Mathematical Notes, Vol. 63, Ha. 1, 1998

O n G r u s h i n ' s E q u a t i o n

Nguyen Minh Tr] UDC 517

ABSTRACT. The paper deals with nonlinear problems for equations of Grushin type. We prove some nonexistence results via Pokhozhaev's identity. In the rest of the paper we prove some results on smoothness near the boundary of eigenfunctions by using an explicit formula for fundamental solutions and the Kelvin transform for the operator.

KEY WORDS: Grushin's equation, Pokhozhaev's identity, Kelvin transform, hypoelliptic operator.

w I n t r o d u c t i o n

In this paper we consider degenerating elliptic equations

02 02

in R 2 . If ~02(xl) = z~ k , k is an integer, and two vector fields

0 X 2 = X1 = Oxl 10z2

satisfy the HSrmander condition, then L is a hypoelliptic operator. Consider the l imiting case of degen- eracy: ~0(z,) e C~176 ~0(zt) ~ 0 if z , r 0, ~0(zl) takes real values, and

= 0 dz[

for any n ; the operator L is also hypoelliptic (although the HSrmander condition is not satisfied; see [1]). A detailed s tudy of smoothness of the solutions of these equations near the boundary was given in [2]. In [3] it was noted that (a + 2 ) / a behaves as the critical Sobolev index for the operator

0 2 12~ 0 2 c o = + ':' > o.

Similar results were obtained in [4] in connection with the study of nonlinear problems for E]b on the Heisenberg group. Let us recall the main results from [3].

T h e o r e m 1, Let f~ be Ga-star-shaped wi~h respect to ~hepoint (Xl, ;/;2) ~-~ (0, O). Then the boundary value problem

G~u + Au + [u['fu = O in ~, u = 0 on OQ,

where A < 0, 7 > 4 / a , does not have any nontHvial solutions in H2(~).

84

Translated from Maternalicheskie Zametki, Vol. 63, No. 1, pp. 95-105, January, 1998. Original article submitted June 27, 1996.

0001-4346/98/6312-0084 $20.00 C)1998 Plenum Publishing Corporation

D e f i n i t i o n 1.

0u t 0u 0xl �9 LP(n), Ix1 0~,~ �9 LP(n).

Let us define the norm in this space as follows:

a Ou p dx211/p. + ,+ , f

If p = 2, we can define an inner product in S~'2(f~) by

(OU 011 ) + ([Z110~O~2 [Z 11,.,, O'O ~ (u, ~)s:,~(n) = (u, . )L.(n) + O-~,' O~ L'(n) ' 0~2 }

It is readily shown that S~,V(f~) is defined as the closure of C01(fZ) in

Theorem 2. 1) Iir 1 _< p < a +

By S~,p(~), 1 < p < oo, we denote the set of all functions u �9 LP(~'I) such that

L2(fl) 1,p a Banach space and S1a,2(f~) is a Hilbert space. The space Sa,o(fl ) is

the space S~'P(s

2, then

S~,o(fl ,p C L((a+2)p)/(a+2-p) --r(fl)

for any positive suf~ciently small r , and this inclusion is totally continuous. 2) //" p > a + 2, then SPa.o(I2) C C~

Theorem 3. Suppose that f (u ) satisfies the following conditions:

1) f �9 f(0)= 0; 2) If(u)l < C(1 + lul = ) with 1 < m < (~ + 4) /k; 3) f ( u ) = o(u) as u --, 0; 4) there exists a number A such that ff ]u[ > A, then

fo " f ( s ) d s < p f (u)u ,

u = 0 on OfZ,

: w h e r e # e [0, 1/2 ).

Then the boundary va/ue probIem

G~u + f (u) = O in fl,

1,p always has a nontrivial solution in Sa,o(f~ ) .

w Nonlinear equations for Grushin's operator

Note that (a + 2) /a --* oo as ~ --* O. This illustrates the well-known fact that the limiting case (the Laplace operator in R 2) does not have a critical index (see [5]). As a --, oo, we have (a + 2)/r ---, 1. Let us give some results that illustrate the above.

