.e.

is a continuous

E(T) E(T)

lies in the

is a piecewise continuous function of

T

having a countable numb-er of salti, as shown in Fig. 4 ( b ) , (i.e. E(T)

lies in the space M[tO,t]). We shall assume that the linear functional dependence of

~ ( t )on E~(T)

E(T)

is c o n t i n u o u s in the following sense.

E~(T)

and

be two strain histories such that

Let vl(t)

and n2(t)

the strain histories

*

Let

be the stresses at time t (T)

and

E~ (T)

corresponding to

respectively.

We assume

We remark here that in functional analysis, functionals which have the property we describe here as linearity are often called a d d i t i v e functionals, while the term l i n e a r is reserved for functionals uhich have an additional property which we call c o n t i n u i t y .

that the functional L is such that [al(t)

-

~ ~ ( t ) l+ 0

as

6

+

0;

(3.4)

i.e. L is a continuous functional of ~ ( r )with respect to the supremum norm.

of

We now consider the rod to undergo a strain-history E(T) the type illustrated in Fig. 4.

We divide the interval [tO,t]

into n sub-intervals [tO,tl), [tl,t2),...,[tn-l,tn=t],

the salti

if they exist occurring at one or more of the times tl, ...,tn. Let E(r)

be a strain-history which is constant over each of the

intervals and is equal to ~ ( r )at the initial point of each interval, i.e.

. ..,n) .

(i=O,l,

B(ti) = €(ti)

(3 5)

This strain-history is shown schematically in Fig. 4. It follows from the assumed continuity of L that

1

as n is increased in such a way that sup. t.-t. 1 1-11

Let u(6,r)

+

0.

be a unit step-function defined by 0 for

uCS,r)

=

11

for

to

(

5 < r

r 5 E

and let

We assume that the material is such that for any specified value of t, g(r, t) is of bounded variation,

This condition is certainly

satisfied for a material with instantaneous elasticity. The strain-history c ( r ) may be regarded as the superposition of a number of step-functions with appropriate amplitudes, as illustrated in Fig. 5.

Thus, noting that e ( t O )

=

0,

Fig. 5

Fig. 6

=1 2'

Fig. 7(a)

'C

Fig. 7 ( b )

Fig. 8

5

From the linearity of the functional L, together with (3.9),

it

follows that n

L[E(T)I

=

1 {€(ti) i=l

-

~(t~-~)~g(t~,t).

(3.10)

Now, let n tend to infinity in such a way that sup.lti-t i-11 We obtain, with (3.6), a representation for L[E(T)]

+

0.

in the form

of a Stieltjes integral thus

I

t

n(t)

= L[E(T)]

=

(3.11)

g(~,t)d~(~). 0

The function g(r,t) for the material.

in (3.11)

is called the memory function

From its definition in (3.8),

we see that it

is the stress at time t resulting from a strain-history which is zero up to time

T

and unity in the interval [~,t].

It may be argued that if the material is such that it is not possible to produce in it a saltus in

E(T)

of finite magnitude by

means of a finite stress, then the procedure we have adopted here is not valid.

This argument can be met by invoking the Hahn-

Banach theorem.

It follows from this theorem that if .the func-

tional (3.1), with E(T)

in C[tO,t], is continuous in the sense of

the supremum norm, we can construct a functional nf(t), E(T)

n'(t)

say, for

in M[tO,t], continuous in the same sense, such that = n (t)

when E(T)

is in C[tO,t]

.

The argument given above

*

then goes through when applied to the functional n (t), and hence to ~ ( t ) if we restrict the argument functions to C[tO,t]. 4.

An alternative procedure In this section we outline an alternative procedure for

arriving at a representation for the linear functional in (3.1) equivalent to (3.11). E(T)

Instead of approximating the function

by means of step functions, we approximate it by a number of

adjacent flat-topped pulses, as shown in Fig. 6.

Consider now a

pulse in the interval [ T ~ , T ~as ) shown in Fig. 7(a).

For this

pulse

1 for rl

( T

< r2

(4 .I)

=

E(T)

0

for

T < T

and

T 2 T ~ .

We can construct such a pulse by superposing two step functions, in which the steps have height -1 and take place at times T

T~

and

2'

Let 1 u(5,~)

=

E < T

for

C?T

0 for

and let h(~,t)

t L [u(S,~)l.

