Some Applications of Functional Analysis in Mathematical Physics (Translations of Mathematical Monographs)

,

(3.15)

Let tO and w be two unit vectors. The sum (cp + w)/2 represents the midpoint of their chord, the length of the chord being equal to IID - wil Clarkson's inequality enables one to assert that the midpoint of each chord of length a will have its norm strictly less than some number ,7(b) < 1 , i.e., will actually lie in the interior of the unit sphere. This property may be called the uniform convexity of the unit sphere. 3. Theorem on the general form of linear functionals. THEOREM. Every linear functional on L. may be represented in the form

If = 1 cpwodv, n

(3.16)

where

W0EL°' (+).=ll
{cik} ,

converges strongly.

Assume the contrary. Then there exists an to > 0 such that we may find pairs of numbers nk and Mk (nk -' oo , Mk 00, ask -. oc) such that II>Eo.

Applying Clarkson's inequality (3.7) to the functions

9,n}

and tone

and (3.15) if I < p < 2, we obtain 2

II°+II`m42

II

IID/(D-1)

cimk I

Vn4

2 +

cimk

+ II

cin.

2

(p ? 2), 9n'II°< IID/(P-

1) < 1

(1 < p < 2).

if p > 2,

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

22

It follows from this that (I Fmk

2 ink II <

(3.17)

where ry > 0 and does not depend on k. We consider Xk = (9mk + 9n )/II9mk + 9nk11 We have IjXkll = 1. From the distributivity of linear functionals, we have 1

1Xk = 2[lFmk +1/pnk]11 V

1

+V k 11

>

1-n

g, 1 conk -. g, from which it follows that for sufficiently large k, we will have However, 1 fomk

1

>(I-rl)g=g,

1-q which contradicts the fact that sup,,,,,_119 = g. Consequently, the sequence {4'm} converges strongly and by virtue of the completeness of Lo it has a limit element go E LP L. Obviously Ilpoll = 1. REMARK. From this argument follows the uniqueness (to within equivalence) of the function loo E Lo such that Ilcoll = I and !90 = g, since otherwise we could construct a divergent sequence for which lim 147k = g which is impossible. We shall show that xk

Ig = g f [Ipoi°-'sgncpo]gdv n

(3.18)

or, putting glcvolP-'sgngo = wo !gyp = f wocpdv.

n LEMMA 4. If for an arbitrary function yi E Lo

J[11P1 sgn q'o]4v dv = 0,

(3.19)

then !yi = 0. PROOF. We consider y(A) =1((co + Aw)/Ilfoo + A vII) , where w # Since II(wo+Aw)/II90+2wl1 II = 1 and since for A # 0, (go+Aw)/IIcpo+ AvIl # go, therefore y(A) < g, while the equality holds only for A = 0. As cgo.

a result y(A) has a maximum for A = 0. If y'(0) exists, then y'(0) = 0. It is not hard to convince oneself of the existence of this derivative. Differentiating formally, we have

d Ilivo+A

II = d- [L1o+Awrdv] 1-1

P

fn

[Igo

+zwl"dv]° pf

JVo+Awl°-'sgn(mo+A )ydv,

§1.3. LINEAR FUNCTIONALS ON Lo

23

and since the integral J n I0P0+).VJP-'sgn(90+Aw)w dv converges uniformly ([294], Lecture VII), the formal differentiation with respect to 7. is valid. Further,

dlb'o +Aw1I (d

i_0

lJn I`°ol'dv]P

fa

[Iipol°-'Sgn 4,01wdv,

and therefore,

Y'(0) = 1

10fn

w

119011'-P

- 11070,12

[Ilaol°-` sgn q,0JV dv,

from which by virtue of (3.19) it follows that 1w = 0, and the lemma is proved. We now establish (3.18). Let 007 E L. be an arbitrary function. We put sgn g0]0p dv.

a=

Then w = rp - acpo satisfies (3.19), since [I,poI"-1 sgn co]

(9, - )0070) dv = a - a fn I9;0I°-1 I9PoI dv = a - a = 0.

Consequently, IV =19, - al go = 0, i.e., 1So = a10p0 = S f [1c01'-' sgn 90)x° dv,

n

and (3.18) is established. Since 0070 E LP , it follows that 11901'-I sgn9POl' =

[10001'-I)'/(P-I) =11)01°,

and as a result, w0 = 81901'-' SP 0070 E Lo

(P + p = 1)

,

from which IIw011Le = g. In this way, we obtain from (3.18)

19= f w00Pdv, n

wOELL-,

and the theorem is proved. In addition, obviously 11111 = IIw011L,

REMARK. It follows from (3.16) that every linear functional on L. corresponds to a function w E Lo , and conversely, every w E L. generates a linear functional on L.. The spaces Lo and LP, are said to be mutually conjugate function spaces.

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

24

4. Convergence of functionals. The concept of convergence is easily extended to function spaces. We shall say that a sequence of functionals {lk } is weakly convergent if for each element rp from L° there exists the limit

klm lk9 =10g. The limit obviously has the distributivity property, since l0(a1p1 +a2S°2) = urn lk(a1S01 +a2g2) = lim (allkcol +a2lkco2) = a1 lim IkS°1 + a2 I'M 1k9'2 = alloy! + a210q'2'

koo

k-.oo

If the limit function l0 is bounded, then it will also be continuous, i.e., linear. We establish the theorem. THEOREM 1. The space LD , i.e., the space offunctionals for L°, is complete

in the sense of weak convergence. In other words, every sequence which is weakly convergent has its limit a linear functional (1 < p < oo). Obviously it suffices to show the boundedness of every weakly convergent sequence of functionals. There will follow from this the boundedness of the limit functional. We show the correctness of a proposition from which this will follow. THEOREM 2. If a sequence of linear functionals (3.20)

1 1 , 12, ... 11k I ...

is unbounded, i.e., if it takes on the unit sphere in L. arbitrary large values, then one can find an element coo of L0 on which this sequence diverges (I < p < oo). PROOF. The idea of the proof consists in choosing from the sequence of functionals (3.20) a subsequence

ml,m2,...,m3,...,

(3.21)

where

(s = 1, 2, ...) ,

mJ = 1

k,

and

kJ -oo for s

oo

r

m3fo = J w3rpdv,

WJ E Lp ,

n Corresponding to the ms, we introduce the system of wJ E L°: Iwsl° _I SP wJ '

WJ 11ww11°-'

Obviously,

WJEL°,

I1WJ11=1,

(m5WJ)=IIwJ11L1

(3.22)

§1.3. LINEAR FUNCTIONALS ON L,

25

We form a series with the functions 00

(3.23)

WO = F, Cksw, , s=1

where a, = Ilwsll-1/2

The series will converge in L. if we have convergence for the series - Ij2

As we show for a suitable choice of the {m,} , the series (3.23) turns out to be strongly convergent to the element wo , and the sequence Ilwsll

lkwo will be convergent.

The sequence m, will be constructed inductively. Suppose m, has been chosen: we shall show how m,+1 must be chosen.

We consider the sequence (11w3), (12w,), ... , (ljw,), .... If it is unbounded, then the theorem has been proved. If the sequence is bounded, we put A, = sup 11jwkl k=1,2,....s

We have

Al 5 Each functional 1k of the sequence (3.20) corresponds to a wk E L. such that the norm of 1k is equal to 11 Wk IIL , Since the sequence (3.20) is un0

bounded and a fortiori these { Il wk ll } will be unbounded, therefore as m,+1 we can always choose an Ik which satisfies the inequalities:

IlwsllLo >1 (s=1,2,...),

(3.24) (3.25)

I1 ws+111

Iiws+1IIL,, >

32(s+1wjIIL,

U 5 s).

,

(3.26)

By virtue of (3.24) and (3.26), we have Il ws 11 Lo > 3'(-'-'), from which it

follows that the series E a, = Ell ws 11-1j2 converges and thereby the series (3.23) converges in L. L. Therefore s-1

00

j=I

j-s+ I

Im,wp1= F, ajm,wj+a,m,w,+ E ajm,wj s-1

> asmsws -

00

ajm,wj

ajmswj j=1

(3.27)

j=s+1

We have

a,m,w,

=11wsil-'j21Iw3II

(3.28)

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

26

By virtue of (3.24) and (3.25), we have s-1

$-1

s-1

F, ai mswi <>Im5W;I <EA;
(3.29)

J-1

i=1

From (3.26), changing notation, we easily find

a; =

llw;ll-1/2

i > S.

sllwsll"2,

< 3;'

from which 00

E aimswi E

<E

3

11w,ll-1/211ws11

j-s+I

i:s+l

00

=

IlwJll1/2

E i-1

1/2

IlwJll 2

=

1

3

(3.30)

Then from (3.27), (3.28), (3.29), (3.30), we obtain s

Im5WOI >

llw,lll/2 > 34

I

,

whence clearly it follows that lkw0 cannot converge to any limit with increasing k. Theorem 2 is proved. Hence, if a sequence {lk} converges weakly, it cannot be unbounded and therefore for all p E Lc we have IIacI = Ik1m 10 = k] n

11k91 :5 AIIwII,

(3.31)

where A is an upper bound on the norms of all the 1k . Thus the functional to is bounded and therefore continuous. Theorem I on the completeness of L.* is thereby proved. WEAK CONVERGENCE IN Lo. A sequence of functions {9'k} is said to be

weakly convergent to the function q0 if for an arbitrary function l E LP we have

km 'Pk =190.

(3.32)

In the space L. the set of functionals coincides with the space Lo.. Therefore the formula (3.32) will hold whenever the functionals 1k E Lo corresponding to the 9k converge to the functional to a LP corresponding to 90 . We have finally in the space L. two forms of convergence: strong and weak, the latter being written

ck~c0' 90 there follows the weak Obviously from the strong convergence 9k convergence ck --+ co . The converse is not always true. We give an example of a weakly convergent sequence which does not converge strongly.

§1.4. COMPACTNESS IN Lo

27

EXAMPLE. Let Q = 10, 2n), p = 2, rpk (x) = sin kx. Then sin kx -+ 0 since for an arbitrary function yr E L2 , 2n

yi(x)sinkxdx=nbk

0,

in view of the fact that 2x

2

2

n1: bkl yr dx, 00

k=1

0

i.e., the series >k I bk is convergent. But sin kx

0, since

r2a Jf

sin2 kx dx = n -» 0.

0

REMARK. Theorem 2 may be formulated in terms of the weak convergence

of functions as: a norm-unbounded sequence of functions cannot be weakly convergent.

§1.4. Compactness in LP 1. DEFINITION OF COMPACTNESS. A set M is called compact if from each

infinite subset one can choose a convergent sequence. EXAMPLES. 1. Every bounded set of points of the plane is compact (the Bolzano-Weierstrass principle). 2. Compactness in the space of continuous functions is established by Arzela's theorem: if a family of functions {tp} is uniformly bounded and

equicontinuous, then it is compact (i.e., if Ici < A and if for e > 0 one can find d(e) > 0 such that for all tc in the family I w(v + Q) - tp(p')I < e whenever IQI < 6(a), then the family {lo} is compact). COMPACTNESS IN LD . Corresponding to the two forms of convergence in LP , we distinguish between strong and weak compactness. 1. A set of functions {c} c Lo is called weakly compact if any infinite subset of it contains a weakly convergent sequence. 2. A set of functions {f} c Lo is called strongly compact if any infinite subset of it contains a strongly convergent sequence. 2. A THEOREM ON WEAK COMPACTNESS. In order that a set X c L should

be weakly compact it is necessary and sufficient that it should be bounded. PROOF OF NECESSITY. The necessity of the condition follows simply from the theorem on the weak completeness of Lp L.

If the set X is unbounded, then from it we may choose a sequence {ck} such that lick II ~ oo, and from which on the basis of the remark in §3, item 4, we cannot choose a weakly convergent sequence, and consequently, the set X cannot be weakly compact.

I. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

28

PROOF OF SUFFICIENCY. Suppose that for all go E X C Lo ,

(4.1)

II(vII < A.

We consider the conjugate space LP . It is separable Q2, item 4). Let us denote a countable net which is everywhere dense in L. by (4.2)

WI, W2, ...,Wk,....

Let {V } be an arbitrary infinite sequence c X . We shall show that from it we can choose a weakly convergent subsequence. Take WI and consider the sequence of numbers (cok , WI) = fn 9k VI dv = ak') . This sequence is 5 All WI II , and from it we may choose a convergent subsebounded: quence, corresponding to a subsequence a(kI )1

I

u) Oak WI dv -» a(')

I (u1 2 (I), ... 19k(I) , ... i

for k

oo.

W2) is bounded and from it we may choose a convergent subsequence, corresponding to the suboo. sequence 9(2) , 9(22) , ... , 9k) , ... , fn 0,(') W2 dv -» a(2) for k Consider W2 . The sequence of numbers a(2k) _

Continuing this process, we obtain a series of convergent sequences of numbers ak) and corresponding sequences of functionals, all obtained from the sequence {fok} and such that each of them is a subsequence of the preceding

{Sok))C{9k-')}C{9k} and ak)=ln9k)Wjdv-a(j)

for k-oo and s= 1,2,3,.... By the diagonal process we may select the sequence { 92k ) } , which is weakly

convergent on the whole countable dense net IV/,}, i.e., (k)

(s)

IkWs= a9k WJdv

fork - oo

IWj

and s=1,2,3,...;

Ilwsl
The convergence of {Ik} on the set {Wj} implies its convergence everywhere.

For in fact, every element W E L., may be represented in the form W=

where II W'II < c, and Wj is an element from our net (4.2). Then by virtue of the boundedness of all the Ik , we have IlkW-IkWkl < Ac, II,,,W

for arbitrary k and m.

