-·

Proceedings of the Royal Society of Edinburgh, lOOA, 39-52, / 985

Relaxation for sorne dynamical problems

B. Dacorogna Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne,

1015 Lausanne, Suisse

(MS received 20 September 1983. Revised MS received 17 July 1984)

Synopsis In this article, we study the functional H(u)=f6 Jn[~iau/a t(x, r)i2- F(Vxu )]dxdt where !l ciR" is a bounded open set and u: !l x (0, T )--> IR"' and when F: IR""' -->IR fails to be quasiconvex. We show that with respect to strong convergence of au/a t and weak convergence of V xu, the above functional behaves as H(u) = J;j ln[~ lau/atl2

- QF(V xu)] dx dt where QF is the lower quasiconvex envel ope ofF.

Introduction

Consider the functional

H(u ) = IT t [~~~~(x, t) r-F(Vxu(x, t)) J dx dt

where (i) D, c IR" is a bounded open set and T > 0,

(ii) u(x, t): IR" x IR~ IRm and thus

Vu = (vu · au)EIR"m x iRm x ' at '

(iii) F: IR""'~ IR is continuous and satisfies

• a+b \A\P ~F(A)~c+ d \A\P

for sorne a, c EIR, d ~ b > 0, p > 1 and for every A EIR"m. We can associate with (0.1) the functional

E(u) = t F(Vxu(x)) dx

where u(x, t) == u(x).

(0.1)

(0.2)

(0.3)

Before describing the results of this article, one should remark that the functional H is rarely studied from the point of view of the calculus of variations, since H has undesirable properties such as unboundedness from below. However, we will see that this is not a major obstacle to obtaining a relaxation theorem.

The usual hypothesis imposed on F in (0.1) or in (0.3) is the so called quasiconvexity condition, i.e.

L F(A + Vcp(x)) dx~F(A)measD (0.4)

40 B. Dacorogna

for every bounded domain D c IR", A E IR"m and cp E W6·=(D; !Rm) (i.e. cp is locally Lipschitz and cp = 0 on an).

Note that in the case m = 1 (or n = 1), the notion of quasiconvexity reduces to the ordinary convexity condition. Furthermore, (0.4) implies the well known Legendre-Hadamard condition, i.e.

(0.5)

for every À EIR", IL EIR"' and A EIR""'. If one assumes F to be quasiconvex and thus (0.5), then the Euler equations

associated with (0.1) (respectively (0.3)) are hyperbolic (respectively elliptic ). In the case m = 1, the Euler equations reduce to a single equation, i.e.

a2 u - 2 -div f(gradx u) = 0, at

div f(gradx u) = 0, if u(x, t) == u(x),

(0.1')

(0.3')

where f = grad F (provided Fis C 1). Thus, in this case, the quasiconvexity ofF is

equivalent to the monotonicity of f. The purpose of this article is to study (0.1) when F fails to be quasiconvex,

therefore · the Euler equations are of mixed type (hyperbolic-elliptic). We will show that if QF denotes the lower quasiconvex envelope of F, i.e.

QF = sup { <l> ~ F: <l> quasiconvex} and

(0.6)

then the following theorem holds.

THEoREM. For every ü E W = {u(x, t): V xu E L~m(n x (0, T)), au/at E L~(n x (0, T))}, there exists a sequence uv E W so that

uv(x, t)= ü(x, t), (x, t)Ea!1x(O, T),

uv(x, 0) = ü(x, 0), x EÛ,

auv aü - (x, 0) = -(x, 0), x E !1, at at

V xu v ~ Vxü in L~m(O x (0, T)) ,

au " aü . -~- m L~(n x (O, T)), at at

H(u") ~ H(ü).

(0.7)

Furthermore, in the case m = 1, if Fis C 1 and if we let f = grad F and f = grad QF, then

a2 u" a2 ü -- 2--div f(gradx u") ~~-div f(gradx ü) in the sense of distributions (0.8) at at

(where ~ denotes weak convergence in LP).

Relaxation for some dynamical problems 41

The proof of the theorem is based on that of [1] valid for the functional E and on a representation theorem (Theorem 5) for the quasiconvex envelope QF.