Consider the problem

~2U 2 ~2'U' L u + f ( u ) = ~ + q o (x l )~x~ 2 + f ( u ) = 0 i n • , (1)

u = 0 on Of~, (2)

where f(O) = O, f (u ) E C(R) and f~ is a bounded domain containing the origin with smooth boundary in R 2. Set

Z" F(,~) = f ( s ) as,

and let v = (vl , v2) be the outer normal to Off. By C we denote general constants that not depend on the functions involved, but do not necessarily have the same value in all the estimates.

85

D e f i n i t i o n 2. A domain f/ is said to be L~,a-star-shaped with respect to the point (0, 0) if the inequality

(w,~ + ~ ( x , ) - ~ ) (x , .~, + ( # + 1)x~ - ~ ) > 0

holds almost everywhere on 0~.

Def in i t ion 3. A domain f~ is said to be L~,,oo-star-shaped with respect to the point (0, 0) if f~ is L~,#- s tar-shaped for any fl > 0.

E x a m p l e 1. The unit ball B~ = {(xt, x2) [ x 2 + z~ < r 2 } is an L~,oo-star-shaped domain.

L e m m a (Pokhozhaev 's identity). Suppose that u(x) is a solution o[ problem (1), (2) in Hz(f~). Then u(x) satisfies the identity

f(u)u dx, dx2

for any fl > 0.

P r o o f . By Sobolev's embedding theorem for bounded domains with smooth boundary, we have the inclusion H2(fl) C V~ where 0 < a < 1. Note that

0 Ou Ox, (z,r(u)) = F(u) + x,f(u)~xl , O~(x~F(u)) = F(u) + z~f(u) ff~Ux2 .

From the Gauss-Ostrogradskii formula we obtain

whence

f F(u)dxldx2 = - f z l f ( u ) ~ x l d x l d x z ,

/S f~ F(u)dx~ dx2 = -fl f x2f(u) ff-~2 dxl dx2.

Therefore,

Ou "~ f u

Again using the Gauss-Ostrogradski i formula, we obtain (the detailed calculation can be found in [6])

Ou

=--5 ~O.l / d'ldX2 + ~ (2XlqOt(Xl)~O('l)"~o2(Xl))kO;g21

/. J. i ( . , . . , ) [ o . ~ ' d , i ( . , .~ ' ( . , ) . . , ) ~ d.

f~ Ou a~, d~ # /~ ( ~ ~

,j, j, o.

86

Since

we have

Hence

0 0 0 0 ~=, = "' " ~ ' a=--T = "" ~ '

/. (I +/~) F(u)dx, dx2 = T ~ dx, dx2

-{- ~ { zl~t(zl)~(zl) + ( T ) ~ (Xl)} (-~'~2 ) dx, dx~

i +~ ~\o~] (~ +v~(=')"g){='~' +#=~'~}ds

+ f~ (=,r #~,(~,) ) ( a,, ,~e=, at,. [] \ a=: J

The following two theorems readily follow from the lemma just proved.

T h e o r e m 4. Suppose that ~ is L#-star-shaped with respect to the point (0, O) and

1) (/~ + 2)F(u) - -~f(u) < 0 when u 9s O;

2) = , . r ___ (# + z)~(=,) in ~.

Then there is no nontrivial solution u E H=(~) of problem (1), (2).

T h e o r e m 5. Suppose that f~ is L#-star-shaped with respect to the point (0, O) and

1) (fl + 2)F(u) - -~f(u) < 0 when u > O;

Then there is no nontrivial positive solution u E H~(f~) of problem (1), (2).

E x a m p l e 2. Consider the cases in which ~o(z,) = e -I='1-' (6 > 0) and f(u) = Au + lu l '~ , where A < 0, 7 > 0. Let us show that for any 3' > 0 there exists an r(7 , 6) such that problem (1), (2) has no nontrivial solutions u E H2(n ) , where s = Br(,,s) = {(z l ,z2) I z~ + z~ < r~(7 ,6)} . Indeed, we have

~o'(xl ) = sign(x, )[xx [-s-Xge-I=t I-S. Therefore,

XIWt(x,) ~ (/~-4- 1)~O(Xl) ~ 61=,l-%-t-,r ' >_ (# + 1)~-t-,t-"

. = . 61=,1-6 > (# + 1) -: > I=,1 <- k ~ - - y ] �9

If we choose r(7 , 6) = (67/(4 + 7)) '/s , then x,~'(x,) >_ ((4 + 7)l"7)r in B.(.t,, ) . By setting