IWjl < Ac,

§ 1.4. COMPACTNESS IN LP

29

Furthermore, IImi - IkWI 5 1lkWs - ImWsl + 2AE,

and choosing sufficiently large m and k, we obtain I/m W - lk WI < 3Ae,

from which follows the convergence of the functionals 'k on V. As shown earlier, the limit 1W will be a linear functional on L. and may therefore be written in the form IV = f cpo yi d v . Thus, for every W E LD. , we have f (k)W dv -. f n q ' dv , and consequently, for every functional in Lo it follows that (lVk )) -. (I90) . Therefore the set X is weakly compact, as was to be proved. 3. A THEOREM ON STRONG COMPACTNESS. In order that a set X C L. (I < p < oo) should be strongly compact, it is necessary and sufficient that (1)

Ilwll
(2) The set X should be equicontinuous in the large, i.e., for an arbitrary e > 0, one can find 8(e) > 0 such that for all q1 E X,

J,(Q) = f

w(P)I'dv < e

(4.4)

(let p = 0 outside fl). LEMMA. The set X c L. (1 < p < oo) will be strongly compact if and

i f 1 0 1<6 ( e )

only if for every e > 0 one can find a finite a-net in X, i.e., a finite number of functions rp1 , 92 , ... , rpN(E) such that for arbitrary q' e X, one can find among the functions of the net a function los for which IIv - (pSII < e PROOF OF NECESSITY. Suppose that there existed an to > 0 such that one

could not construct a finite a-net for it. This means that for any element V, E X, one can find qq2 E X such that II91 - P211 > eo . The elements c1 and 92 do not form an eo net; consequently, one can find q,3 E X such that

IIVI -93II>eo and

1192 _9311 > CO'

Continuing this process indefinitely, we construct an infinite sequence { Itik } C

X such that III,, - 07k II > to for i # k. Obviously from this sequence one cannot choose a strongly convergent subsequence, which contradicts the assumption of the strong compactness of X . The necessity of the condition is proved. PROOF OF SUFFICIENCY. Suppose that X is such that in it one can construct finite a-nets for arbitrary e > 0 (and for which, moreover, this is possible for each of its subsets). Let Y1 C X be an arbitrary infinite subset. We construct in it a finite (1/2)-net, 44Z1), ... , p"). For arbitrary E Y1 we can find pk') such that 119' II < . Let Y(s) be the set of

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

30

functions E YI and at distance from psl) less than 1/22. At least one of the Y,W is infinite since Y1 is an infinite set. We denote it by Y2 and construct on it a (1/22)-net. Repeating this process, we obtain on the kth step c Y2 c Y, and on it a (1/2k)-net lO1,kl, 92 , .. , 1PNk' . Let Yk c Yk_I c

p', p"CYk. Then 1/2

lip'-p"II<_lip'-pk11+Ilp"-pkll_(1/2k)+(1/2k)=

k-I , i.e., any two functions from Yk differ from one another by not more

than 1/2k-1. The sequence of functions {pk} , where pk E Yk , will be strongly convergent since 111ok. m - pkll < 1/2k-I < e for sufficiently large k (pk+m E

Yk+m c Yk for arbitrary m). Thus from YI c X we have extracted a strongly convergent sequence, and consequently X is compact. 4. PROOF OF THE THEOREM ON STRONG COMPACTNESS.

1. SUFFICIENCY. We consider a family {p} = X satisfying the conditions (4.3) and (4.4). We construct a family of averaged functions for each p E X by means of the family of kernels w(Q' ; h) (§2, item 4):

ph(P) = h°X f

p(PI)w(P -

PI , h) dv1.

(4.5)

By the estimates of §2, item 4: 11pk - p11 < KJ'(h), where K does not depend upon the particular function of the family or upon the parameter h of the kernel, and JP(h) = sup,o, 0 one can find o(e) > 0 such that for h < 3(c), (4.6) Ilpti - pll < 2 for all p E X. Then an (e/2)-net for {pk} will be an a-net for {p} . If we now establish the compactness of the family of averaged functions,

then the lemma will imply the existence for the averaged functions of a finite

e/2-net. This means that in the set X, we may construct finite a-nets for arbitrary c. By the lemma once more, X will be compact. In order to show the compactness of {pk} , we show that {py) for given h are uniformly bounded and equicontinuous. Then by Arzela's theorem, this set of functions

will be compact in the sense of uniform convergence and a fortiori in the sense of convergence in LD . But for sufficiently small IQI , by virtue of the continuity of the kernel,

lw(P1-P'-Q; h)-w(P'1-P;h)I<>7, from which

-P'-Q; h)-w(P1 -P; h))dv1l

Ipti(P+Q)-ph(P)I

1

Xh I fa

lp(PI)I° dv,

=Xhq Ilpll<_A

C-hq

J

/D

§1.4. COMPACTNESS IN Lo

31

and for sufficiently small IQI and fixed h, I Ph(P + Q) - 9vh(P)I < e ,

i.e., {rph} is equicontinuous. The uniform boundedness follows from the estimate I vh(P)I < {

f

Xh lfn

Iw(P'1

I /P

- P; h)I° dvl]l

[L1P,)rdv,] } < h .

Thus, the family {loh} is compact, and by the lemma, we may construct on it a finite (e/2)-net. This will be a finite a-net for X and, as a result, by the lemma, X is strongly compact as was to be proved. 2. NECESSITY. Let X be strongly compact. Then it is weakly compact and by the theorem on weak compactness for all (p E X we have 11911 < A, i.e., the necessity of condition (4.3) is established. By the lemma there exists a finite (e/3)-net {tpk} (k = 1, 2, ... , N). Each one of the functions c'k is continuous in the large, and since there are finitely many of them, for a given e we may find 6(e/3) > 0 such that 1/p

[fa

19k('# + Q) - 9k (P)I° dv]

<

e

(k = 1, 2, ... , N)

if IQI<6. Take an arbitrary function V E X and choose the nearest 9k to it from the (e/3)-net. Using the Minkowski inequality, we obtain 1 /v

Va

I V (P + Q) - Sv(P)I° dv]

- lJn19Op

+Q)-Vk(P+Q)+Vk(P+ _

- 9'k(16) + fvk(F) - 9(P)I° dv

1/P

1 1 /P

(r <_ [j

I /P

+ [f, IVk(P+Q)- rk(P)I°dv]

(f +

I/P

IDk(P) - 9(P)IP dv]

<3+3+3=e for1Ql<6, i.e., for all rp E X for the given e, we may give b such that J',(Q') < e whenever IQI < 6, and consequently, the necessity of condition (4.4) has been shown.

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

32

§1.5. Generalized derivatives

1. Basic definitions. Suppose that q' is given on the whole space and summable on every bounded domain G with 'G C 11. We consider a function

w continuous together with all its derivatives to order l and equal to zero outside some bounded domain 1,, with I7r, c a. If rp has continuous derivatives, then r r r 1+1 rp J

n

axe' ... 194-

+

(-) 1

waxi'

dv = 0.

ax."^

(5.1)

Suppose now that we know nothing about the existence of derivatives of which is summable on any and suppose that there exists a function coo Q bounded domain G with ?; c 12 and satisfies the equation

dv = 0

(5.2)

for all functions w from the class considered. Then we set a1w

(5.3)

0X 0X2= "' axe a generalized derivative of the function P. and call mQ Q REMARK. The equation (5.3), by the basic lemma of the calculus of variations, is true in the ordinary sense almost everywhere if w admits the corresponding derivatives and the integration by parts is justified. of the function There can exist at most one generalized derivative wQ rp . If there were two: and o)(2)..Q , then on the basis of (5.2) we would have (1) (2) dv = 0,

from which by the arbitrariness of w we would have

woe.

almost everywhere. Thus generalized derivatives are uniquely defined (up to a set of measure zero). The new definition of derivatives does not coincide with the definition of derivatives almost everywhere, as is shown by examples. EXAMPLE 1. Let 9(x) be continuous on [0, 11 but not absolutely con-

tinuous. If ;P(x) has a generalized derivative w(x), then for every w(x), continuous together with its first derivative and vanishing outside (e, I - s) (0 < e < 1), we have the equation r1

9dx dx = - fo m(x)w(x)dx. Let i1(x) = f co(4) d4 . Then

-

fI J0

dd

dx, w(x)w(x) dx = I i2(x) 'P x 0

§1.5. GENERALIZED DERIVATIVES

33

and consequently, we have

f[c-f(x))dx=0, d V/

from which, by virtue of the arbitrariness of dW/dx satsifying only the condition fo d yi = 0, will follow to - f2(x) = C, i.e., go = Q(x) + C, and consequently rp(x) is absolutely continuous, which contradicts the assumption. Thus, g(x) does not have a generalized derivative. For this reason, if rp(x) is continuous and has a derivative d 1D/dx almost everywhere but is not absolutely continuous, then it does not have a generalized derivative. Thus, from the existence of the derivative almost everywhere, does not follow the existence of the generalized derivative. EXAMPLE 2. We consider the function of two variables q'(x, y) = f1 (x) + f2(y) , where neither f1 (x) nor f2(y) is differentiable; then F(x, y) does not have derivatives in the ordinary sense, but the generalized derivative 49 2V/ax ay exists and is equal to zero. Indeed

f2(y)ax

Oy

ay

But (ayi/ax)(ay./ay) = 0 on the boundary of 0, and therefore

J(x)axa y dv = f I,(x)f axay y-d f'(x) =f `ax IY.C dx = 0, dydx

b

a

and analogously 2

ff2(Y)dv = 0. Consequently,

J4JJ_dv = 0, as was to be proved. 2. Derivatives of averaged functions. Existence of generalized derivatives.

THEOREM 1. Suppose that the function 9 has a generalized derivative Ox -" a p on the domain fl. Then the derivative of the averaged function for 9 is equal to the averaged function for its generalized derivative: 8a

ax;,ax2 .. 8xx - axI

L9 a

x2=

9

ax

(5.4) b

on the set f1h of points P E i l at a distance greater than h from the boundary of Cl.

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

34

PROOF. Since 1

vh(P) = Xhn o rp(P'1)w(P'1

- P; h) dv1,

where P _ (x1 , x2 , ... , xn) and PI = (yi, y,, ... , yn) , it follows that

a°9h(P) ax°' ax°2 2 ...

Xhn

&X,,'

I

P; h) A(PI) axI ' 8x2 ... aX n

dV 1

a°w(PI

-P; h) dv I . I)ayllay2=...ayn. By the definition of the generalized derivative, the equality (5.2) holds for (-1)° Xhn

P

every a times continuously differentiable function Vi vanishing outside some

bounded domain v,, with V. C Q. If P E fl,, then the function w(PI - P3 ; h) vanishes outside the ball (P1- P'l < h contained in fl. Taking yr = w(P1 -,P; h), we obtain a°rph(P)

a°rp(P )

f

1

ax,'ax22...ax'" - Xhn ow(P1 -P; h)ay,'ay2,...8ya dv1 ax°' ax°' ... ax°" 2 n 1

h

as was to be proved. COROLLARY. If V is a summable function and ex ... E Lo on Q, then aa 80 Ph 41 in LP as h .- 0 axial ax2'.y

ax:.

axe' ax" .. axe"

in Lo on any domain G C Q at a positive distance from the boundary of fl. PROOF. Since G c Q. for sufficiently small h, this follows from properties of the averaged functions. THEOREM 2. If a given function 9 summable on any bounded domain G with G C i2 can be approximated by a sequence of continuously differentiable functions 9pk (k = 1, 2, ...) in the sense that for every function 'v continu-

ous and such that y/ = 0 outside a bounded domain v. with U. C fl,

kim foVk(P)W(P)dv and if in addition,

fno0(P)y/(P)dv,

a ° k(P)

f0l8X" 8X2= ... aXn"

dv < A ( i2), I

v

with A(12) independent of k , then the generalized derivative

ax,ax2=.. axe" exists in C.

(5.5)

(5 . 6)

§1.5. GENERALIZED DERIVATIVES

35

PROOF. From the boundedness of the integrals (5.6) follows the weak compactness in L. of the sequence {a°fk/(ax;'8xe= Ox,*,-)}, and consequently there exists a weakly convergent subsequence a°4)k °,°2...°^

ax°'ax°2 ... ax°^

(where w° °

n

2

1

E La) . We have the equation

1, f

a...axn^ °w

°a

+(-1)°+1w

,ax;'

,

°

} dv=0

ax1, ..axn^ J

k

for all w continuous with all derivatives up to order a and vanishing outside Vv, CO.

oc, we obtain

Passing to the limit for k,

L{ox'ox

+(-1)°+1ww° ...°^}

dv =0,

n

l

which establishes the theorem. REMARK. By virtue of (3.31), fa Iw°. °^ 10 dv < A(fl), since and is the weak limit of a°cok/(axI' axn^) .

E Lo

COROLLARY. If the set of derivatives of ath order of the averaged functions is weakly compact, then the given function has generalized derivatives of ath order.

3. Rules of differentiation. From the definition of generalized derivatives follow the assertions

1. If 101 and q'2 have generalized derivatives of a given order ° and

ax; 1

° aaxa^ 9

ax;

and if cl and c2 are constants, then c11o1 + c2P2 has a generalized derivative of the same order, and a°r1 -c ax°^ - 1 ax°'ax°= . axn n 2

a°(c1g1 +c2P2)

ax°'ax2 1

a°P2

+C2

1

ax°'ax°=

1

2

axn

The proof is obvious. 2. If V has the generalized derivative ox','B ' and w(x1 , x2, ... , x,,) has the generalized derivative a1w

-A(x1,...,xn),

w(x1, x2 , ... , xn) ,

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

36

then ).(x, , x2, ...

is the generalized derivative of P of order a +'0:

,

_

aQ+f9

.

ax°'+p' OX02+02 ... axG.+p. 2 n 1

PROOF. Let yi have continuous derivatives up to order a+# on the whole

space and be equal to zero outside some domain V,, c Q with V,, c fl. Then ap w/(ax,#' . . has continuous derivatives up to order wp a and is equal to zero outside of V1, . From the definition of generalized derivatives there follows: r /O.lwdv

-

(-I)p

p8xp

r

fnwaxpa I

o+pj

dv

= (-I)p

;

I

a wp,...pe

A

w

dv

x .+p.

1A

f

,

which in view of the arbitrariness of w establishes the assertion.

3. If fp, and afo,lax, E L. 92 and aq2/ax, E LP, , then the product of 92 has the generalized derivative

+ aVI ax, 2 axi PROOF. Indeed, P1 E L. and 92 E L. , and therefore P, ip2 is summable. Analogously, q),(acD2/8x,), 920011ax0 , and 9',(8q'2/8x,)+92((991/ax,) are summable. Let rp,h and 92h be the averaged functions for P, and 92 . By the properties of the averaged functions, we have on any G with ?i C f2 that '9(9192) OX,

9 1 1 h9 1 1

,

_ 9'

ax; a in Lp,

92h *

92,

8z'h

a in L,,,.