By rephrasing the theorem, one can say that with respect to weak convergence, the functionals H and H are " equivalent"; note however that the Euler equations associated with H are hyperbolic in contrast to those associated with H. Besides, in the case m = 1, one can consider the weak solutions of the equation associated with H if they exist (which is obviously an open problem) , as generalized solutions of (0 , l') in the sense of (0.7) and (0.8) .

The article is divided into three sections. The first recalls basic facts about quasiconvex functions and relaxation theorems. The second section gives the proof of the above theorem. In the last section, we see how to apply the above result to nonlinear conservation laws of mixed type

{u, - vx = 0,

v,-a(ut = 0,

where a is not necessarily an increasing function of its argument.

1. Quasiconvex envelopes and the relaxation tbeorem

We recall fust sorne elementary properties of quasiconvex functions (the notion of quasiconvexity was introduced by Morrey [5], [6]). For more details about the history and the proofs of the next theorems, see [2] .

THEOREM 1. Let F : [Rnm ~ IR be continuous and let

I (u, fl) = L F(Vu(x)) dx. (1.1)

(i) Let F satisfy

a ~ F(u)~b+ c \u\P

for every u E !Rnm and for some a, b E IR, c ~ 0 and p ~ 1. Then I is (sequen­tially ) weakly lower semicontinuous in W 1

·P, i.e.

lirn inf I(u v, fl) ~ I(u, fl) v~=

for every uv ____,. u in W 1· p if and only if Fis quasiconvex, i.e.

L F(u + V'cp(x)) dx ~F(u) meas D

for every bounded domain D c !Rn, u E !Rnm and cp E W5·=(D ; !Rm). (ii) If Fis quasiconvex, then it is rank one convex, i.e.

F(Au + (1 - A)v) ~ AF(u) + (1- A)F(v)

(1.2)

(1.3)

(1.4)

for every À E [0, 1], u, v E IRnm with rank(u - v) ~ 1. Furthermore, if Fis C2

42 B. Dacorogna

then (1.4) is equivalent to the Legendre-Hadamard (or ellipticity) condition

a2F(u) L ÀiÀj/.LoJLs ~ 0 i ,j ,a, f3 aU;a aujf3

for every À E ihf, 11- E IRm, u E IR"m. (iii) If n = 1 or m = 1, then F is quasiconvex if and only if F is convex. (iv) If n = m and there exists G: IR~ IR continuous and such that

F(u) = G(det u),

then F is quasiconvex if and only if G is convex.

Remarks. (i) One has in general

F convex :::;> F quasiconvex :::;> Frank one convex,

(1.5)

the converse being false for the fust implication and an open problem for the second.

(ii) The case n = 1 corresponds in (0.3) to the case of a system of ordinary differentiai equations while that of m = 1 corresponds to the case of a single equation.

A special class of quasiconvex functions are the so called quasiaffine functions (i.e. functions <t> such that <t> and - <t> are quasiconvex).

THEOREM 2. The following properties are equivalent: (i) <t> is quasiaffine,

(ii) let adj , u denote the matrix of all s x s(1 ~ s ~ inf {n, m}) subdeterminants of the matrix u E IR"m, then

inf { n ,m}

<l>(u) =A + L (B, ; adj , u )u(s ) (1.6) s = l

where

( ) (m) (n) m ! n! as= s s =s!(m-s)!s!(n-s)!'

(., .)u(sl denotes the scalar product in IRu(sl , A E IR and B, E IRu(sl are constants.

We now show that quasiconvex functions must be Lipschitz, analogous to convex functions [ 4, Corollary 2.4, Chap. 1]).

PROPOSITION 3. Let F: IR"'n ~IR be quasiconvex. Then F is locally Lipschitz.

Proof. Step 1: We first show that Fis continuous. Observe that there is no loss of generality if we prove the continuity at u = 0 and if we assume that F(O) =O.

For fixed E E (0 , ~) , we want to find ô > 0 such that

l vl~ô :::;> IF(v)I~E. (1.7)

Let us put

B.., ={xEIR": lxi<TJ}.