A= = I=1~+ 2 F(u) = T + 7 + 2

and taking .f(u) in condition 3) of Theorem 3 with # - (4 + 7 ) /7 , we obtain

/~+=~<r=(,, , ,{ ( 4 ~ 2 7 ) ( ~ + [ : [ ' t / : ) _2 (Au2+ [u l . y+2 ) }dx , dx2

= 1 2 z s/~ ( 4 ~ ' y ) 2 /~.{_Z~=r2(.7,6)(~---~) (/''12+ " /222){ xl "/"SlAt- - - 2~2" /Y2} ds

+/~+=,<r=(..t,,s){XlCp,(Xl)~O(Xl)_ \'---~/(4 + 7~ ~2(xl) ~J (\~]Ou .~:2 dxldx2.

87

Since ~I~O'(Xl)~O(~I ) ~_~ ((4 + 7)/7)C(~,) in Br(7,$) , it follows that

2 / , 4 + 7 ~ x 2 11'2d'Tl dx2 ~ 1 (~--~v) (2~21 "{'2~81/'Y'~2){~g2aff ~ R T ; 2t dS"

For A < 0, we have

which leads to a contradiction. If A = O, we obtain

Therefore,

2 , , , +,;=, ,c , . ,~ ~ 0~ j " ~, - 7 - / = / ~* = 0.

_=0

By the uniqueness theorem, u = 0.

Now let us return to the case q02(xl) = x 2k .

w T h e f u n d a m e n t a l so lu t ion a n d t h e K e l v i n t r a n s f o r m

Let us define the polar coordinates as follows:

pk+l z~ = p s i g n ( s i n 0 ) [ s in0[ 1/(k+~), z2 = k + 1

pk+i dxl dz2 = k +-----~] sin0]-k/(~+l) dpdO.

We also use the following coordinates:

~COSO,

p(~l, ~2)= (~+~ +(k + 1)~.~) '/r w = I sin 011/(k+1) sign(sin 0) = xt

Gk = co 2k -57~p=+ -~ ~ + (1- oo,5 &

T h e o r e m 6. Suppose that E~(x l , x2) = p-k ; then

2k , /~ r ( ( k + 1)/(2k)) a (x , ,~2) . G k E k ( x l , X 2 ) - k +~ r ( ( 2 k + 1)/(2k))

P r o o f . First, let us show that Ek(xl , z2) e Llok+2)/~-"(R 2) for any sufficiently small positive r . Since Ek( z l , x2) e V~176 2 \ 0), it suffices to prove that Ek(x l , x2) E L(k+2)/k-~(Bp), where B a = {(xl , x2) I p(zl , z2) < p}. We have

L. k/xl /0 /0 2~ i sin0[-k/(k+l) dO sk+l(s -k) (k+2)/k-r ds , x2)} (k+2)/k-~ dxl dx2 = k + 1

< C s - l+rk ds < oo.

88

l~( k'sr2)/k ([I~ 2~ Ek(x~, x2) ~ ~loe ~ . /. It is readily seen that GkE~(xl, x2) -= 0 if (x], z2) ~ (0, 0). Also note that Denote ~ , = {(xl, x2) e R 2 ] p(xl, z2) > z}. In view of Green's formula, we have

- t E~,(x,,x2)Ckv(xx,x2)dxl dx2 = [ CkE~(xl,x2)v(x,,x2)dxl dx2 Jll

q- V(Xl,X2)'Y2" x21k" OEk(Xl,X2)Ox2 } ds

+ E~(~l ~2). ~2 .:1 ~- 0 - ( ~ 1 , ~ ) ~ d~ ' 0~ , 2 J

~ OE~(z,, z2) Jr t/(,~l, X2)" l/1 "

r pro,, ~, 02:1

Z ( - E~,(xl, x2). v l . Oz~

for = y ~ 6 C~(a). Since

and

1 (p21sine1-2~/(~+1) cos 2 0 + p2~+2 sin 2 e)112 ds = k +----f

=~+i (~ + ~)22 u=(u l ,u2 ) = (z~ ~+2 + (k + 1)2~g),/2 , (z~ ~+2 +(k + 1)2xg)II=] '

we have

( 3 )