Suppose that w is continuous with its derivatives of first order on the whole space and is equal to zero outside V., C Q. Suppose I wI , I0 w/ax, I <

A. Then f (PIhP2h

-

91924-VI dvI < A I I9Ih92h

- 07,921A

AZv 191h(92h-92)+92(91h-9I)Idv


§1.5. GENERALIZED DERIVATIVES

I.

n

37

2-Of

9,02h ax1 -j dv -+ fn 192 ax, dv

for At -. 0.

Analogously, one shows

fa(91hi9xi 9'2h+V2haa )WdvfnY/( pl1+9zx991 'l

dv.

I

Then from the obvious equation

a a I 9'1h42h ax dv - - fn w (Plh ax) + z>, Oxf dv for h

0 follows

In 9)1920x A - fn w

ax1) dv' (2+92L

and this means that aw

ax, +

92

a1 _ ax, -

(9192)

ax,

REMARK. If Ip, and 92 have continuous derivatives up to order C1, then ')°(9)192) ax*l'axz2 ... axO°

_E

aP(P 1

a°-B92

...Oxn. axe,-P,

-o E s,=a

where C are the binomial coefficients. This same formula is correct if to, and 92 lie in L. and Lo , respectively, and have all the generalized derivatives occurring on the right-hand side, with

the derivatives of 9, lying in L. and the derivatives of 92 lying in Lp. . The proof is analogous to the one just given. 4. Independence of the domain. Let ip and A be two functions on Q summable over any bounded domain G with c 0. If A is the generalized

derivative of 9, aX 1 ... axn 1

on Q, then A is the generalized derivative of 9 on an arbitrary subset of 0, as easily follows from the definition of generalized derivatives. Thus, although the definition of the generalized derivative is "in the large," its character in an arbitrary neighborhood of any point is defined by the local properties of the function lO . We consider the possibility of extending the domain of the initial definition of the generalized derivative. Let us first prove a lemma.

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

38

LEMMA. Suppose that the functions to and A are given on the domain i2 and are summable on any bounded domain G with G c i2, and that A is a generalized derivative off in each ball having radius less than some a > 0 and lying in 0,

A=

aX

aXn"

Then A is a generalized derivative off in n. PROOF. Suppose that 0 < h < S1 < J. According to Theorem 1 in item on 06 . We now consider any function V continuous with all derivatives up to order a and identically equal to zero outside some bounded domain > with V. c Q. Then Vy, c Q for some 0 < a, < a ; 2, Ah

consequently,

fwAhdv=(_-l)°J,9hex''.aX:, dv. Since A. = A and iph limit that

f

Jv

V in L,, on V,, as h -. 0, we get by passing to the

yrAdv -

r

(-1)°

Jv

9°ax°'

8.

ax

dv,

f8o100dv, (-1)°8X01100

InVAdv -

and the lemma is proved. Let Q(') and fl(2) be two arbitrary domains. Obviously 0(l) + f2s2) c (0(I) + 0(2) )6.

If for some 8 > 0, we can find S' (0 < 8' < a) such that (0W + f(2)), c SZa') + i2a ) ,

then the pair of domains 0(') and 0(2) will be called summable. THEOREM. Let d" and 0(2) be a summable pair of domains having the

intersection Q(1)Q(2) . Let qP I be defined on d"" , 912 on f2(2) , and io, = 92 on i2(1) n(2) (with the exception of a set of measure zero).

Suppose that p, has on O(1) the generalized derivative 8°9,l8xi'. 8x.*° and 92 on 0(2) has the generalized derivative 8°(o2/8x, ' . . axn*°. Then the function l0 defined on n(1" + 0(2) by

(tp, in d'>, 102

in Qc21 ,

§ 1.6. PROPERTIES OF INTEGRALS OF POTENTIAL TYPE

39

has on a(') + a(2) the generalized derivative equal to a°91

ax ... ax

a°

in f1 p I

in 0. 2

PROOF. By virtue of the fact that 91 = toe on 0(1)0(2), we have a°co / ax° = a°92/ax;' 8x°^ on !n(1)0(2), which follows from the local character of the generalized derivative. Thereby the right-hand side of the formula (5.7) is consistent on !Q(I)!n(2). Obviously it suffices to show that A is the generalized derivative of p in (12111 + S2(2))a , where b > 0 is arbitrarily small. Since the pair of domains 0(') and L2(2) is summable, (i2(1) + i2(2))a C 526 + fIT for some 6' > 0, and consequently each point f E (f2(1) + i2(2))a lies in at least one of the domains f1 , i1a2 . Suppose P E Then the ball Ca, (P) c f"1, where 2 coincides with 8°ipl/8x',' ax°^ and ip coincides with 9'I . Therefore A is the generalized derivative of rp on Ca. (P') for any P E (0(') + 52(2)), . By the lemma, A is the generalized derivative of to on (52") + n(')), . By virtue of the arbitrariness of 6 , A is the generalized derivative of ip on 52('' + 12'2} (4)

ax;'

§ 1.6. Properties of integrals of potential type

1. Integrals of potential type. Continuity. Suppose f E LP (p > 1) on the unbounded space of n variables, while f = 0 outside some bounded domain Q. We construct the function U(Q)

=1
(6.1)

where A is a number, A < n , and 0

r = I f - QI = Dxi - y1)2 , i=1

where r < R is a sphere including the domain Q. THEOREM. If 2 < n/p' ((11p) + (11p) = 1), then U(Q) is continuous and satisfies in any bounded domain w the inequality I U(Q)I < KIIIIIL' ,

where K is a constant depending in general on w but not on f.

(6.2)

I. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

40

PROOF. We have Ap' < n. Therefore the integral f,
'f

U1°dvp)

Jr-Af(f)dvp1 'if

f 5 IUIIL,

l Lh

r-"' dvA)

l I/°' 'P' dvp } <e r-

(6.3)

h
if h is sufficiently small, independently of the position of the point Since

r-zf(P')dvp+ f r f(P)dv#

U(Q)= f

and the first term on the right side is a continuous function of Q, U(Q) is continuous as a uniform limit of continuous functions.' Applying the Holder inequality to (6.1), just as in (6.3), we find IU(0)15 IIIIIL,

IIP

If l

r-A°'dv1

l./

=KllflILL

i.e., the estimate (6.2). The theorem is proved. (5)

2. Membership in LQ . THEOREM. Consider the hyperplane of s variables y,,, = ys+2 y,, = 0 and suppose that Qls1(y1, y2 , ... , yy) is a point of that hyperplane. If

1 (n/p'),and (s/p)>A-(n/p'),i.e., s > n - (n - A)p, then U(Q($)) is summable (on an arbitrary finite domain Es of the hyperplane) to

the power q' , where q' < q, (s/q) = A- (n/p'), i.e., q = sp/(n - (n - A)p), and the inequality IlU(d('))IILa. < K,IlfllL,

(6.4)

holds, where K, is a constant independent of f.

(In those cases where we shall simultaneously meet the spaces Lq on Euclidean spaces of a different number of variables, we shall sometimes designate these spaces by subscripts in the form La. s .)

PROOF. From the definition of q, it follows that q > p. Suppose that q' is some number satisfying the inequality

p 0. From (6.1), we have

IU(Q)I 5 J (Ifl

r

v

*`)(III°s°_Q 1)(r-v

1 See 12941, Lecture VII, for more details on integrals depending on parameters.

§1.6. PROPERTIES OF INTEGRALS OF POTENTIAL TYPE

41

and applying the Holder inequality for three factors, with Al = l/q', A2 = (q* -p)/q*p = (11p) - (1/q'), and A3 = 1/p' (obviously Al +A2 +).3 = 1), we obtain

IU(Q)I < {fr
f

III°r

III°dvp}°"q


IfJr
ILR III

I U(Q)1 <- K111f11

°r-°+eqdvpq

(6.5)

Raising (6.5) to the q'th power, integrating over the domain Es in the hyperplane

3's+1=ys+2="'=3'n=0 and interchanging the order of integration, we find

f IU(Q`3')I° dv, ,

1IfI° I fE

dvp.

(6.6)

We shall show that the integral r-s+cq

dvQ,,

fE,

is bounded. Indeed in polar coordinates in the hyperplane of Q(s), we have

r=

p2 +h2,

(6.7)

dvd,,, = ps-1 d p d w(s) ,

where h is the distance of the point f to the hyperplane, d w(") is the element of solid angle in the plane around the foot of the perpendicular dropped from f to that hyperplane. If Xs is the surface area of the ball in s-dimensional space, and r < R for Q E Es , P E Q, then

f r-s+cq d v Q(.) <- Xs E,

f

R2-h2

(/T) h2

0

R peQ.-1 :5 X,

dp=K2.

s+eq ps-1 d p

42

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

Substituting in (6.6) and denoting KIK21q' by K, we find I

dv,,,ll;

11U(d15)11L'.

= { fE IU(Q''')Iq


which was to be proved. Suppose oj 1 < q'. Then U(&l)IgI dv&,

EI fE, I U(Q

q

(s1 4 q )I

j dvQ1,1

11

9

fE dvti,,,J ,

from which it follows that IIU(Q)IIL41 < CIIU(Q)IIL'. ,

and the inequality (6.4) is valid for arbitrary q' . Consequently, the condition that q' > p may be dropped and the theorem is completely proved ((6) , (7)) .

REMARK. The constants K and KI in the theorems in I and 2 depend exclusively upon the form of the domain and the numbers A, p, s, n, and q' but do not depend upon the function 1(P). If the s-dimensional manifold is not a plane, then the corresponding theorem may be reduced to the preceding by a change of variables. One must assume that there exists a coordinate transformation which introduces only a finite distortion of distance (i.e., such that on bounded parts of the space, one can find constants M > m > 0 such that m < p/r < M, where r is the distance in the old system and p in the new) and which carries the manifold under consideration into a plane. §1.7. The spaces L(l) and WP(1)

1. Definitions. 1. The linear manifold of all functions 9 summable on any closed bounded set contained in it having all generalized derivatives of order l summable to power p >_ I, will be called Wor)

a', ax°I aX°= 2

aX,1

E LP in Q;

Ea, = 1.

2. By Lp1j we will mean the set, the elements of which are the classes of elements in W(l) having all derivatives of order 1 the same, i.e., c1 and {v2 will be said to lie in the same class in Lot) if ar92

aX; ar918x ' - 0 X;, ...aX

Ma

_ 1)

§1.7. THE SPACES /1i AND Rd'

43

almost everywhere in (. The elements of LP(l) will be denoted by the letter y/. Functions to of the same class yr will be said to be mutually equivalent. The elements of L,' may be added and multiplied by real numbers. Thus Lp1j becomes a vector space. For the multiplication of the element W by a constant, it suffices to multiply by the constant all the functions 1° lying in the class t y. It is not difficult to see that in this way we will obtain elements of one and the same class. The class y/, + W2 is obtained if we add in pairs all the elements from the

classes W1 and v,. It is not hard to see that thereby there will be obtained elements of one and the same class. 2. The norm in L(1). By the norm of an element w in Lp1) we mean the number 2

n

IlwllLr = I /!

fn

=

1/P

p12

a

dv

ark

1/p

)21 P/2

dv

a (axl'...axo:.

(7.1)

where cp belongs to the class W . In some cases, where it causes no confusion, we shall write IIwIILo for

rp E Wprl , meaning by this the norm of the class W to which f belongs. We note some of the simplest properties of the norm: 1.

Ilasoll = lal IIVII if a is a constant.

II. II91 + 8211 11c1II + 119211 (the triangle inequality). 111. 11g'11 ? 0, and if 11911 = 0, then rp is equivalent to zero, i.e., it is

given by a polynomial of order at most 1- 1 . IV. The norm is invariant under every orthogonal transformation of the space of x1 , x2 , ... , x .

The first property is completely obvious. We prove the validity of the triangle inequality, for which we employ the first representation for the norm. Then using the fact that (by the Minkowski inequality for p = 2) z

19191

\

ax.... ax. + ax... ax. a192

_8j

(

p1

Ox....Oxi r, ,

r

2

'9192

+

(ax....Ox. ) r, it

2

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

2

f

ax;

1/P

dv

ax; /

ax;

- ax;

P

I2

r

fn

Applying the Minkowski inequality once more, we obtain 2 p12

af

I/P

f

Ilw, + 9211L')

(

Jn

If

r

dv

/

\8x; a91,P2

2 P/2

dv

ax; ... ax;

= III,

UP

IIro211L,I1

The third property of the norm follows from the fact that if all the generalized derivatives of lth order of some function to are equal to zero, then all the generalized derivatives of order I of the averaged function P. are equal to zero on t2h . This means that the averaged functions are polynomials of degree at most 1 - 1 . However, the limit of a sequence of polynomials of degree at most 1- I can only be a polynomial of degree at most 1- 1 , and it follows that q is a polynomial of degree at most 1- I and is equivalent to zero.

The invariance of the norm under orthogonal transformations of coordinates follows easily from the first representation of the norm, since the expression 2

ap ax.

ax. ax-

is one of the invariants of the tensor afro

ax. ax.... ax = v; 1,

1=

We note also two inequalities: II911t,+, 5 K1 max II

of axi' ...a X"'..

84 1 arro ... &x.^ 5 IIwIILD

L,,

(7.3)

§1.7. THE SPACES L,"' AND W,"'

45

In fact, the inequality (7.3) is obvious. The inequality (7.2) follows from the fact that a!,P

n

a'V

ax.... ax

)

2

ax.

ax.

<

Applying the Minkowski inequality to the first part of the last inequality, we obtain 11

i,....,i+=1

a, axi,

ax'a

Lp

K max +,,...,y

a axti ... ax+

Lo

where K1 is the number of terms in the last sum. Thus (7.2) is proved.

After the introduction of the norm, the manifold L' becomes a function space. Later we shall introduce a norm in WDl). Therefore we may thenceforward consider W(1) as a normed function space.

3. Decompositions of W(i) and its norming. We consider also the space S1 of all polynomials of degree at most 1- 1 . This space can be considered, obviously, as a subspace of the space W(1) . The space LP(1) appears, speaking

algebraically, as the factor-space of W, by S1. By a projection operator in the space W(1) is meant an operator whose square coincides with itself

n2/Q=nnc=nc. If some projection operator n1 carries the space W«) onto the whole space S,, then it will be the identity operator on S,. With the aid of each such projection operator it is easy to construct a decomposition of the space W,t1 . We put

n;,=,-n1,, i.e.,

n, =E-n1,

where E is the identity operator. The operator n, will be in turn a projection. In fact,

n', n*, = (E-II,)(E-171)

=E-2n1+n, =E-n1 =n,, which was to be proved. An arbitrary element io of W(1) may be represented in the form

rp=nlq'+n;q.