Relaxation for sorne dynamical problems 43

We then define

x<*Fix~+l and let cp E W6·=(B 1 ; !Rm) be defined as

cp(x) = -vx(x)x. (1.8)

Since F is quasiconvex we have that

F(v)measB 1 ~ J F(v+Vcp(x))dx B,

= l F(v+Vcp(x)) dx+ l F(v+Vcp(x)) dx Bl- E B.~BI- o;

~F(O)measB1_0 + J F(v+Vcp(x))dx B I- B l-e

~ J F(v + Vcp(x)) dx, B 1- B 1- o:

(1.9)

since F(O) =O. Observe theo that there exist K and K' such that

( K') K \v+ V cp\~ \v\+ \V cp\~ \v\ 1 +--;- ~-;\v\, almost everywhere in B 1 . (1.10)

We then define

a= sup {F(z): \z\ ~ 1},

and let

8 = E/K

so that

\v\~8 ~ K/E \v\~1 ~ \v+Vcp\~1 almosteverywhere

and hence

F(v + Vcp(x)) ~a almost everywhere in B 1 - B 1_0

•

On combining (1.9) and (1.12), we obtain

( meas(B 1 - Bt_o)

F v)~ a <naE. measB 1

Similarly, if we choose

t{!(x) = vx(x)x

and use the quasiconvexity of F, we have

(1.11)

(1.12)

(1.13)

F(O) meas B 1 = 0 ~ l F(Vt{!(x)) dx ~ F(v) meas B 1_ 0 + J F(Vt{!(x)) dx. BI BI - Bl - e

(1.14)

44 B. Dacorogna

As in (1.10)-(1.12), we also find that

lvi~ o ~ F(Vt{i(x)) ~a

and thus since e E (0, 1),

F( ) meas (B 1 - B 1_E) nae

v ~-a >- > -2"nae. (1.15) measB 1_E (1-e)"

By combining (1.13) and (1.15), we obtain

lvl~e/K ~ IF(v)l~2"nae. (1.16)

Step 2: We are now in a position to show that Fis actually locally Lipschitz. Let

and let

For v En, we define

m = inf {F(v): v E !1},

M=sup{F(v): vEn}.

G(u) = F(u +v)- F(v),

SO that if U, U +V E fl we have

G(u)~M-m for uEfl

and using (1.16) we get that

lul~e/K ~ IG(u)I~(M-m)2"ne.

On !etting u = u +v and choosing e = K lu- vi, we get from (1.20) that

(1.17)

(1.18)

(1.19)

(1.20)

lu-vl~e!K ~ IF(u)-F(v)I~K(M-m)2"n lu-vi. (1.21)

Now let w En and consider the segment [v, w] c n and choose points ul, ... ' UN

on [v, w] such that

with

(1.22)

By using (1.21) and (1.22), we get

and thus

IF(w)- F(v)l ~K(M- m)2"n lw -vi

for every v, w E n. D

..

Relaxation for some dynamical problems 45

We now turn our attention to quasiconvex envelopes of F and we mention here a result obtained in [1]. Let

QF = sup { <P ~ F: <P quasiconvex}.

THEo REM 4. Let D c !Rn be a bounded domain and suppose that

a ~F(u) ~ b(1 + \u\P) ,

aEIR, b?;O, p?;1 and UE!Rnm. Then

QF(u)=inff 1 DJ F(u+V<p(x))dx:<pEW6·=(D;!Rm)}.

"' ~eas 0

(1.23)

(1.24)

Remark. The question of whether FE C 1 implies that QF E C 1 is still open, as Proposition 3 just ensures that QF is locally Lipschitz. However if n = 1 or m = 1, Theorem 1 shows that the notions of convexity and quasiconvexity are equivalent, thus QF = F** (where F** is the convex envelope of F) and in this càse a direct application of the Hahn-Banach Theorem shows that F** is also C 1

.

We now obtain a more explicit form of the quasiconvex envelope in sorne special cases.

TH:EoREM 5. Let u(x, t): !Rn x !RN~ !Rm, Vu(x, t) =(V xu, V,u) and let

F(Vu(x, t)) = G(V xu(x, t)) + H(V,u(x, t)) (1.25)

where G: !Rnm ~IR, H: IRNm ~IR are continuous and satisfy hypotheses of the type (1.24). Then

QF=QG+QH.

Proof. The proof is in three steps. Step 1. We first show that

QF~QG+H.

(1.26)

(1.27)

Step 2. A sirnilar proof to that of Step 1, but inverting the rôles of x and t will give

QF~G+QH. (1.28)

Step 3. Assume that Step 1 (and thus Step 2) have been established and let us demonstrate the theorem. Apply Step 2 to

'l'= QG+H to get

Q'l' = Q(QG+H)~QG+QH.