=

T=-m

We note that u(~,T) is the particular choice of ~ ( 6 )shown in

Fig. 7(b).

The pulse (4.1) is then u(S,'r2)

-

u(S,T~).

As in 53, we divide the interval [tO,t] into n sub-intervals [tO,tl) , [t,,t2),

..., [tn-,,tn=t] , the

salti in E(T),

exist, occurring at one or more of the times tl,

E(T)

if they

...,tn.

Let

be a strain-history which is the superposition of pulses of

height

E (ti-l)

. ..,n)

(i=l,

occurring in the intervals [ti-l,ti]

Then, we may write

With (4.3),

it follows that

.

We now let n

+ m

in such a way that sup.lti-ti-l

1

-+

0.

It k '

follows, with the assumed continuity of the functional L,that t ~ ( t )= L[~(r)l = ~t L[i(~)l = ~(~)dh(r,t) + g(t,t)~(t). n-

j

(4.6)

The relation between the representations (4.6) and (3.11) can be established in the following manner.

We note from the

definitions of h(~,t) and g (T,t) given in (4.3) and (3.8), bearing in mind the different definitions of u(5,~) in the two relations, that h(T,t)

+

gCr,t) = g(-m,t)

-

-

(4 7 )

Using this relation to substitute for h(~,t) in (4.6), we have, with €(to) = 0,

5.

Additive functionals The procedure used in 14 to arrive at a representation for

the stress, in the case when it is a linear functional of the

*

The result given in (4.6) could, in principle, be read off from Riesz's theorem, if the latter is properly interpreted. However, the proof of Riesz's theorem given in many of the standard texts on functional analysis is somewhat incorrect and leads to the result (4.6) with g(t ,t) = 0.

strain history, can also be used to obtain a representation for the stress in certain cases in which it is not a linear function-

al of the strain-history. In Fig. 8, we show a strain-history ~ ( r )consisting of two disjoint strain-histories E~(T) and E~(T).

Let al (t) and a2 (t)

be the stresses at time t corresponding to the strain histories E~(T)

and

E~(T)

taken separately and let a(t)

be the stress at

time t corresponding to the strain-history E(T).

for all disjoint

E~(T)

If

and E~(T), we say that the stress is an

additive functionaZ of the strain history E(T).

We now suppose that t = F [ECT)I, T=tO

where F is an additive functional.

We divide the time interval

[tO,t] into n sub-intervals and approximate the strain-history by a strain history E(r) consisting of n adjacent, but dis-

E(T)

joint, rectangular pulses.

The strain history u ( ~ , t ~ - ~ , t ~ )

corresponding to the pulse in the interval [ti-l,ti) is for , ' ( u

ti-l 5 T < ti

...,n)

(i=l,

ti-l,ti)=

(5.3) 0

We define

otherwise.

~ I (ti-l), E ti-l,t}

by

(ti-ti-l)kI~(ti-l),ti-l,tl

=

F[u(~,t~-~,t~)l ;

(5 - 4 )

i.e. kI~(t~,~),t~-~,t) is the stress at time t associated with the pulse-like strain history (5.3), per unit duration of the pulse.

It follows from the additivity of F that

where g{~(t),t,t)

is the stress at time t associated with a step

in the strain from zero to ~ ( t )at time t. Now, allowing n

with sup. ] ti-ti-ll

+

that F is a continuous functional of

E(T)

-+

0, and assuming

with respect to the

supremum norm, we obtain

6.

Hereditary materials For many of the materials with which we are concerned, it is

a good approximation to assume that their properties do not change purely as a result of the passage of time.

Such materials are

called h e r e d i t a r y materials. Let cl(r)

and

E~(T)

be two strain histories in a hereditary

material, such that E1(d

=

c2(T+T),

where T is some constant positive time.

Let nl(t)

and r2(t)

be the corresponding stresses at an arbitrary time t.

Then, if

the material is hereditary,

Also, from ( 2 . 3 ) ,

Since the material considered is hereditary, the time to in each of equations (6.3) may be arbitrarily chosen prior to the time at which the strain becomes non-zero. We can accordingly

r e p l a c e (6.3)

by t+T

For each o f t h e s t r a i n h i s t o r i e s E ~ ( T and ) E ~ ( T ) l, e t s d e n o t e t i m e p r i o r t o t h e i n s t a n t a t which t h e s t r e s s i s measured. We c a l l s t h e l a p s e d t i m e .