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

46

The elements of the form qp' = rI, g constitute a subspace of the vector space WD1) , since the sum of two such elements III V. + rI*, 92 may be put in the form f i (rp1 + q') and thus again lies in this subspace.

It is not hard to see that the space of elements of the form 171,9 is isomorphic to the space L(l), because each class in LP(l) will correspond to only one element of the form rl *, f . This follows from the fact that III f = 0

if 9 E S1. A sum of elements of the form ni9 corresponds to a sum of classes in L(1) and conversely. Analogously, multiplication by a constant of corresponding elements leads again to corresponding elements. The space S, can be normed, as can every finite-dimensional space. It is convenient to define a norm on it in the following manner. Suppose that P is a polynomial of degree less than I having the form 1-1 a

P

k.OEa,=k Then we put

}P12

1-1

IIPIID S,

=

E E aa,...a^a !.. .a.! k=0 Ea =k k!

2

I

The so-defined norm will be invariant under all rotations of axes of coordinates, while in distinction from the norm on L(1) it will no longer be invariant under the translation of the origin. Indeed, the quantity k!/(a I! ... a !)a2

a,

is one of the invariants of the tensor as

a^ and therefore this quantity is preserved under orthogonal transformations. We may verify that for such a norm all the three basic conditions hold:

(a) IIPI + P211 <_ IIPI II + IIP2II , (b) IIaPII = Ial IIPII , (c) IIPII > 0, and if IIPII = 0, then P = 0.

The validity of the conditions (b) and (c) is obvious. We establish the triangle inequality. Suppose !-1

(1)

as ... a"XIa,

PI =

x" ;

k=0Ea,-k 1-1 (2)

P2 = E k=0

as

a,.k

... a"x1a, ... X"^.

§1.7. THE SPACES L... AND WP

47

We have Ip/2

-1

P_

k1

k=0

k

1

(1)

(2)

2

n' 1/2

1/2

7- l

< 2 t=k [a(2

+ > [a(2)

k=0

P

]2

La'=k

IIP, + P211 < k=0

+

]2

ar=k

p/2

1/p

1/2

1/2 I

k

]2

1J0

!- I

I-1

k=0 E ni=k

E k=0 E

p12

1/p

ell.

which was to be proved (here E' denotes summation with the weights k!/a 1!.. .an!) The establishment of a norm on S, enables us to carry through also the '

norming of the space Wpr) . This norming may be carried out if we are given any projection operator whatever. It is natural to set II9II (')=Iln,c1155,+lln;wlip L PM =Iln,cllps,+11911P'

(7.5)

This method of introducing a norm depends upon the given projection operator. Later we consider the question of what relations will hold be-

tween norms constructed with the aid of different projection operators. In the meantime, it is necessary for us to verify that the three basic properties of the norm are satisfied for our definition. In fact, it is obvious that IIQ(p11,,0,,,) =

Ial

Il'llwul

Furthermore, if II rP oll w') = 0, then to = 0 almost everywhere. The triangle inequality follows in an obvious way from the Minkowski inequality. 4. Special decompositions of WP(l) . It is useful for our purposes to study one special form for the operator II1. Let us concern ourselves with this

form. To begin with, we impose some restrictions upon the domain in space on which we consider our functions.

Let Q be a starlike domain with respect to a ball C of radius H lying within fl, i.e., the segment joining any point of C to any point of i2 lies

48

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

in 11. For convenience, we assume to begin with that the center of this ball lies at the origin of coordinates, and that C e Cl. Let P' and Q' be two arbitrary points of Cl. We set r = If - Q) and let

= (Q - P)/r be the unit vector having the direction from f to 0. Each function of two variable points µ(Q, P) may be represented as a function of f , 1', and r, setting Q' = P + r1', and

u(Q,P)=u(P+rI,P) where the bar over u indicates that Q is replaced by F, r, I. Conversely every function µ(r, 1, P) may be represented as a function of Q and P. We consider the function QR2/(R'-H2)

V(j)

-{0

for R < H; for R > H,

where R is the distance of the point Q from the origin of coordinates. The function v(Q') is continuous with its derivatives of all orders and differs from zero only in the ball C. We form a new function of two points f and Q by setting

X(r, 1, P) = X(Q, P)

v(P+r11)rn,-' dr1 v(r1 , 1, P)ri -' d r1.

(7.6)

The integral clearly degenerates into an integral within finite limits, since v is nonzero only in a bounded domain.

Note that as r1 runs through the interval (r, oo) the point f + r1I runs through the ray emanating from the point Q in the direction of the vector r. If this ray does not intersect the ball C, then X(r, 1, F) = 0. Thus, for

a particular P the function X(r, 1, P) is nonzero only for those r and I for which Q = P + rl lies interior to the domain consisting of the points of all intervals joining f to a point of C (Figure 5). Obviously, the function X(r, 1, P) is continuously differentiable. We introduce the additional function W(r, 1, P) = (1 1 l)Ir1-'X(r, 1, P)

Now for any p continuously differentiable up to order 1 in the domain Cl we can construct a corresponding function by the formula a1-1_ 1-21-1 _ _ w a7a w +... + (-1) r-,a Or/-2 Or arr-I 8rr-' W Obviously, we have

-

r

OT5 a=r81V

+(-1)1-'a

w.

(7.7)

§1.7. THE SPACES Lr") AND ND')

49

FIGURE 5

In addition,

aw

of-2w

=0.

art-2

ir-O

F-0

Calculating a'-1 w/ar!-I , we obtain

a`-lw art-1

=X (0, 1, P)= - f 00 v(r1,1,P)rl-Igdr1, r=o

0

from which it follows that

foWF(r1,1,P)r'-l

i5(0c,/,P')=0.

dr1,

x(0,1,P)=-oo(P)

0

Integrating (7.7) in r from 0 to oo, we find

q(P) j I(r1, 1', P r1 dr = fo )

'

oo

a rt

I

+

(-) 1

arI

-J dr.

(7.8)

Multiplying (7.8) by the element of solid angle ddr and integrating over the unit sphere, we obtain

qi(P) f dwrf ,* v(rl

f

,

11, P)rn

_1

dr1

o

= f dcor

f

00

dr.

T

Taking into consideration that r1 d r, d cor = d vo , where dvo is the volume element at the point Q, we find f.dco1JoccF(r1 , T, P)rl-I dr1

=

f
50

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

i.e., the size of this integral does not depend upon the position of the point P. In this fashion, we obtain ao

+0 dr

0I

r0(P) = X f

r

(9r i

or, introducing dvd into the integral on the right, we obtain

(P) = X fn 8 V' r°

dvd

(7.9)

Xr-I L"Or'rsT_I dvd. We now show that the first integral in (7.9) is a polynomial of degree < 1- I in the coordinates x, , x2, ... , x,, of the point P. In fact, from the definition of 7, it follows that a1

=

k

k=I

_

aIr-k

Ckrk-I k=I I

ekk Or

k-1

= k=I s=I I

+

(

L91-kr1-1 ak

I

arr

1

k-

Dk.sr

ark

f v(r1, 1,

-11(rn-1 v(r,

l , P))

8rs

n-I+s8'v(r, 1, P)

I

(7.10)

Ors

k=I s=0 $

dr1

k-I rn-k+sasy(r, 1, P)

D

where Bk , Ck, and Dk tions of them).

P)r,-1

r

are constants (binomial coefficients or combina-

But v(r, !, P) = v(Q), with the substitution Q = P' + r1-, Q = (Y1,Y21...,y"). Therefore

av 8r

av 1

y.

where 1, _ (y, - x,)/r are the projections of a$v

Osv I, ..,

,

asv

=

C0,...0

E°,=s

8rr

=r n-I

0,

0=

x1 x2 E0 <1-I

2

av ...Ov ' (y1 - x,)

Substituting in (7.10), we find aSv

and we find analogously

aYl1lf...1.

ay., ay,, ...

n

ar=

jr

O

J.

1-1

X.

.9v

y.) 8

0

°,

(y" - x")

§1.7. THE SPACES Lp' AND R""

where

yn)

51

are bounded and continuous functions of

,Y.),

(YI,

°,...an(Y,,...,yn)=

L.

C°,....o"

ay" i..

yn

6"yl

ayn

Therefore the first integral in (7.9) takes the form

X

fns a

xi' ...x°"X f

rn' dvo =

0(a)

E° a-1

i.e., it represents a polynomial of degree 1- I in x1 , x2, ... , xn . We note that r° °"(Q) = 0 outside the sphere C, since this property holds for v(Q) and all its derivatives. The operator n1

1r a' =X n

1

arr rn-I

dv

has the property that it carries an arbitrary polynomial of degree not higher than 1-1 into itself. This follows easily from the fact that if one substitutes such a polynomial in (7.9), the second term vanishes. Consequently 111 rp is a projection operator and formula (7.9) gives the decomposition of interest to us. We turn now to the investigation of the second integral in (7.9), i.e., to the study of the operator Il, lO . For this purpose, we note that arrp

art

_rC

(YI - X1)°' ... (yn - X")°"

°,=r

ark ay1

rr

... ay" n

This last formula is obtained like the one for asv/ars . Furthermore, it follows from this that I

rn-1

arrp-

1

arrW=rn-1+r

rL

a, C0',...,n"(YI-XI)

° ark ..(yn-xn)"dayi...ay°"

µ° .....°"(Q> P) ay" ..Oyn"

n

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

52

We have (y1 - X1)°'...(y. - xn)°' _

C."

7

rn-1+1

C

10,12=...1n.

rl-I

where w°

(r, 11, P) is a bounded and infinitely differentiable function of its arguments. Considered as a function of (Q1, F), it is a bounded function of its arguments. Then the second integral in (7.9) takes the form

(-1)1-l x

f-pr

(Q, P)

W

°,

dv . ° ax sax 2= ... axn

(7.11)

1

Absorbing the coefficients 1 /X and (-1)1-1 /x into the functions { and w , we may rewrite (7.9) in the form

9,(P)= E

xI'X2'...xn"!cZ°,.....°,(Q)!D(Q)dvd

w°,, ..°,(Q> P-)aya v(

+f _

d)

dvd.

(7.12)

The formula (7.12) was established by use for functions having continuous

derivatives to the lth order. It is not hard to see that for almost all x E n it remains correct for an arbitrary function from WDl°. Indeed, let t E W(l) and let 9. be its averaged function. For Ph , (7.12) is correct, i.e., we have

X''

Ph(P) _ II'

x,-,- f

a9'h(Q)

1

+In rn-I E w°,,...,o,ay'i...ayn, dv

F°,=1 = 5(h) + P.M.

(7.13)

We set

x2"...xn^Jc

,!n r"-

w°, ,...,°,(Q,

ayn

dvd.

§1.7. THE SPACES Lo' AND W

53

dv dv as h - 0, and consefc ra, quently, S(h) -+ S uniformly on Q. We assume first that rp E W(r) on S2 and that rp is equal to zero in a neighborhood of the boundary of S2 . Then, by the theorem in 2 of §5,

It is obvious that fc

II

ash ayla,...ayn,

a9 ayoli.aynin 11L, -

0,

h0,

Q. Consequently, if lp > n, then it follows from the estimate (6.2) that V'(h) -+ g' uniformly in fI, and if 1p < n, then it follows from (6.4) rp' in LP on fl (since p < q = e that g'(h) for s = n). Moreover, according to the properties of averaged functions, rph tO in LI on Q. Therefore, passing to the limit as h 0 in (7.13), we get that (7.12) holds almost everywhere in Q.

Note further that since the functions w (Q, P) vanish outside the "cone" formed by the union of all the intervals (P, Q), where Q E C (see Figure 5), it follows that for any b > 0 there exists a al > 0 such that if f E Q8 , then all these cones are contained in Q., . Consequently, for P E Qa the integral over f2 can be replaced by the integral over Q., in the second term on the right-hand side of (7.12).

For any 9 E W(l) the equality (7.12) for almost all P E Q for any d > 0 (and hence almost everywhere in Q) follows from the fact that for each domain Ray there exists a function rp3, E Wpr) equal to zero in some neighborhood of the boundary of Q and coinciding with 9 on fla, The case when the center of the ball C lies at some point f(y1, ... , y,,) can be reduced to the case considered by a translation of the origin into the point f (i.e., by the application of the formula (7.12) to the function w defined by yi(P) = rp(P + f). Here it follows from (7.12) that

go(P) = E (X1 - y)a' ... (Xn - yn)aJC ra, .....0 (0- ?)go(Q) dv Ea,5r-1 + Jo

dvQ.

rn-r

(7.14)

E a,=/

If the point f is the origin, then (7.14) passes into (7.12). Further, we set

0(P) = S(P) +

(7.15)

Thus, for functions ip having continuous derivatives up to order 1, tD coincides with rp on 0, and for functions r0 E Wpr) it is equal to 0 almost everywhere on Q. ((8) - (12))

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

54

An important class of spaces is made up of the normed function spaces having the property of completeness, i.e., the property that Cauchy sequences converge. With the norm introduced, the space W(l) is complete. We establish this later, but now we prove some important theorems. §1.8. Imbedding theorems

1. Imbedding of Wo`) in C. Denote by C the space of functions continuous and bounded in the domain f2, and let II47llo = sup, lip[. THEOREM 1. If f2 is a bounded domain starlike with respect to some ball, 9 E W(`) on i2, and n < Ip (1 < p < oo), then 0 E C, and (8.1)

Il0llc < Mllgllw('

where M is a constant independent of the choice of (p.