By using (1.27) and (1.29), we obtain

QG+ QH ~ Q( QF) = QF ~ Q( QG+ H) ~QG+ QH

and thus the result.

(1.29)

It therefore remains to show (1.27). Let u"'!Rnm and vEIRNm and use Theorem

46 B. Dacorogna

4 to get

QF(u, v)= i~f {L J [ G(u +V xcp(x, t)) + H(v + V,cp(x, t))] dx dt:

n cpE W6·=(DxD;IRm)}

(1.30)

where D c IR", n c !RN are unit hypercubes. Let e > 0 be fixed, theo Theorem 4 implies that there exist u E W6·=(D; IRm)

such that

L G(u + Vp(x)) dx ~ e + QG(u). (1.31)

On extending u by periodicity from D to IR", we trivially have that for vEN,

L G(u + Vp(vx)) dx ~ e + QG(u).

Let nv c n be a hypercube with the same centre as n and such that

dist (D; DJ = 1/v.

We theo define t{! E W6·=(n), 0 ~ t{!(t) ~ 1 such that

We now choose

t{!(t)={1 ~f tEflvcflc[RN, o 1f tE an.

1 cp(x, t) =- u(vx)t{!(t)

v

and observe th at cp E W6·=(D x D; IRm ). On using (1.30), we get

QF(u, v)~ L t G(u+t{!(t)Vp(vx)) dxdt

(1.32)

(1.33)

(1.34)

+ L t H(v+;u(vx)®gradt{!(t))dxdt, (1.35)

where u(vx)®grad t{!(t)EIRNm denotes the tensor product. We next use (1.32) to estimate separately the two terms on the right hand side

of (1.35), recalling that meas D = meas n = 1. Thus we have

L t G(u + t{!(t)Vp(vx)) dx dt

= i i G(u+Vp(vx))dxdt+ J f G(u+t{!(t)Vp(vx))dxdt n " o n - n " o

~meas Dv(e + QG(u)) +meas (0-DJ sup {\G(u + t{!(t)Vp(vx)\}. (1.36)

Similarly,

L t H( v+; u(vx)®grad t{!(t~ dx dt

Relaxation for sorne dynamical problems 47

= H(v) meas av + J f. H(v + _.!:_ a(vx)®grad tfi(t)) dx dt o n - n " v

~ H(v) meas av +meas (!1-!1J sup { IH( v + ~ a(vx)®grad t/l(t)) 1}. (1.37)

By combining (1.36) and (1.37) and choosing v sufficiently large, we deduce that

QF(u, v) ~ QG(u) + H(v) + 2s.

Since s is arbitrary, we have indeed obtained (1.27) and thus the theorem is proved. 0

We conclude this section with the relaxation theorem established in [1].

THEO REM 6. Let ac IRq be a bounded open set with Lipschitz boundary. Let r.p: IRqs ~IR be continuous and such that

a + b \A\ 13 ~r.p(A)~c+d \A\13 (1.38)

for every A E IRq', for sorne a, c E IR, d ~ b > 0 and {3 > 1. Then for every u E wu(n; IR 5

) , there exists {u v}, u v E W 1•13(!1; IRS) such that

uv = u on an, u v ~ u in W 1•13,

L r.p(V'u v(x)) dx ~ L Qr.p(V'u(x)) dx.

(1.39)

(1.40)

(1.41)

Remark. In fact , one can allow a much weaker coercivity condition than (1.38). One only needs

J J

a+ L bi \<Pi(A)\ 13; ~ r.p(A) ~ c + L di \<P/A) \13; (1.42)

for every AEIRqs, for sorne a, cEIR, 1~1 (an integer) , {3i > 1, di~bi>O and where <Pi : IRqs ~IR, j = 1, . .. , 1 are quasiaffine functions. Then the conclusions of the theorem hold with (1.40) replaced by

2. Main result

We now restate the hypotheses.

(1.43)

(i) Let 0 c IR" be a bounded open set with Lipschitz boundary and let !1 = 0 X(O, T)ciR"+1 where T>O.