Then, f o r e q u a l v a l u e s of s t h e

s t r a i n s a r e e q u a l and we may w r i t e

el ( T ) = E ~ ( T + T =) 2 ( s ) Then, from (6.3)

and (6.4)

, we

, say.

(6.5)

have

and

i is

where

a f u n c t i o n a l o f t ( s ) and an o r d i n a r y f u n c t i o n o f t i n

(6.6)1 and of t + T i n (6.6) 2 .

Since t h e r e l a t i o n ( 6 . 2 ) i s v a l i d

f o r a r b i t r a r y p o s i t i v e T , i t f o l l o w s from (6.6) t h a t

?

i n (6.6).,

must be i n d e p e n d e n t o f t f o r a h e r e d i t a r y m a t e r i a l . We c o n c l u d e t h a t i f t h e s t r e s s s ( t ) i n a h e r e d i t a r y m a t e r i a l , c o r r e s p o n d i n g t o a s t r a i n h i s t o r y E ( T ) , i s g i v e n by ( 2 . 3 ) ,

then

i t may be e x p r e s s e d i n t h e form

where ;(s)

= E(T),

S

=

t-T,

and t o i s any time such t h a t E ( T ) = 0 f o r T L t o . We now a p p l y t h i s r e s u l t t o t h e c a s e , e x p r e s s e d by ( 3 . 1 ) , when ~ ( t i) s a l i n e a r f u n c t i o n a l of ~ ( r ) . Then,

where

Writing g(r,t)

=

g(s),

and using (6.8), we find that for a

hereditary material, the constitutive equation (4.8) may be written as n(t)

(s)di(s)

=

+

(0) i(0)

,

(6.11)

0 where so=t-to. Similarly, the constitutive equation (3.11) may be written as

Equations (6.11) and (6.12) may, of course, also be written in the forms

-/

t

n(t)

=

~(~)di(t-T) + ;(o)E(~),

(6.13)

0 n(t) = lt;(t-~)dp (r) 0 respectively. 7.

Implicit constitutii*e equations We take as our starting point the constitutive equation

(6.14) with to

=

and write a =a(t),

--.

For convenience, we omit the roof over g

thus: n

=

/

t g(t-r)d~(-c).

-0

We now assume that

E(T)

is differentiable, so that (7.1) may be

re-written as

We approximate the kernel g(t-r) by

where Ck and ak are constants. Differentiating (7.2) i times with respect to t, we obtain

Taking i=O,...,n we obtain from (7.4) of which is (7.2)

.

n+l equations, the first

With (7.3), we obtain

-

J

w

We can eliminate the integrals in (7.5) from the (n+l) equations obtained by taking i=O,l, ...,n.

We obtain

where S1,S2,,..,Sn are the sums of products of the a's taken 1,2,

...,n

at a time and SO=l. This equation has the form

where Pn is the operator

and Qn is an operator of similar form with different coefficients. When n=l, we have

and the relation (7.6) becomes

The material modelled by (7.10) is called a MaxweZZian fluid. Of course, the differential equation (7.6) for n (which may equally be considered to be a differential equation for

E

if r is

given) does not provide a complete constitutive equation unless n appropriate initial conditions are specified. We may generalize (7.6) to the non-linear ease heuristically as

8.

Alternative constitutive assumptions It was remarked earlier in 52 that there may exist materials

with memory for which the stress a(t)

is not strictly expressible

as a functional of the strain history up to and including time t, i.e. by a relation of the type ( 2 . 2 1 .

This is the case if the

stress at time t is a function of the instantaneous value, at time t, of the rate of change of strain, which may have a saltus at time t.

For example, in the case of an incompressible New-

tonian fluid, we have ~ ( t )= 3Q&(t)/{l+E(t) where

Q

1,

(8.11

is the viscosity of the fluid and i(t) denotes

I

[ d (~~ ) / d ~ lT=t.