PROOF. Indeed, the continuity of 0 follows from (7.15) and the theorem in I of §6 with A = n - I < n/p'. Further, we conclude from (7.15) that 101 < ISI + Iw'I

Letting max Ixi ' .. x°^ I = A (x1 E 0, F I a1 < 1 - 1), we get by using

the Holder inequality (1.20) and denoting by N the number of different monomials

xR^

,I a1 < 1- 1 ,that 1-1

ISI <_ A

v=0 E n v

Iao,...tr^I

1-1

1/2

UI

v=0 E -v

I

^

1-1

Using (6.2), we find an estimate for I9' 1. Writing sup lw0,

II,QEC

,

Eai=1),weget lip* I <- KB E

, =t

8x1 a.. ... axR Lo

L,

o^

I = B (P E

§ 1.8. IMBEDDING THEOREMS

55

By (7.3), I(p'I
where NI is the number of different derivatives of order 1. This gives us that 10J:5 (K'+K")II9llwul, = M1107 113; , and the inequality (8.1) o

is proved. (13)

2. Imbedding of W(l) in La. . THEOREM. If fl is a bounded domain starlike with respect to some ball, cp E W(1) on fl, and n > lp (1 < p < oo), then 0 E LQ. on the section of S2 by any s-dimensional hyperplane, where s > n-lp and q' < q = sp/(n-1p) . Moreover,

(8.2)

11011Lo. < M11911wo, ,

where M is a constant independent of the choice of Ip . PROOF. Indeed, the fact that rp E LQ follows from (7.15) and the theorem

in 2 of §6 with .1= n - I > n/p' and s > n - (n - 2)p. It remains to prove (8.2). According to (7.15), II OIIL,. < IISIILo + III' IIL,.

Writing A = maxi a,
11x;'

x no- II Lo , we find just as in the proof of the

IISIILP <- KII,OIIwol

Again writing sup 1w0 o I = B (P E Q, Q E Q, E a1 = 1), we find by using the estimate (6.4) for Z = n/-1 that I

I

1

L..

BKI

n

Lv .

8tV o

< BKINIIVIILO,
11011L,. < (K + K2)II9IIwu = MII9IIwn, ((14)_(15)) i.e., the inequality (8.2) holds. The theorem is proved. The theorems in items 1 and 2 mean that the imbedding operator carrying a function c as an element of Wpt) on 12 into the function 0 as an element

of C on fZ (n < 1p) or an element of LQ. on the section of i2 by an sdimensional hyperplane (n > 1p, s > n-1p, q' < spl(n-lp)) is a bounded operator. It will be proved in § 11 that it is also a completely continuous operator.

I. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

56

3. Examples. We have shown that if rp E WD1) and n > 1p, then 9 E Lq.

on any hyperplane of dimension s, where s > n - lp and q' < q = sp/(n 1p), and then the inequality (8.2) holds. We present two examples showing

that in the theorem in 2 the number q* may not be replaced by q + a, no matter how small we choose e > 0. Thus we will have shown that the given exponent q is precise and may not be increased. EXAMPLE 1. Let R < 1. We consider the half-ball Sl of radius R in n-dimensional space

x2=r2
xn>0(n>3).

i-1

Then u = (r(/2)-' lnr)-' E WZ') on Q, since

1(,_l)2+n-2

2

8u

"

1

(ax;)

r"

IL

1

1

(in r)' J

(ln r)3

(In r)2

and

T1-R

JE'au'2 8x; d

+

1

2 f o { r" f (in r)2

(in r)3

Tin r)4,

}

r"-'dr

converges.

Here a" denotes the surface area of the sphere in n-space. We consider the section Cl1 of a half-ball Cl by the plane x1 = 0 and write E 2 x = p2 . By the theorem in item 2, u E Lq. on i21 , where

q
fa

ulq+e dvn_1

_2(n-1)

n-2

diverges if e > 0. Indeed,

2

u q+e dv = a1-1

2

R pl]q+e p

dp

R p-I-e(n-2)/2

an-1

=a"_2I

n-2

1

(p(n-211211n

o

Jo - In R

plq+e d p I in e{(n-2)e/2 q+e

which implies that the integral diverges no matter how small e > 0 is. Thus, u does not belong to Lq+e on fl1 .

EXAMPLE 2. Let 6 > 0. Denote by ra the distance from the point (0, ... , 0, -6) to the point (x1 , ... , xn) , i.e., 1/2

n-I

r= 1-1

,

§1.9. GENERAL METHODS OF NORMING

57

Then in the half-ball it the family of functions

I-R)

0<6 <

U6 (Inra) -1 a =rl-("/2) a

J

2

1)

is uniformly norm-bounded in W2 . Indeed, the integrals of l ua1 and E"_1 1aua/ax,12 over a half-ball of ra-

dius R+6 < (I +R)/2 < 1 about the point (0, ... , 0, -6) are uniformly bounded, since the integrals (I+R)/2 r"l2dr

Ilnr1

(I+R)/2

and o

dr rl In

rIz

converge.

Therefore the integrals of 1ua1 and E", Iaua/ax12 on IZ are also uniformly bounded (i2 is a part of each half-ball of radius R+6) and by virtue of (7.12) and (7.5) 11ua11W= is uniformly bounded.

On the other hand, it is not difficult to see that uala+`

In I and, consequently, 11ua 11

L.,.

dv,,_1 -. oo for 6 -+ 0

on the hyperplane x, = 0 will not be bounded,

i.e., for q' = q + e the inequality (8.2) will not hold. (16)_(Is) §1.9. General methods of norming W(r) and corollaries of the Imbedding theorems

1. A theorem on equivalent norms. Let n, and 112 be two projection operators mapping W (l) onto S,. These operators generate norms on WD I ) which we designate by '1Q11 and 211Q11 We shall say that these norms are equivalent if one can give two positive numbers m and M such that

m<jIl9oll<M

for all 47 E W (1) .

(9.1)

IIQII

We consider the operator y,2w = (Il, - n2)?. This operator possesses two properties.

(1) E12S = 0 , i.e., _,2 carries into zero every polynomial in S E SD . This follows from the fact that D,S = II2S = S. It follows from this that "I2 puts in correspondence with one and the same polynomial S all the functions of one and the same class w E L()

(2) ?129 = 0, i.e., the square of the operator '12 is the annihilation operator. Indeed, =12Q = S E S,, and consequently _z °,21° = .C.12S=O.

I. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

58

THEOREM. For the equivalence of the norms

111911

and 211911 generated

by the operators nI and n2 it is necessary and sufficient that there exist a number M > 0 such that for all c E W(l).

II°12911s, <- MII9IILo

(9.2)

PROOF OF NECESSITY. The proof proceeds by reductio ad absurdum. Sup-

pose that the condition (9.2) does not hold. Then we may find a sequence of functions {r/pk} , 'Pk E W(') , for which

(k =1, 2, ... ).

N"I29kI1s, > k1l9k11

Normalizing 9k, we may assume that IIcPkIIL')

(9.3)

1

We consider the functions y/k = rI,9k = 9k - nIVk and calculate both norms of Wk IIIvkII=IIwkIILDU,=1.

Since 1711'k E S1 , rI2nl pk = 111 9k

In addition, taking (9.3) into account,

we obtain 211vkII

= Iln2vkIIs, + I19'kllq)

= Iln2pk - n2nlpkIls, + 1 =11112pk-III 9klls,+1

=II=129'kIIs,+1 >k+1, from which it follows that 211 vk II/' II vk II - oo. This contradicts (9.1) and the norms 1119 11 and 2111P 11 are not equivalent. The contradiction shows that (9.3) is false, and therefore that (9.2) holds for all In E W(t). PROOF OF SUFFICIENCY. Suppose that (9.2) is satisfied. Then 1101=111],9IIs1 +Il0lto)

[IIn2vlls, + II(n, - n2)glls,l + 1I911gI D <- II1129IIs, + MIIOIL,, + IIwOIL')

= IIn2tPlls, + (M + 1)IIAL'I < (M+ 1)(IIn2911s, +

Analogously, it may be shown that < (M + 1) 'IIiPII. These two inequalities together are equivalent to (9.1) and the theorem is proved. The condition (9.2) is conveniently written in some other forms. We have 2II1PII

12/P=n,(P-1'2V=n,9-n,n2V=n,op-n2g)=nn2gP and analogously

!129 = -n2n;,.

§ 1.9. GENERAL METHODS OF NORMING

59

Using these, we see easily that each one of the following three inequalities below is a necessary and sufficient condition for the equivalence of the norms: 11=129lls, <- MIIVIILu, ;

(9.2)

IIn1n;q,lls, < MIIwflLy, ;

(9.4)

IIn2n;s011s, <- MIIcoIILu.

(9.5)

We shall make use of this result below.

2. The general form of norms equivalent to a given one. We shall now concern ourselves with a more detailed study of the structure of projection operators yielding norms equivalent to a given one. Let n, be given. The general form of a projection operator 172 will be

E

Eu, I-1 where I., .. are functionals with the distributivity property Q3). It is not difficult to see that these functionals also have the property

- f;)2 = 0, if ° - , 1 0

za" - J 1 if E(a,

(9.

6)

.

"

THEOREM 1. In order that the norms 111p11 and

2II9?lI should be equivalent,

it is necessary and sufficient that all the functionals 1,

should be Q bounded in the sense of the norm defined by the projection operator 171 . PROOF. It is easy to see that the condition (9.5) for the equivalence of the norms is equivalent to the system of conditions (9.7)

<- MII4,UILo,1.

It remains for us to show that (9.7) is equivalent to the condition of the theorem.

. ,1 *are bounded in the norm

Suppose that all the functionals 1 1II(DII

Then I/o,,a,,....o"ni l s MI11n;911wv1) = M1191ILv1,1,

i.e., (9.7) is verified.

Conversely, if (9.7) is verified, then taking (9.6) into account, we find Ilo,....,0 Q1=

ml

IInlvlls, +MIIVIIL01

(MI =max(I,M)),

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

60

and the i.e., from (9.7) follows the boundedness of the functionals 1, theorem is completely proved. on W, satisfying Thus, an arbitrary system of linear functionals 1,, the equations (9.6) yields a projection operator n2 and consequently a new norm on W(l) , and this norm is equivalent to the norm with respect to which the boundedness of the functionals was determined. If we now consider an arbitrary system of linear functionals h, p^ , the number of which is precisely equal to the number of monomials of power not higher than l - 1 , and taken so that the determinant of the matrix A

is non-null, where each row of the matrix consists of the values of one and the same functionals and each column of functionals acting on one and the same monomial, then from such a system we may always construct linear which will satisfy the equations (9.6). combinations la Each such system of linear functionals enables one to define a norm on Sr by the formula I1IP lls, =

IE(he,.....a^rp)ZIII2.

(9.8)

Such a norm, as we saw above, will be equivalent to the initial norm on S1.

The condition IAI # 0 may be reformulated in a form suitable for applications in two ways. It is well known that if the determinant of a square matrix is not equal to zero, then it is impossible to have a nontrivial linear combination of the rows

of the matrix or of the columns of the matrix which is a null vector. Conversely, if the rows or the columns are linearly independent, the determinant must be different from zero. From this follow two remarks. (a) The condition Al I# 0 is satisfied if the following holds: for any nonzero polynomial P of degree not greater than I - 1 , we can find a functional such that he C^P is not zero. (b) The condition Al I# 0 is satisfied if the following is valid: for any linear combination of functionals p=

2

Aa, .....0 ha, ..... o

Aa....0 # 0,

we may always find a monomial

x ... x0^ n for which pxr'

xn^

0.

3. Norms equivalent to the special norm. Up to now we have considered the equivalence of norms arising from two arbitrary projection operators,

§ 1.9. GENERAL METHODS OF NORMING Not)

61

and all the arguments worked for an arbitrary domain Q. We return now to domains starlike with respect to some ball C for which we constructed a special projection operator defined by formula (7.12).

For the norm given by this operator, the imbedding theorems in 1 and 2 are valid, i.e., inequality (8.1) for the case n < Ip and (8.2) in the case

n>lp.

Let ha a (E of < 1 - 1) be a system of linear functionals on C (if n < !p) or on L.. (if n > !p) satisfying the condition: if the functions 9 and yr coincide almost everywhere on fl, then ha, ..... a 9 = ha, , ....

v.

(9.9)

Then, for example, in the case n < lp we find using (8.1) that K1110I1c
i.e., ha is bounded on Wp1! with respect to the norm with the projection operator (7.12). An analogous assertion is valid if ha, is linear

on L9. (n > 1p). From this we get the following assertion by using the preceding theorem.

THEOREM. Suppose that it is a domain that is starlike with respect to a ball, {hai

an} (E a; < 1 - 1) are linear functionals on C (n < lp) or

on L.- (n > lp) satisfying condition (9.9), and at least one of the ha . .a is nonzero for each nonzero polynomial S E S,. Then the norm defined by (7.5) and (9.8) is equivalent to the norm (7.5) generated by the operator (7.12), and (8.1) or (8.2) hold if the norms of 1D on the right-hand sides of the latter inequalities are assumed to be defined in turn by the formulas (7.5)

and (9.8). 4. Spherical projection operators. It will be assumed that the domain it is starlike with respect to the ball C of radius H centered at the origin. The formula (7.12) determines the projection operator II in the form

no =

X"

... yn+I H,...,a,(Q)V,(Q)dv

Ea,
assigning to each rp E W(1) a polynomial S E S1, where the coefficients of this polynomial are determined only by the values of 9 in the ball C (the in (7.12) can be expressed in terms of the function v, function Ca, which depends on H ; to underscore this dependence we denote it here by

Let ?(y1, ... , yn) be a point of C such that the ball C, of radius H, about T is contained in the domain it (it is not assumed that R is starlike

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

62

with respect to C1) . We construct the operator

E

(XI

-y1)",...(xn-yn)°" f C,

E. '<1-I

SG)dvd. T)o(/

(9.10)

The operator f . assigns to each function IP E WP(l) a certain polynomial S E S, . Moreover, flc carries each polynomial S E S1 into itself, since (7.14) is valid for it with C replaced by C1 and f2 by f21 , where Q1 is any domain starlike with respect to C1 , for example, Q1 = C1 (here the second term is equal to zero). This implies that fIC is a projection operator. The operators fIC, defined by (9.10) will be called spherical projection operators. THEOREM. Let f2 be a domain starlike with respect to some ball. Then the norms constructed with the help of arbitrary spherical projection operators are equivalent.

PROOF. As in 4 of §7, it can be assumed without loss of generality that f2 is a domain starlike with respect to a ball C about the origin. To prove this theorem it suffices to establish that the norm defined with the help of an arbitrary spherical projection operator is equivalent to the norm IIwII'wl+w defined by the formulas (7.5) and (9.8) with the functionals linear

on C (n < lp) or on L.- (n > lp, q' < np/(n - 1p); s = n). We show that these norms are equivalent. Indeed, the functions 1 H. are bounded, and hence the functionals hot

(p=

4n

?)9(0)dvo,

(9.11)

,

are bounded on C (n < lp) or on L.- (n > lp, q' < np/(n - lp), s = n). As mentioned above, fI.' is the identity transformation on the set St ; therefore, the functionals hog o" , Z a, < 1- I , have the property that for every nonzero polynomial S E S, at least one of the functionals is nonzero. Consequently, by the theorem in item 3, the norm generated by the operator nc is equivalent to the norm Thus, the norms generated by arbitrary erical projection operators are equivalent to the norm 11911 k u, , and hence spherical are mutually equivalent, and the theorem is proved. 5. Nonstarlike domains. We may now free ourselves of the restriction of starlikeness imposed upon the domain Q in the proof of the imbedding theorems.