(ii) Let F: IR"m ~IR be continuous and such that

a+b \A\P ~F(A)~c + d \A\P

for every A E IR"m and for sorne a, c E IR, d ~ b > 0 and p > 1. (iii) Let p = min {2, p} > 1 and let

W = { U E Wl,jj(!1; IRm) : ~~ E L~(Ü) , V' x U E L~m(!1)}, We then have the main theorem.

(2.1)

48 B. Dacorogna

THEO REM 7. For every ü E W, there exist uv E W so that

uv= ü on an,

V xU v~ V xÜ in L~m(n),

j_ uv ~j_ ü in L~(n), at at

L F("ilxuv(x, t)) dx dt~ t QF(Vxü(x, t)) dx dt.

(2.2)

(2.3)

(2.4)

(2.5)

Remarks. (i) Obviously if F satisfies coercivity conditions similar to (1.42) instead of (2.1) then the theorem remains valid provided (2.3) is replaced by the natural weak convergence, i.e. (1.43).

(ii) (2.4) and (2.5) imply in particular that H(uv) ~ H(ü). (iii) An elementary change in Theorem 6 allows the replacement of (2.2) by the

more interesting conditions

Proof. Let

[

uv(x, t)= ü(x, t), (x, t)Eaüx(O, T),

uv(X, Ü) = ü(x, Ü), XE Ü,

a a - uv(x, 0) =- ü(x, 0), xE O. at at

cp(Vu(x, t))=cp(Vxu, :tu)=~ l:t ur +F(Vxu), (2.6)

I(u) = t cp(Vu(x, t)) dx dt, (2.7)

Ï(u)= t Qcp(Vu(x, t)) dxdt= t [~ l:t ur +QF("ilxu)J dxdt, (2.8)

where we have used Theorem 5 in (2.8). We then apply Theorem 6 to such a cp. We deduce that there exist uv E W so

that

uv= ü on an,

Vxuv ~ Vxü in L~m(n),

a v a _ -u ~-u inL~(n), at at

I(uv) ~ Ï(ü).

(2.9)

(2.10)

(2.11)

(2.12)

Observe furthermore th at sin ce the functions u ~ 1 u 12 (IRm ~IR) and QF: IR"m ~IR are quasiconvex, they are weakly lower semicontinuous (Theorem 1) and thus

lim J li uv(x, t) 12

dx dt?; J \i ü(x, t) 12

dx dt, v~= n at n at

(2.13)

lim J F(Vxuv(x,t))dxdt?;J QF(Vxü(x,t))dxdt. v-~ n 0

(2.14)

Relaxation for sorne dynamical.problems 49

On combining (2.12) with (2.13) and (2.14) we get that (2.13) and (2.14) are actually equalities; thus the theorem is proved. D

We now conclude this section by a refinement of Theorem 7 , in the case m = 1, in particular the Euler equations are reduced to a single equation and OF= F**.

CoROllARY 8. Let a be as above, F: IR" ~IR be C 1 and let f = grad F, f = grad OF. We suppose furthermore that

lf(A)I~a+b IAip-\ F(A)~c+d IAiv,

(2.15)

(2.16)

where a, b ~ 0, c E IR d > 0 and p > 1. If ü E W, then there exist u v E W such that

uv = ü on an, gradx uv~ gradx ü in L~(ll) ,

~uv ~~ ü in L 2(a), at at

(2.17)

(2.18)

(2.19)

~Uv ~Ü -- 2- - divxf(gradx uv) ~-2 -divxf(gradx ü) in the sense of distributions. (2.20) at at

Proof. From Theorem 7, there exist v v E W su ch that

v v = ü on an, gradx v v ~ gradx ü in L~(O) ,

(2.21)

(2.22)

a a -v v~- ü in L 2 (0), (2.23) at at

I(vv) == L F(gradx v v(x, t)) dx dt~ Ï(ü) == L F **(gradx ü(x, t)) dx dt. (2.24)

Observe that since F** is convex, then so is Ï. Furthermore, I and Ï are Gâteaux differentiable and we denote their differentiai by I' and Ï' respectively.

Let p =min {2, p} > 1 and

V= { u E W6·il (O); ~~EL 2 (0), gradx u E L~(n)}.

Let V' be the dual of V and <· ; ·) the bilinear canonical form on V' x V. We then define for every v E V,

J(v) = I(v + ü)- i(ü) - (Î'(ü); v ),

Ï(v) = Ï(v + ü)- i(ü) -(Î'(ü); v).