OEk(xl,X2) z~k OE~(=I,==)~,I . , ~ - ~ , . + 2 (k i)==~) -I12 �9 . . = x z 1 ~ k z a + +

ul Ox, + v2 0z2 J l .= ,

= --kek[ sin 0[ 2k/(k+l) (~4k+2[ sin 0[ (4k+2)/(k+l) + e 2k+2 cos 2 0) -1/2

Therefore,

t {=" Vl" OEk(xl ' x 2 ) + tP2" X ~ k 0 x l " OE'(xl'x2)} v ( z 1 ' z 2 ) d s O x 2

--~02" v(e,O)( k+k 1)r [ . n,2k/tk+l~/r sin0l-2k/(k+x) cos 2 0 + r sin 2 x12 -- sln~, . . . . k e--~,+--~ i sin 0] (4 k + 2 , / ( t c + ~ l T ~ 62t~+------~ cos----~ 0 ) dO

/0" = v(e,o) k + l I sin01~/(k+l) d0"

Letting e --* 0, we obtain v(e, 0) = V(e, 0) + v(0, 0), where V(e, 0) --* 0 uniformly in 0 for e --~ 0. This shows that

k + 1 v(e, 0)[ sin 01 k/(k+l) dO --* -----~v(0,k+ 0) I sin 01 k/(k+l) dO

2k F((k+l)/(2k)) v(0,0) for ~--*0. (4) - k +---i ~r((~k + 1)/(~.~))

Ov(xl,Ox, z2) + Ek(x, , x2)" v2 " x~ ~ " Ov(xl,Ox2 x2) } ds.

Let us now estimate the expression

- - t = r , "~2) " Vl "

< C

ov(~, ~) ~,~ o,,(x,, x~) } d~ ] 0;~1 -{- Ek(,~l, x2)" 1/2" " 0z2

= , = , (=4.+= + (k + 1)2z~) '/2

I jr0 2" + I ~k]sinOl2k/(k+1)(~IsinOll/(k+l) + ~k+1] c~ sinO]--k/(k+1) dO

I o~o 2~r + 1 IsinOlkl(k+t)(ElsinOI 'l(k+l) +~k+llcos0[) dO --. 0 for e --* 0. (5)

We have

k

k

89

In view of (3)-(5), we find that

(GkEk(Zl x 2 ) , v ( x , , x 2 ) ) = (Ek(x, x2) Gkv(x,,x2)) = lira [ Et~(x~,x2)GkV(Xl,X2) ' ' ' ~ - .o j ~ > ~

__ 2k , /~ r((~ + 1) / (~ ) ) v(0,0). + i ~ "' r((2~ + ~)/(~))

R e m a r k 1. Since G~ is invariant with respect to the transformation (Zl, Z2) --~ (Zl , Z 2 "4- C), we also hD, v e

2k , /~ r((k +_~)/(2k)) ~o -~) . GkEk(Xl ,X2§ k- f - lV"r( (2kT1) / (2k)) ' '

R e m a r k 2. If Xl # 0, the kernel behaves as the logarithm; however, it experiences a jump in the vicinity of the segment xi = 0.

Let us introduce the Kelvin transform

yl Y2 ~' = {y~+' + (k + l),y~} ~/(~+~ ' x~ = {y~+2 +(k + l l oyd}

It is readily seen that

Yl X2

and { ~ + 2 +(k + 1 )~ ]} =

The Kelvin transform takes the curve {y~k+2 + ( k + 1)2(y2 - 1) 2 } _< (k + 1) 2 to the curve x2 >_ 1/(2k + 2). Indeed,

1 z2 > 2k + 2

y2 > 1 ~" ~ {yp+2 + ( k + 1)~y~} - 2k +-----~

This indicates that {y~k+2 + (k + 1)2(y2 - 1) 2 } < (k + 1) 2 . Note that

C~2 2

O~T2 03~1 § t t " ~ "4- OX 10Z 1

{ Ou Ov 2, 0u Or} = vGku + uGkv + 2 cgxl CgXl + x~ cgx2 cgx2 "

2k f 02u 0% Ou Or}

Let v(y) = pk(x(y))u(x(y)), and let G~ y) and G~ *) denote Grushin's operators with respect to the variables y and x. A detailed calculation (see [6]) shows that

v~,)~(y) = pk+4(x)v~%(~).