Let f2 be an arbitrary domain containing some sphere C of radius H. Formula (9.10) defines a projection operator on W(t) and we define a norm by formula (7.5). It remains for us to show that under some restrictions upon Q, the imbedding theorems in I and 2 are still valid as well as the inequalities

§1.9. GENERAL METHODS OF NORMING W"'

63

(8.1) and (8.2). We remark that if for some domain the imbedding theorems hold, then for that domain the preceding theorem is valid. LEMMA. Suppose that the domain Q has the form 12 = f21 +f)2, where 121

and fl2 are two domains for which the imbedding theorem holds. Then the imbedding theorems are valid for the domain 0 (the function 0 is defined by (7.15) on each of f21 and 02), and the norms defined for all spherical projection operators are equivalent. PROOF. Let to E W(1) in Q. Then Sv E W(i) in i21 and in f22 and therefore c E C (if n < Ip) in 121 and 02 are therefore in Q1 +f22. One treats the case n > Ip analogously. It remains to prove (8.1) and (8.2). Assuming n < Ip, we shall prove (8.1). Let C12 be a sphere in f21f12 and nc 1a its spherical projection operator. Then 1')II9IILU ] in f2;

II9IIC <

(i = 1, 2),

from which it follows that in 12 = 121 + Q2 we have the inequality 11911c <-

since `0II9IIL+, 5 II(pIILa) and where K3 = max[K1, K2] .

As a consequence (8.1) is valid. Analogously for the case n > 1p, (8.2) may be proved. On the basis of the theorem in item 4 there follows the equivalence of all norms defined by spherical projection operators. THEOREM. Suppose that the domain f2 is a sum of finitely many domains 121, ... , ilk, each starlike with respect to its own ball. Then the imbedding theorem holds for f2 ( the function tb is defined by (7.15) on each of the domains Q), and the norms of all the spherical projection operators are equivalent.

PROOF. This theorem follows by the repeated application of the lemma to the domains f21 + C121 f21 + Q. + C13 , etc.

In the following we shall consider only such domains, and will not speak of this on each special occasion. We may without loss of generality consider only such norms on the space W U) as are equivalent to the norms obtained from spherical projection operators. We will call such norms natural. Each time that we speak of a norm or of convergence in the space Wot) , if no special provision is made, we shall have in mind an arbitrary natural norm and convergence with respect to an arbitrary natural norm. 6. Examples. We introduce two examples, illustrating applications of the above theorems.

EXAMPLE 1. Let p =2, 1= 1, s = n >3. Since q = 2n/(n - 2) > 2, we may take q' = 2, i.e., W21 c L2 . Since 1 = I, for the definition of the norm it suffices to take one functional. Let hip (h , q') = fn 9 dv .

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

64

It is obvious that (h, 1) # 0, and the functional h is linear on L2. Therefore on the basis of the theorem in 2, we have

(2,P2 I

1 ff,191'dv}l/2<<M

Ifnopdv1

I/2 {1E\ax.)dvJ

+

from which there follows

<MI I

)z

r }Z+Ja E

fnipdv

\8x

dv]

(9.12)

,

where M and MI are constants. The last inequality is well known under the name of Poincare's inequality.

ExAMPLE2. Let p=2, 1=1, s>n-2; s1=n-1, s2=n. Since q1 = 2(n - 1)/(n - 2) > 2, we have by hyperplanes of dimension (n - 1).

W21)

c L2 on sections S of S2

We put (h, c0) = f rpdv._1. s

We have (h, 1) # 0, and the functional manifold S.

is bounded on L2 on the

At

On the basis of the same theorem in 2, the imbedding theorem holds for the norm

I+ fdv_1IIwIILu

,

an d in particular, since W(1) c L2 in 0 (s2 = n), f 10PI2 dt)

1/2

<M

[fdv_1 I +

from which follows

f

< MI

{jdV_12 + is

IIRDIIL=1]

f ()2

dv

.

(9.13)

n;_I 8x; It is obvious that the theorem in 2 can yield many inequalities for suitable choices of the functionals h_

_

.

§1.10. Some consequences of the imbedding theorems

1. Completeness of the space W(/) . We consider a Cauchy sequence (''k } , 9k E WD I) . Suppose that the norm in W «) is defined by an arbitrary spherical projection operator II . We have II

m-

VkIIWW') --+ 0,

m, k

oo.

On the basis of the imbedding theorem we conclude that IINm - SDkIIL,

0.

(10.1)

§1.10. SOME CONSEQUENCES OF THE IMBEDDING THEOREMS

65

Let the limit of the functions be q.. Setting r19k = Sk, we obtain from (10.1): IISm - SkjIs,+Il m- VkIILvIY) --..0,

m,k

from which follows:

m,k-+oo;

(10.2) (10.3)

Ilq7m-(pkIIL/1-.0,

From (10.2) it follows that the coefficients of Sk converge to finite limits and therefore that Sk converges uniformly to some polynomial So. Obviously, we have

IISk - Stills-. 0,

k

(10.4)

From (10.3) it follows that aIIPm ... ax,,Q,

axQl,

Il

_

axol

apk

v i ... ax,'

-. 0,

I

{I L

Ox,,, converges in LP to some

af90

,....a, =

m, k -. oo,

S

i.e., each of the derivatives a'cok/ax;' function wQ o, E L.. We show that Wa

=1,

2

(10.5) n

Indeed, for each yi having continuous derivatives up to the /th order on the whole space and vanishing outside some bounded domain V. C i2 with 33W C 12, we have

r

Jn9kaxe81

ax,7' du' from which, taking the limit as k -» oo, the correctness of (10.5) follows. Thus, l°o E W(1> (since co a, E LD) . It is not difficult to verify that rlfo = so. In fact, replacing rp in (9.11) by (Pk and passing to the limit, we obtain I'M Sk = SO' k-oo

Since a'rok/ax;'

8xn'

wa

a

in L. and by virtue of (10.4), obvi-

ously, we have Ilq'o-c0kHHWo,l--.0,

k-yoo.

The completeness of the space W' is proved. 2. The Imbedding of W(') in W`k) . Up to this point in §§7, 8, and 9 we have not mentioned the derivatives of order less than 1. In §5 it was observed that from the existence of the generalized derivatives of higher order, there does not follow in general the existence of derivatives of lower order. Now the following theorem will be proved.

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

66

THEOREM. If V E W`') on a domain (

that is a sum of finitely many

bounded domains, each starlike with respect to its own ball, then the function

0 defined in the theorem in 5 of §9 has in !2 all generalized derivatives of order less than 1. For these. (1)

If ip > n, 0 < m < 1- (n/p), then 0m0/8xi' ... axa^ is continuous and (10.6)

am ax a,I <MIIgIIWp«. IOXG1

n

(2) Iflp0,and m>I-(n/p), s>n-(1-m)p,then on every s-dimensional section of S2, am0

axa' ... ax,n -

E Lq

where q' < q = spl (n - (I - m)p), while am0 axe, .9

(10.7)

< M11914 0'). L.-

REMARK. If lp > n, then Will c C!-I"/pl-I , i.e., W(ll is part of the space of functions having I - [n/p] - 1 continuous derivatives after modification on a set of measure zero. This follows from the first part of the theorem. Setting s = n in the second part of the theorem and noting that in this case the possibility of q' = q is established (see (14)), we conclude that if k > 0 and k > 1- (n/p), then W(l) c WQ' (k < 1), where q is determined from

k=I-n(1/p-l/q),that is, q

PROOF. It suffices to prove the theorem for domains Cl which are starlike with respect to some ball. Let q)(P) be continuous and have continuous derivatives up to order 1. Then formula (7.12) holds:

(P) _

X*11 ...x"' "

r

4919(Q)

P)aX l...ax dvd. n

+Jn

1

The theorem will be proved if we show that

a

l

w

Q

e,.....a

al 9

dvQ

1

X..dvd, - Jo

rl+mwu,.....a: ay-1, -

(7.12)

§ 1.10. SOME CONSEQUENCES OF THE IMBEDDING THEOREMS

67

where wfl, are bounded functions. Indeed, let us differentiate m times both sides of (7.12) written for the averaged function cph . The limit of the first term of the right side for h -+ 0 will be the polynomial a'"S

axl'

axn^

the coefficients of which are simply expressed in terms of the coefficients of

S and as a result, a'"S

< MIISIIS,

axp] ... axp n 1

The limit of the second term of the right side will be the sum of terms of the forms of the right side of (10.8). On the basis of the theorem of §6 on integrals of potential type, the assertion of the theorem follows. For the proof of (10.8), it suffices to show that

a

1

\r"

n

!w(Q,

P)) = w( (Q

rn-1+m,P)

(10.9)

where w(m) is a bounded function (for simplicity the subscripts are omitted). We have (§7, item 4): 1

1

rh- w(Q, P) = C°

(x1 -

Y)°,...(x" - y")°.

rn

X(Q, P)

Let us show that each differentiation of (1/r"-')w(Q, P) increases by one unit the order of the singularity of this function. Indeed, for the first factor y")°

(x1 r"

this is obvious.

P)/r, If we are given Q and P, then v(QI), where QI =,P + r11= P' + (r1 /r)(Q - P) . Therefore, X(Q,P)=- fv[P'+r(f2-P)]rI-Idyl.

v(r,1,P)_

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

68

Differentiating with respect to x1 , we find ax(Q, P) = 8x1

IxI -Y, r

v ( Q)rn-

_

x

r2

- 8v(01)r n

1

-r

r

f

-f

n 8v(Q1)

F,

ay;

(xI-y1)(xj - yj)

+

X k lj

00

0 y1

E

,

- ajI II

Jr

-Idrl

(XI - Y,)(x; - YJ)

r2

;_,

00 8'(QI)r', dr11 +v(Q)rn-2(x, -y1) ay;

8v(QI)r"-'dr. ay

Denoting by X 1(Q , P') and X2 (Q' , P) and so on, integrals of the same type

as X(Q, P) (v(Q1) is replaced by a suitably often differentiable function, r_I is replaced by r" , r" +1 etc.) we obtain ax1 =

r

E ;a,

P) +X2(01,5) + v(Q'}r"-'11

,

from which it follows easily that further differentiations with respect to x; will each not increase the polarity in I /r by more than one unit. Thus 8mX(Q, P) _ 1 W(Q, P), rm 8x1 - ax.,

where co(Q', P) is a bounded function of its arguments. Thereby (10.9) is proved and consequently the theorem as well. REMARK. All through §7, the theorems were formulated for planar manifolds of dimension s. These theorems are extendable to sufficiently smooth manifolds of dimension s. Namely, if the manifold of dimension s lies in some domain for which there exists a one-to-one mapping which is continuously differentiable with bounded derivatives and which carries the manifold under consideration into a planar manifold, then for that manifold all the theorems stated for planar manifolds are valid. For the investigation of certain problems, it will sometimes be important to know the behaviour of functions on the boundaries of domains. We introduce a class of domains which for convenience will be called domains with simple boundary. We shall say that the domain 0 has a simple boundary in the case that the boundary can be decomposed into a finite number of manifolds o , ... , Solo of various dimensions and such that S,(,2)2 , ... , S'1 each manifold S'_, by means of a transformation of coordinates defined on

§ 1.10. SOME CONSEQUENCES OF THE IMBEDDING THEOREMS

69

part of the domain S2 and continuous with continuous derivatives up to lth order, can be transformed into a planar manifold. For domains with simple boundary, we may assert that the imbedding theorem is valid also for the boundary manifolds. 3. Invariant norming of W(l) . For further discussion, it will be convenient

for us to introduce a norm on W(l) in still another way. Let cc E W(1) on a domain 12 satisfying the conditions of the theorem in 5 of §9. Then V E LP(l) and by virtue of the imbedding theorem ip EL°. We shall show that the norm II p II µ°,, given by the equality 0

Ilwli .,

+ IIcII_,

is equivalent to an arbitrary norm constructed by means of a spherical projection operator, i.e., it turns out to be a natural norm. The right-hand side of the equality which gives the definition depends neither upon the choice of the origin of coordinates, nor upon the direction of the coordinate axes, and thereby turns out to be invariant under all possible orthogonal transformations. From this we see that a natural norm may be defined in an invariant way. We show the equivalence of Ilwllw, to an arbitrary natural norm. Let 0

be a norm defined by means of some arbitrary spherical projection operator. It is necessary to show that there exist constants m and M such that l9;1

mIIIPIIwu, <_ IIcIIw 0

n

<_

0

MII'PIIW11

Ilniplli+ IInlcll°s, = where n, is the given projection operator. Further Ii9llL'(', <_ IIdHn .

We shall prove also that Iln1 pII(') < KIIPIIL, ,

where K is a constant depending upon the shape of the domain 0. Let the coefficients of the polynomial Ill pp be a0 o,

Ilnlwll(l) < K1 maxIa.,.o,.....o,l

; then obviously (10.10)

I. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

70

may be represented

On the other hand, each of the coefficients as an integral

where C(Q) is a bounded continuous function of its arguments (cf. (7.12)). Applying Holder's inequality, we obtain Ia°,.°2....°^1 < K211wIIL,

(10.11)

By (10.10) and (10.11), Iln,9,11s,' <- KI1wI1L

From this, finally (IIcI 1101

n)° <- MII14,11p n + IhIIPi 1:5 *114,1Ia;, ,)°

,

It remains to obtain the lower estimate. By the Imbedding Theorem we have

W(I) c LP and

In addition, IImIILD, = In;PIIL( <- IIFIIwoo,

from which obviously follows IIipIIwU)

U

,

which was to be proved. (19) - (2s) §1.11. The complete continuity of the imbedding operator (Kondrashov's Theorem)

1. Formulation of the problem. In this section it will be shown that every bounded set in Wpl) (bounded in norm) turns out to be bounded and equicontinuous in C if n < 1p, or in L.. if n > 1p, for bounded domains 11 starlike with respect to a ball. As a preparation we shall establish a lemma on integrals of a special form.