It is then easy to see that, since i is convex, for every v E V,

J(v) ~Ï(v) ~ 0 = Ï(O) = min {Ï(u): u EV}.

Furthermore, from (2.21)-(2.24), we have

J(vv - ü) ~ 0 =min {Ï(u) : u EV},

(2.25)

(2.26)

(2.27)

(2.28)

50 B. Dacorogna

and thus

min {Ï(u): u EV}= inf {J(u): u EV}= O. (2.29)

We finally apply [ 4, Corollary 6.1 , p. 30] to the functional J and to the sequence v v - ü to ob tain the result. 0

3. Applications

In this section, we show how to apply Corollary 8 to nonlinear conservation laws of the type

{u, -vx = 0,

V,- <T(U)x = 0, (3.1)

where <T is not necessarily an increasing function. We only assume that there exist F: ~ ~ ~ C 1 with F'(z) = <T(z) and such that

(3.2)

and every z E ~ and for sorne a, c E ~, d ~ b > 0 and p > 1. Therefore, a function <T of the type shawn in Figure 1 is admissible. (For

another approach to the study of equations (3.1), see Shearer [7] .) By setting u = wx and v= w, it is immediately seen that (3.1) is equivalent to

(3.3)

The above equation governs the one-dimensional motion of a homogeneous nonlinear elastic material under zero body forces ; w(x, t) denotes the displace­ment at time t of a particle having position x in a reference configuration. We also have that

F(z) = r <T(u) du (3.4)

is the stored energy function and since <T is not a monotone function theo F is not convex.

o ( z)

z

Figure 1

Relaxation for some dynamical problems

à ( z)

'

Figure 2

' ..... __ ........ ; 1 1

B

51

We then define QF, the quasiconvex envelope of F, which is in this case (i.e. m = n = 1 and Theorem 2) the convex envelope, F** ofF. We also have that F** is C\ since F is C 1 (c.f. Remark following Theorem 4). We may then define

6-(z) = (F**)'(z). (3.5)

More precisely, if a- is as in Figure 1, then 6- appears as in Figure 2 and satisfies the Maxwell condition, i.e.

J: 6-(z) dz = r a-(z) dz. (3.6)

(The line {6-(z) = a-(A)} is usually called the Maxwell line.) We then imrnediately get from Corollary 8 the following result.

THEOREM 9. Let T>O, a,{3E~ with a<{3, p=min{2, p} and W= {u E Wl,jj((a, {3) x (0, T)); au/at EL 2 , aujax E LP}, then for every w E W, there exist w" E W so that:

w "(a, t) = w(a, t), w "({3, t) = w({3, t) for every tE (0, T),

a a w "(x, O) = w(x, 0),- w "(x, 0) = - w(x, 0) for every xE (a, {3),

at at

w~= u" ~ wx = u in U((a, {3) X (0, T)),

w~=v"~w,=v in L 2 ((a,{3) x (O, T)) ,

w~-a-(w~)x ~Wu - 6-(wJx in the sense of distributions.

Remarks. (i) Returning to (3.1), one may rewrite the last equation of the above theorem as

{ u~-v~= u,- iïx = 0,

v~- a-(u ")x ~v,- 6-(u)x in the sense of distributions.

52 B. Dacorogna

(ii) The problem of solving the relaxed equations (i.e. (3.1) with a replaced by ô-) is still open. For recent results on system of equations of the above type with â­allowed to have only one inftection point, see Di Perna [3].

References

1 B. Dacorogna. Quasiconvexity and relaxation of non convex problems in the calculus of variations. J . Funct. Anal. 46 (1982) , 102-118.

2 B. Dacorogna. Weak continuity and weak lower semicontinuity of nonlinear functionals. Lecture Notes in Mathematics 922 (Berlin : Springer, 1982).

3 R. J. Di Perna. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983) , 27-70.

4 I. Ekeland and R. Témam. Analyse convexe et problèmes variationnels (Paris: Dunod, Gauthier­Villars, 1976).

5 C . B . Morrey. Quasiconvexity and the lower semicontinuity of multiple integrais. Pacifie J . Math. 2 (1952) , 25-53.

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(Issued 3 June 1985)