90

w T h e e igenvalue p r o b l e m

Consider the problem G~,, + .X,., + f (z) = O, (6)

1,2 where A e C, f e L~(~) , and u E S~,~(~). Let us state that the classical Hilbert-Schmidt, Jentsch, and Fredholm theorems for G~.

T h e o r e m 7. 1) The set ofeigenvMues {hi} is not empty, is located on the reM ax/s R + = {x E R [ z > O}, is at most countable and has no finite limit points. Each characteristic number has finite multiplicity, the system of eigenfunctions {r can be chosen to be orthonormal in L2(~), and it is orthogonal and

1,2 complete in L~(n) and so,~(n ). 2) r ,z~) does not change sign in ~ (reca/l tha~ f~ is connected); )q is positive and simple. 3) //" A # Ai, j = 1 , 2 , . . . , then Eq. (6) is uniquely solvable for any right-hand side f E LZ(f~). If

A = Ai, then, to solve (6), it is necessary and sufficient that

(f ,Oy+i)C'(n) = 0, i = 0 , . . . , r i -- 1,

where e j , e j + l , - - - , e j + r i -1 are the eigenfunctions associated with the eigenvalue $1, and rj is the multiplicity of A i .

Proo f . We omit the details of the proof. []

Now we are concerned with the smoothness of the eigenfunctions. From the hypoellipticity of Gk we obtain u E C~176 At all points ( z l , x2) E 0 ~ , za # 0, we have u E Coo(O(zl, x2) f3 ~) , since Gk is elliptical there. Here by O(z l , z2) we denote some neighborhood of ( z , , z~). Moreover, if (0, z2) e 0f~ and u(0, x2) ~o (0, ! ) , then a result due to gohn-Nirenberg [7] implies that u e Coo(O(xl, zz) f3 ~) (at least for k = 1). Let us show some results on the smoothness of the eigerrfunctions for (0, x2) and .(o, x~) ~ (o, t).

T h e o r e m 8. A) Suppose that 1,2 I) Gku + ~a(xl, ~2),~ = o in a, wh~'e a(~1, ~) e Coo(fi) and ,, e So,k(a),

2) if(o, ~2) e 0~, then there exists aneighborhood 0(0, ~2) such that 0(0, ~'2)nO~ = line(zl, ~'2), where - - c , < z , < c i .

Then u E Coo(O(O, "s t3 G). B) Suppose that

- - * , 2 i) Gku + Au = 0 in ~, where a(x,, z2) e Coo(n) and u e So,k(n); 2) if (o, ~,) e o~, then there e~sts ,,. neighborhood 0(0, ~,) su~ that 0(0, ~,)n O~ = {(k +

1)'(:, -~,) ' + :~+~ = ( k + 1)'} n o ~ . T h e n u e Coo(O(O, ~'2) ["'1 ~"~).

P r o o f . A) Since Gk is invariant with respect to the transformation (x , , x2) --~ (xl , x2 + c), we can assume that (0, 52) = (0,0) and Q C {(xl ,z2) t x2 > 0}. Consider the continuation of a(xx,x2), u(xt , x2) from ~ to fl = Q N fl~, where ~ = {(z, , x2) ] (z , , -x~) e Q} is defined as

{a(.,,*~), a(.~,.~)= a(~,,_.~),

~ ( x l , ~ ) = _~(~,,_~),

if (x , , x2) e 5 ,

if ( z l , x2 ) e ~x,

if (xl , x2) E s

if (xl , zz) e ~1"

Then ~(x , ,x2) e Coo(O(O, 0) \ (0, 0)) , a (z l ,x2) E C(O(O, 0)), and

Gk~ + A~(z,, x2)~ = 0 in O(0, 0) \ (0, 0).

91

The Schwartz theorem implies that

Gk~§ Xa(J~l,X2)~= Z Ca~a(O'O)" p,l<_~,,

Note that ~ 6 Ls(O(O, 0)). This implies

c,,~ + ;,'~(=,, =~)~ e H?d (o(o, 0))

On the other hand, 6 = ~ H~o~(O(O , 0)) if I~l >_ 1. Therefore,

This means that Gk(~--CEk) = -A~'~. We also have A~(x~, xs)~" E L2(O(O, 0)); therefore, in view of HSrmander's theorem [8], we obtain ~-CEk E L~or 0)). Therefore, Ek E L~or (O(0, 0)) , which leads

_ r(k+5)/k(m Consider the case k = 1 In view of Theorem 2, to a contradiction if k > 2, because E~ ~ *%c w~ �9 we have ~ ~ L~or 0)) . Fxom the Rothschild-Stein theorem [9] we obtain ~ - CEk ~ L~or 0)). Therefore, E~ e L~o r (O(0, 0)) , which also leads to a contradiction, because Et~ r L~o cO(O, 0).