We introduce some notations. Let P and P + bP be two arbitrary points, Q the point corresponding to the variable of integration, r = if - QI , r, = If +&P - Q1. Let f(P) E Lo on C. We shall consider the function 1(P) to be equal to zero outside of 12 and extend it thereby to the whole space. Let 0 < A < n. We consider U(P,

) = /' ,
(r

+ r,) I1"rf

1

(11.1)

§ 1.1 1. KONDRASHOV'S THEOREM

71

We represent (1.11) in the form

+ ff

U(P, AP) = I

( r'

>Io.'l/2

rarI

For fixed AP, the last integral is bounded, the first integral is bounded by 1 /rl and the second by 1 /rz . Each of these integrals yields a function in

C if n < (n - ;.)p, or a function from Lq. if n > (n - A)p, since the variable domain of integration may be replaced by a fixed domain by means of introducing multipliers which are equal to zero or one on the corresponding

domains. Therefore U(P', AP) also lies in C or Lq. in the corresponding cases.

2. A lemma on the compactness of special integrals in C. If n < (n -,I)p (i.e., if A < n/p'], then for IAPI < 1 and P in a bounded domain w, IAPI I U(P, AP)I 5 CIIfJL,IAPIP ,

(11.2)

where fl is a constant with 0 < P 5 1 , and the constant C depends on w but not on f . PROOF. Since A < n/p', it follows that 1/r2 E L°. on a bounded domain. We have I U(P> AP)I 5

If(Q)I dv

J,
where R is a number such that the ball of radius R about any f E to contains D. Applying the Holder inequality with the exponents p and p', we find 1/P

'1D,-D,

I

P, eP)I <_ IIfIIL

r
(

T + / 1)

rD'D'

= IIIIILv 1"P'.

dv

(11.3)

We investigate the integral JI

(r+rl = ,<,R

rz0 rA0'

dvQ.

1

For this purpose, we pass to new variables, setting x, = IAPI ; , Y, = IAPIni

where the coordinates of the point P are x; and the coordinates of Q' are y,.. Under this change the point P goes over into a point PI , and the point P+AP into a point P2 such that IPI - P21 = 1 . The point of integration Q goes over into a point QI .

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

72

We will have IQ, - P11 = p = r/IAPI, 10, - P2I = p1 = r1/IAPI, dvQ = [API" dvQ. We obtain -v IAP'I'v -v

(P +

J

p 1p' pi IAPI 2'D

v
1

AP " dv I

Q'

(P +

J

=

I

per' PIv'

dv -

Q

-°- J2.

(11.4)

'1p

= IAPI"

Dividing the integral J2 into two parts, we obtain

f5R/ICI

2

=

f

pzd p

p4,4

(P + PI) zp

Q

-v

)zp'-v'

(P + P

dvQi +

pao' pv'

dvQ.

12:5p:5R/jAfj

v<2

The first term does not depend on IAPI, so that this is a convergent integral.

Estimating the second term, we remark that for p > 2 the ratio p1/p is included within fixed bounds, for p - I
1-1/p
f

(P +

2
p2 o p'1p'

dvd< K f '

R/1oP1

p"-p'-4'-1 dp,

2

1

where K is a constant. Thus for n - p' - 2p' # 0 J2 < K, + K2

R (IAPI

(11.5)

Taking account of (11.4) and (11.5), we obtain

J, < C2 + CI From formula (11.3) there follows

IAPI"-; -xd

IU(P, AP)I <- IIfIt[C2 + C1

1API"-D -"D )1/0

from which we find IAPI I U(P, AP)I <- IIIII[C21APr + C11API"-zp')110'.

If we set min(1, (n - Ap')/p') = f , we obtain 1,01 1 U(P, AP)I <- CII11I IAPI" ,

where 0 < f < 1,

and the lemma is proved. REMARK. If n - 2p' - p' = 0, then by a modification of the argument we obtain the desired inequality with 0 < ft < 1 .

73

§ 1.1 I. KONDRASHOV'S THEOREM

From the theorem

3. A lemma on the compactness of integrals in Lq .

in 2, §6, we have that if n > (n - A)p (i.e., A > n/p') and if in addition s > n - (n -1.)p , then U(P, AP) E L. as a function of the point f for fixed OP on an arbitrary bounded domain E, in a hyperplane of dimension s, where sp 9* <4 n-(n.-A)p LEMMA. Under the conditions of the theorem in 2, §6 we have the inequality IAPI

(11.6)

II U(P, AP)IIL,- 5 CIIIII IAPI0. C = const.

where the norm in L. of the function U(P, AP), as a function of

is

taken over a bounded domain E, in a hyperplane of dimension s, IAPI 1, fi = min(1, 2e) if 2e = (n/p') + (s/q') - A # 1, and ft is arbitrary with 0 < fl < 1 if (n/p')+(s/q') -d = 1.

We shall prove the lemma in the case 2e # 1, leaving the proof for the special case 2e = I to the reader. PROOF. We have as before in the proof of the theorem in 2, §6:

s=n-n+2n,

i.e., x=n+s. q' p p We put A = (n/p') + (s/q*) - 2c, where 2e = (s/q') - (s/q) > 0. We choose, as before, q* > p (p < q* < q). First of all, we establish an auxiliary inequality for the function U(P, AP')I" . Namely, we show that

'1>n; p

p

q

')p.]9*1P'

IU(P, AP)Iq 5 IifIIio-P[A, +AZIAPI(` (c+)q'

r + rs

X

J,
I11P (

r' -

q

.

dv - .

(11.7)

In In fact, we obviously have 1

1

[Ifil"p -q )J [IfIQ

I U(P, AP)I 5 I

.
(r+r1)v -`n rv -`rv'-`

X

(r+r1)q -Cr9 rQ

dv-. t1

We apply Holder's inequality to the last integral with the exponents pq*

I 1

p

1

q

=,,._.,

qp.

74

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

As it is easy to see, V + v + c = 1 . We obtain !- 1 II1° dAdI P

I U(P' AP)I < Vr
f

IrI

v (r + r1

}s-(c+)99 dvQ


x

= 11 f lI

4

dv(2

rn-'p 'rn-tD'

_ (r + rl s (c+1)p

44 &)(AP-)

tq

If1D

9

dvQ (11.8)

where

co(AP) =

(r +

r1)n-(a+)d

rn-CP rn-c0

Jr
dvQ

Estimating cu(AP) by the same means as we estimated J1 in the proof of the lemma in item 2, we obtain

IW(AP)I"' <

IAPI(a-

)v'

/ p
(t+I)v'

p1)n

(p +

pn-tv'Pn-ad

dv

1

< 014-IV k,+K21 RlePp(a)P dpi I

From (11.8) and the estimate which we have obtained for Iw(AP)I, (11.7) follows. We remark that in the estimate Ico(AP)I° for a < j the first term is the important one, while for a > 7 it is the second term. Using (11.7), we pass to an estimate of the integral of I U(P, AP)I4 over Es .

Integrating the inequality (11.7) in the variable j5 and interchanging the

§1.11. KONDRASHOV'S THEOREM

75

order of integration, we obtain AP)I9

f l U(P,

[+A2IAPI(t-gym l P

dv, 5 IIfIILD-° Al

'f ii
,

[Al

=

JJ

)q

dvo

+A2IAPI('-7)v

dv,

10

(r+r,)s-(t+I)9" IIIP

" 11,R

r=-tQdv,

fE

dvo. (11.9)

In the inside integral in the variables of the point P, two singular points Q' and Q AP will be possible. Each of these points for various values of Q may fall either on or outside the plane over which P runs. We estimate

-

the inside integral by a device which does not depend upon the positions of

the points Q and Q' - AP. First of all, applying our previously described change of variables of integration, we obtain (r +

r)s-(+)4d v,

fE I

(P + ps

p1)s-(9+1 )Q"

i4p1s-t9

dvs (11.10)

We decompose once more the domain of integration into the portion lying within the ball p < 2 and the portion lying outside that ball J1 = J2 + J3 =

(p +

p1)s-(e+;)9

ps-eg

p-egdv,.

(11.11)

We shall show now that the integral J2 is bounded for arbitrary positions

of the points Q' and Q - AP. We introduce on the hyperplane a system of polar coordinates, taking for its center the projection of the point Q on this hyperplane. Suppose that this projection is Qo . The distance in the hyperplane P, containing E, to the point Q we denote by 9R. Then p = (912 + h2)112 , where h is the distance of the point d from the plane P, .

76

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

Suppose, further, that the projection of Q - AP' on the hyperplane PS is Q1 , the distance on Ps to the point Q1 is denoted by 911 , and the distance of AP from the hyperplane by hI . Then p1 = 9t1 + hi . For h > a , h, > , the integral J2 will be bounded by virtue of the fact that p and p, will both be bounded from above, as well as from below (by the value 1) . For h < I , h, > a , to estimate the integral we remark that .

p+p1 < 5, p1 > a, and p, > 9t. Thereby Its_ 1

J2
91s'z.

9R<2

d9t=KI,

Q

where K, is some constant. Similarly we estimate J2 in the case in which h > h, < o . Finally in the last case, in which h < a , h, < a , the distance between the point Qo and Q1 is greater than I and, consequently, Jz

- K 4:52 9V-,q

I

dvs,

Sts-`°

i.e., J2 will be a convergent integral. We now estimate the integral J3. We remark as before that the ratio p, /p is included within fixed bounds, and, using polar coordinates in the plane P, , we shall have p-s+(8-1)9-9s-1

J3 < K f 2

r

R/IA#I

Ms-1I

a

1

dt

s+(c- )4

912+h2J

`

d9,

where a = 0 if h > 2 and a = (4-h2)1/2 for h < 2. For h > 2 this integral is a decreasing function of 4. Therefore it suffices to verify the estimate for h < 2. For these values the integrand is bounded on the segment a < 91:5 2. Furthermore, if 91 > 2, we obtain by means of simple estimates: R/IAPI

J3

9tle-

C

C3

R

IAPI

d91

+ C4

= C4 + C51AA

From this, using (11.11) and the boundedness of J2, we find JI < C6 + C5JAPI-(4-1)4

j 1. 11. KONDRASHOV'S THEOREM

77

Thereby, on the basis of (11.10), we obtain J < CS + C6I AP'I (`- q )q'

and if we apply (11.9), we obtain fE I u(°P)IQ dv, +A21APlr`-ilo,lq'/n'

[c5 +

5 Ilfllgp [Al JJ

c6iPi}

From this we conclude easily that 11q*

l

U(P, AP)lq

dv3J

I

ife>

B1

<_ IIIIIL,

ife <

B2I0I2z-1

from which follows 1

IAPI [fE. IU(P,

I/q*

AP)lq. dv,1

<_ BIIfIILokAPI',

where fi = min(1, 2e) and the lemma is proved. 4. Complete continuity of the imbedding operator into C.

THEOREM. If I < p < oc, n < 1p, and the domain El is a union of finitely many bounded domains, each starlike with respect to its own ball, then

the operator imbedding Wprl in C is completely continuous, i.e., for every bounded set {c} c W(r) (IIr01Iw o <_ N) the set {rp} , where the functions c are defined just as in the theorem in 5 of g9, is compact in C. PROOF. From the Imbedding Theorems it follows that {gyp} is bounded in

C. It suffices to prove the equicontinuity in C of the set {0}. We have = S + rp* , the decomposition of 13 by the formula (7.12). Since IISII,, + IIS,IIL(o <_ N for all q' E J411, IISII$ < N and, as a result, the coefficients of the polynomials are uniformly bounded. From this follows easily the equicontinuity of the polynomials S. It remains to show that from II9IIL', 5 N follows the equicontinuity of the ql . We have

Jn rnl

1

a =r

ex°'a 1

x° "

dvo.

(11.12)

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

78

Set

r1 = IQ - (P + AP)I. Then

o,(Q, P+AP)

wa

rn-1

Q.(Q, P)

-

rn-1

a^(r1, 11, P'+AP')

wa

rn-1

-

I n ( r,

-

I

wa .... o (r, 1, P) rn-1

II rn-

(r1,11,P+AP)(r,

rn 11{[wo,

+ Af)]

1

+[wo

p^(r,11,P+AP)-wa

+[wo

1, P+AP)-UI0

o^(r,1,P'+AP)] (r, 1, P)]). (11.13)

(r, 1', P) and its derivatives with respect to r, 1, and f are bounded, while in addition jr, - rI < IAPI , (r + r1 /r) > I, and I /(r + r1) > A > 0 (the domain being bounded), it follows that Since wQ

I

ir

"

1[wo...ofr1,I1,P+AP')-11,P'+AP)]

1

1 -< g

A IAPI

(r + r,)"-1- I rn-lrn-1

rr

1

ICI

1

Analogously we find

re < A`IAP'I

!-I
1

Furthermore, since 1 = (Q - 6)/r and I'1 = (Q - (P+AP))/r1 ,

1 -1= rQ-rP-rAP'-r1Q+r1P rr1 _

-(r, -r)(Q-P-AP)-r1AP rrl

from which it follows that

if, - II <

r1 IAPI rr1

+ r, IAPI = r1 r

2IIr

S I.11. KONDRASHOV'S THEOREM

79

Therefore

rn AI Illrn-- lI < 2A 11API

1

rrn1

I

-I

Ap' < - 2A II

(r + r1) n I-1 I

rn-Irn-! 1

and, finally, 1

wa, ..... a

<

(r' (' P) Gn Al

n-I

r°_I rn-I I A

Ir

rn-1)

11

n-I

- r1

I

= rn 1 Ir1 -rl(rn-1-I +r n-1-2

n-I-2

n-I-1

+r1

)

1

AIAPI

r°_Irn-I(r+rl)

n-1-I

1

Substituting these estimates in (11.13), we obtain way

(0, J6 +A,#)

P)


rn-I

(r +

rl)n-I-I

IAP'I rn-Irn-I

and from (11.12) we find

I

x E

a19

aX t ... aX°

d v -. Q

In the derivation of the inequality (11.14) we never used the assumption that I p > n . We apply the lemma in item 2 to (11.14), putting A = n -1. We obtain n - 2 = 1, (n - A)p = lp, and by the hypothesis of the theorem, Ip > n ; consequently the inequality n < (n - A)p is satisfied. Then

Irp'(P+AP)-I'(P)I
fl =min (1,1-p

(11.15)

from which follows the equicontinuity of p* for a set {{o} bounded in W(I)

since this estimate of the oscillation does not depend upon the particular member io of the family of functions {Ip} . The theorem is proved for 1:5 n (the proof for I > it is analogous, with essential simplifications).