It is evident that z(t) and consequently ~ { i '

cannot be expressed as a functional of

E(T)

~ i t hsupport

(-=,TI,

if :(T)

has a saltus at time t, i.e. if ;(T)

= O

(~

=

I,(v0)

( a = l ,...,v )

(5.6)

has as solutions only the column matrices v given by

...,v )

i.e. only if Ia (a = 1,

determine the argument tensors com-

pletely apart from a transformation of the symmetry group. In order to prove that (5.6)

does imply (5.7),

we first de-

fine the distance D * ( V ; V ~ ) of v o from the orbit of v by

t

(5.8) D R ( v ; v O ) = min. { ( s A v - v o ) ( s X v - v O ) } . X We note that D* so defined is an invariant of v and is also a con-

tinuous function of v.

From the Weierstrass theorem, it can be

approximated by a polynomial in v , say P ( v ; v o , ~ ) , such that for any E > O IP(v;v~,E) - D * ( v ; v ~ )1

Materials with Memory (C.I.M.E. Summer Schools, 74)

Shape Memory Materials

Stereodynamics (C.I.M.E. Summer Schools, 56)

Pseudodifferential Operators with Applications (C.I.M.E. Summer Schools, 75)

74

Controllability and Observability (C.I.M.E. Summer Schools, 46)

Stochastic Differential Equations (C.I.M.E. Summer Schools, 77)

Dynamical Systems (C.I.M.E. Summer Schools, 78)

Wave Propagation (C.I.M.E. Summer Schools, 81)

Wave Propagation: (C.I.M.E. Summer Schools, Vol. 81)

Teoria Della Turbolenza (C.I.M.E. Summer Schools, 14)

Differential Topology (C.I.M.E. Summer Schools, 73)

Algebraic Surfaces (C.I.M.E. Summer Schools, 76)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Potential Theory (C.I.M.E. Summer Schools, 49)

Spectral Analysis (C.I.M.E. Summer Schools, 64)

Stability Problems (C.I.M.E. Summer Schools, 65)

Theoretical Computer Sciences (C.I.M.E. Summer Schools, 68)

Relativistic Fluid Dynamic (C.I.M.E. Summer Schools, 52)

Thermodynamics of Materials with Memory: Theory and Applications

Economia Matematica (C.I.M.E. Summer Schools, 40)

Statistical Mechanics (C.I.M.E. Summer Schools, 71)

Materials with Rheological Properties

Приключения-74

Materials with memory: Initial boundary value problems for constitutive equations with internal variables

بحار الأنوار - الجزء 74

Schuldig in 74 Fällen

74 Seaside Avenue

Ποίηση 1971-74

74 Seaside Avenue

Materials with Memory (C.I.M.E. Summer Schools, 74)

Shape Memory Materials

Stereodynamics (C.I.M.E. Summer Schools, 56)

Pseudodifferential Operators with Applications (C.I.M.E. Summer Schools, 75)

74

Controllability and Observability (C.I.M.E. Summer Schools, 46)

Stochastic Differential Equations (C.I.M.E. Summer Schools, 77)

Dynamical Systems (C.I.M.E. Summer Schools, 78)

Wave Propagation (C.I.M.E. Summer Schools, 81)

Wave Propagation: (C.I.M.E. Summer Schools, Vol. 81)

Teoria Della Turbolenza (C.I.M.E. Summer Schools, 14)

Differential Topology (C.I.M.E. Summer Schools, 73)

Algebraic Surfaces (C.I.M.E. Summer Schools, 76)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Potential Theory (C.I.M.E. Summer Schools, 49)

Spectral Analysis (C.I.M.E. Summer Schools, 64)

Stability Problems (C.I.M.E. Summer Schools, 65)

Theoretical Computer Sciences (C.I.M.E. Summer Schools, 68)

Relativistic Fluid Dynamic (C.I.M.E. Summer Schools, 52)

Thermodynamics of Materials with Memory: Theory and Applications

Economia Matematica (C.I.M.E. Summer Schools, 40)

Statistical Mechanics (C.I.M.E. Summer Schools, 71)

Materials with Rheological Properties

Приключения-74

Materials with memory: Initial boundary value problems for constitutive equations with internal variables

بحار الأنوار - الجزء 74

Schuldig in 74 Fällen

74 Seaside Avenue

Ποίηση 1971-74

74 Seaside Avenue

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