B) As above, since G~ is invariant with respect to the transformation (x~, zg) ~ (x~, =5 § c), we can put (0, ~5) = (0, 0) and

O(0, 1) n 0f~ = {(k + 1)5(=5 - 1) 5 + =~+5 = (/~ + 1)5} n 0f~.

Let v(y) = pk(x)u(x), where y~, y2 are the same as as in the Kelvin transform. Then we obtain the following equation for v(y):

~+'(~)c~')~(~) + ~p~(~)~(~) = 0.

This implies that Gkv(y) + Ap-4v(y) = 0 in some neighborhood

0(0,1/(2k + 2)) n flux, u2) I u5 _> 1/(2k § 2)}.

From part A) we find that

(( 1)( . (y ) e c ~~ 0 o, 2k ~- 2 n (yl , ys) I ~5 > 2k +----~ "

Th~efore, ~ (= , ,=5 ) = "(y)p-k(= , ,=5) e C~176 1 ) n d ) . []

E x a m p l e 3. Here let us give some explicit calculations for the eigenvalues in the square. Let f~ - ( -7r/2, ~r/2) x ( -7r /2 , r / 2 ) . We try to determine the eigenfunctions in the form u(xl , x2) = f(xl)" g(x2) with f ( - r r / 2 ) --- f(vr/2) -- 0 and g( - r r /2 ) -- g(zr/2) = 0. We have

f ' (xl)g(x2) +x~kf(xl)gtt(x2)+ Af(xl)O(x2)=O

o r

Therefore,

g(x2){f"(xl) + Af(x~)} = -x~kf(x~)g'(~2).

f'(xl) + ~f(xl) g'(z2) x~kf(x,) g(x2)

Thus we come to the following eigenvalue problems for f ( z l ) , g(x2):

(7)

(s)

92

It follows from (7) that C = n 2 and g ( z 2 ) = C s i n ( z 2 - r/2)n.

For each C = n 2 , problem (8) involves eigenvalues A{ subject to limj_eo A{ = oo, and each eigenvalue has finite multiplicity. It follows from the property of the eigenvalues for Grushin's operator that the eigenvalues of the problem

f " ( z , ) - n2x~f (x , ) + Af(xl) = 0

never coincide infinitely many times. It is readily seen that the system

sin(z2 -Tr12),~/~(zl) (n = 1, 2 , . . . co, j = 1,2, . . . c o )

is complete. Therefore, the eigenvalues for Gk in the square coincide only with A j .

The author wishes to express his thanks to the International Center for Theoretical Physics of Trieste for support and hospitality.

References

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(1981). 3. N. M. Tri, Sobolev Imbedding Theorems for Degenerate Metric, Preprint ICTP, World Sci. Publ., Teaneck, N.J. (1995). 4. N. Garofalo and E. Lanconelli, "Existence and nonexistence results for semilinear equations on the Heisenberg group,"

Indiana Univ. Math. J., 41, 71-98 (1992). 5. S.I. Pokhozhaev, "Eigenfunctions for the equation Au +Af(u) = 0 ," Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 165,

No. 1, 36-39 (1965). 6. N. M. Trl, Super//near Equations for Degenerate Elliptic Operators, Preprint ICTP, World Sci. Publ., Teaneck, N.J.

(1995), pp. 1-17. 7. J. J. Kohn and L. Nirenberg, "Non-coercive boundary value problems," Comm. Pure Appl. Math., 18, 443-492 (1965). 8. L. H~rmander, ~Hypoelliptic second-order differential equations," Acta Math., 119, 147-171 (1967). 9. L. P. Rothschild and E. M. Stein, "Hypoelliptic differential operators and nilpotent groups," Acta Math., 137, 247-320

(1976).

INSTITUTE OF MATHEMATICS, HANOI,

AND INTERNATIONAL CENTER FOR THEORETICAL PHYSICS, TRIESTE E-mail address: triminh~ictp.trieste.it

Translated by N. K. Kulman

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