80

1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS

5. Complete continuity of the operator of imbedding in La. .

THEOREM. If 1Ip, s>n - lp, q'<sp/(n-Ip),and the domain (I satisfies the conditions of the theorem in item 4, then the operator

imbedding the space W(' in Lq- on a section of 11 by any s-dimensional hyperplane is completely continuous, i.e., for every bounded set {w} c W(l) ((III') 5 N) the set { qi } is compact in L. on this section. PROOF. The proof of this theorem is essentially a repetition of the proof of the theorem in item 4. It all reduces to the proof of the equicontinuity in the large of the functions 47' in L. if IIcPIILp) <- N. We need to show that Iltv'(Pca)

+AP) - v'(P(s')II". < e

(11.16)

if IAPI < 6(c), where 6 - 0 when a -+ 0. For this purpose, we apply Lemma 2 to the right side of (11.14), putting A = n - I ; then the inequality n > (n - A)p is satisfied since by the hypothesis of the theorem n > I p . We obtain, using the inequality (11.6): +A,#)

-

=min

< KII,pIIL(, IAPI' ;

II;9 -(

(11.17)

r

_1)],

from which there follows (11.6). Theorem 2 is proved. REMARK. Suppose that on the domain Cl there is given a summable func-

tion 9 (x, , ... , x,). Let E c 0 be a smooth manifold of dimension s. We shall say that p(x, , ... , is continuous in the sense of L9. s if

f p(P+AP) - o(P) dE -.0 for Ifli -+0,

(11.18)

for any manifold E for which only the translation of E by the vector AP' is contained in Q. From the theorems proved above it follows that for every rp E Wot) the function 0 is continuous in the sense of L. s if s > n - lp and q' < sp/(n - lp). If n - Ip < 0, then go is continuous in the ordinary (29)_ (43) sense after modification on a set of measure zero.

CHAPTER 2

Variational Methods in Mathematical Physics §2.1. The Dirichlet problem

1. Introduction. As is well known, the equations of mathematical physics often appear as the Euler equations for certain variational problems.

In the calculus of variations, when looking for the extremum of some functional of the form

faF

(u az , XJ di2+

is

I u, 8x. ° xi

dS,

where S is the boundary of the domain 0, in some class of functions, one finds the solution of some boundary value problem for the corresponding Euler equation. However, these variational problems for the extrema of functionals may sometimes be solved by direct means. It is natural to ask whether conversely in those cases where the basic equation is an Euler equation, one can reduce the given boundary value problem to a problem of the calculus of variations which may then be resolved by means of the direct method. This is precisely (44) the idea of variational methods in mathematical physics. We begin the investigation of variational methods in mathematical physics with the study of the simplest equation of elliptic type, namely the Laplace equation: Au=0. (12.1)

We consider for this equation the Dirichlet problem, i.e., the problem of finding the harmonic function taking on given values on the boundary. Let 1 be a bounded domain of n-dimensional space bounded by a surface S which is simple in the sense defined above in Chapter 1, §10, item 2.

We consider on 0 a function u(x,, x2 , ... , x,,) which is summable and has square-summable generalized derivatives of first order. Let D(u)

r" i_

1

8xi

df2 < oo.

(12.2)

This means that u E W.0), and consequently, by the imbedding theorems, 81

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

82

u E L2 on every (n - 1)-manifold since 2
n-2 Moreover, one may assert by virtue of the complete continuity of the imbed-

ding operator that if a piece SI of some (n - 1)-dimensional manifold is translated by a vector A7 so that it remains within fl, then

f Ju(7+A7)-u(7)12dv7

JsI

(12.3)

if

JAPE- 0. It is obvious therefore that not every function q E L2 given on the surface S can be the limiting value of some function v E W(I) given in the interior of the domain. Indeed there also follows from the imbedding theorem the summability of the limiting value v on the surface to any power less than 2(n - 1)/(n - 2). Later we shall see (cf. item 5) that such summability and even continuity of the limiting values is still not sufficient in order that such a function may appear as the limiting value of a function in W.(') . Let us agree to call a function 9 given on the boundary S of the domain permissible if there exists a function v in W(I) for which SD is the boundary value. (45)

The Dirichlet problem consists in seeking a harmonic function in WZ I) which assumes on the boundary the given permissible value ip in the sense indicated above: (12.4)

ups = 10.

Let us proceed to the solution of this problem. For this purpose we first solve a variational problem and then show that the solution of the variational problem turns out to be a solution of the Dirichlet problem. 2. Solution of the variational problem.

We denote by WZ I)(cp) the set

of functions v in W(I) which assume the value ? on S. Since p is a 3' (r,) is not an empty set. For each v e W(I)(lo), we have 0 < D(v) < oc, and therefore there exists a greatest lower bound for the values of D(v) :

permissible function,

d =

inf

D(v),

d > 0.

(12.5)

vE W,"(U)

From the set W (I) (rp) , we may choose a sequence {v k } for which

lim D(vk) = d,

(12.6)

as follows from the definition of greatest lower bound. The sequence {vk} is called a minimizing sequence.

§2.1. THE DIRICHLET PROBLEM

83

THEOREM 1. The minimizing sequence {vk } converges in W21 1 ; the limit

function lies in Wlll ((y) and yields a proper minimum for the functional D(v) among all such functions. PROOF. Indeed, we define the norm in

W213

by the formula

{[fvds}2+D(v)}h/2,

IIvlIWzs> =

(12.7)

obtained from formulas (7.5) and (9.8) for

(h, v) = jvdS. From the equality f(Vk_Vm)dS=0

we obtain IN - vmIIW I) = [D(vk - Vm)}1/2.

The convergence of {vk} in WO) will be proved if we can show that

D(vk-vm)-0 fork,m

oo.

Let t > 0 be given. We can find N > 0 such that D(vk) < d+e if k > N. Let k and m > N. Obviously, (vk + vm)/2 E W21)(P) ; and therefore D

(Vk

2vm) >d.

From the obvious equality D

(vk 2 V-) + D (vk

follows the inequality

\

2 Vm)

= D(vk) + 2D(vm)

/ d+D(vk


2vm) DI Vk 2vm) <

or D(vk - vm) < 4e,

and, consequently, D(Vk - Vm)

0

fork , m

oo.

From the completeness of the space W21) it follows that {vk} converges to , i.e., Ilvo - vk II W= some function vo in 0, k - + oo. We shall show W21 >

84

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

that D(vo) = d. For this purpose, we note that ID(vo) - D(vk)I =

8 v° axi fn?{(:i

C `((

fn?R8xj

avk ] dQj ax

2

I

- ex;) (ax; avk

ax; )

x; `If(8v° 8vo

[In (ax;

-

+ 8vk

8x;)J dQ

avk

(\av° ax; + ax; )

da l

8vk 1vk 2J [i 8xi) dS2J

n

2

W2')

k-1 <-

rn(ax \

["

,

2

8x; + ax;) d I/2

ex l/ df2

IN - VkIIWwu) ,

from which there follows

D(vo) = kim D(vk) = d. -oo

It is not hard now to show that vo E W, ("(,P) .

Indeed, vo E W ( ' ) and therefore, the function vo has a sense on every (n - 1)-dimensional manifold and on every such manifold lies in L2 . The value of the function vo on the surface S will be equal to c . Indeed,

f(vk

- vo)2 dS < IIvk - v°II

3;O11

and consequently

(vk - vo)2dS - 0, but vk IS = t° , and thus,

j(c- vo)2dS=0. The function vo takes on its boundary value p by converging to it in the mean as was shown in the imbedding theorems. Thus vo E W ( I ) is such that 1)

2)

v°IS=41 D(vo) = d.

Therefore the variational problem is solved.

(12.8)

§2.1. THE DIRICHLET PROBLEM

85

3. Solution of the Dirichlet problem.

THEOREM. The function giving a minimum to D(v) on W(1 )(o) is the solution of the Dirichlet problem. PROOF. We shall show that vo in the interior of S2 has continuous derivatives of arbitrary order and satisfies equation (12.1). Let E W(1) , js = 0. Consider (12.9)

D(vo) + 2AD(vo,

where

ev e axi ax

fo

By virtue of the fact that vo + 9 E W(1)(q) , we have D(vo + A) > d = D(vo) and therefore (12.9) has a minimum for d = 0. By the theorem of Fermat, we have

D(vo,) = 0.

(12.10)

We shall choose a (x1 , ... , x^) of a special form. Let w(,) be such that w(>r) = I for 0 < s ) < , vol) = 0 f o r , > 1 , and {v(q) is monotone on [11 , 11 and has continuous derivatives of arbitrary order for all q E [0, oo]. For example, put

- 317

(

{v(q)=211+tanh

2
Consider some interior[[point M. of the domain i2; the distance from it to an arbitrary point we shall denote by r ; suppose Mo is at distance b from S. We choose two numbers h 1 and h2 ,

0
r nLw\h1I-w\hz

(for n = 2 the proof is analogous, with

set equal to

r Inrlw\h1/-W\h21J/ r

It is obvious that js = 0 by virtue of the choice of h, and h2 . In addition, 4 = 0 for r < h1/2 and consequently the function 4 is continuous and has W,11). For such 4, (12.10) continuous derivatives of all orders, and thus 4 E holds. By virtue of the definition of the generalized derivative 8vo/8x; , we have 2

fn avadf2 ;

;

fvoe2dSl. in 8x

Therefore the equality (12.10) gives

JvodQ=O.

(12.11)

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

86

But =A/r2_"V

11\hi//-Mr

"W

(

\h2

11 hI w

l

(

\hl/

hi

to ( l \h2

where hnA (r2

-"y, \h;h'/

\hr/

[(r)2

and it is obvious that the right-hand side is a function only of r/h; . Using the fact that yI(r/h1) = 1 for r < h,/2 and Are-" = 0, we obtain

w(h =0 forr< 2' andr>h;. ,/ Thus g is the difference of two arbitrarily often continuously differentiable functions on the whole space, and equation (12.11) gives

1" r h,

o

/

v°w I

r) dig = h2 In vow (r 1 Al. h2 n

(12.12)

hl

Multiplying both sides of (12.12) by 1 /[(n - 2)o"] where o" is the surface area of the unit sphere in n-dimensional space, we obtain

fn vow (i-)

(n -

i

dig(n -

2)o"hfnv°(h)

d.

(12.13)

The function ((n - 2)oh")-Iw(r/h) may be considered as an averaging kernel (see Chapter 1, §2, item 4), since its integral over the whole space is equal

to 1. In fact,

(n-2)oh" fw l h) dn- (n-2)a" (r

1

_

1

8(r2 "W(>;))

1

(n - 2)a" Jr.h 1

f 8(r2 "w(x)) dS

(n - 2)0" r.4

Or

O" 1

n

f

/r.4\

r

2 jA(r

Or

di2 dS

rl -" dS

2 II " (h J"_l o" = 1.

Using this, the equation (12.13) may be rewritten in the form (vo)h, = (vo)h=.

(12.14)

We see that the averaged functions for vo do not change with a change in h

(if h < 8) at points lying at a distance greater than d from the boundary, and consequently the limit of (vo)h coincides with (vo)h , i.e., (vo)h = vo Since (vo)h has continuous derivatives of all orders, the same is true for vo.

§2.1. THE DIRICHLET PROBLEM

67

Suppose now that is an arbitrary function continuous with its first derivatives in S2 and null outside some interior subdomain. Then obviously an integration by parts gives .

fa

4AvodC1=0,

from which by the arbitrariness of c , one concludes Avo = 0,

i.e., vo is a solution of equation (12.1) and, as was shown earlier, assumes on S the values of rp (in the sense of L2,(._,)). Thus vo is a solution of the Dirichlet problem. (46) 4. Uniqueness of the solution of the Dirichlet problem. THEOREM. The solution of the Dirichlet problem in the indicated formulation is unique.

We establish as a preliminary one important lemma. Define the function

_

in i12h ,

I

2h

0 outside f22h , where Std is the collection of points in S1 whose distance to S is greater than 6. We form the averaged function for `P2h by means of the kernel W(r/h), where W is the function introduced above, and

K=an

n-I

(I

Jo

n

W(n)dn.

This averaged function is denoted by Xh :

--

-

1

Xh(P)= xh" 1.
E W.(') is continuous along with its partial deriva-

tives of first order in Q, and js = 0 in the sense of L2 _I . Then for any function v E W21) , we have the equality

D(v,

limD(v,X6).

(12.15)

PROOF. Obviously Xh has continuous derivatives of all orders, Xh = I on f13h , and Xh = 0 outside II.. From the equation 8X6

8xi

-

r

1

Jr

Xh"
8r

1 h)8xihdvP.

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

88

we conclude that

axhl < h

(12.16)

The function tXh has continuous derivatives of first order in 0, and is equal to zero outside 11h and equal to in i23h . We need to show that

forh-+0,

(12.17)

for any v E W. M. We estimate this expression. We have

19V a( Xh) df2

D(v, - Xh) =

fEax;

ax;

av al(1-x8)- aX ex; [ax; ax; J

_

-n -

)di2,=I lr

av a(1

L03.

dil

ax; ax;

aA_n3h

i=='

ax; ax;

Iz I/2

As is easy to see, I 1 - Xh I < 1 , and therefore the first integral on the right side does not exceed

(Ov)2

f

r

da

( ax, E i=j

dig

and therefore converges to zero as h - 0 by virtue of the convergence of the

integrals D(v) and D(4). Let us consider the second integral. We have by virtue of (12.16): "

InA-n,A

av

E i-I 8x;

a8Xh

x; h

<_ h

<

h'

dt2

nA-nH

ICI

;=I

f

illn nA-nJA

AVII Al

/ l

lal \

8x)

df2

1

42dil f

I

nh-n1A

n

2

(aX) Al.

Because of the convergence of D(v) ,

f

(PiL)2dn_o fn1=I ax; It therefore suffices to show the boundedness for h -s 0 of the expression

In z

h

-am

2dQ,

(12.18)

§2.I. THE DIRICHLET PROBLEM

89

and the lemma will be proved.

Let S' be a domain in the plane yn = 0 and S2 a cylindrical domain in the n-space of (y1 , y21 ... , yn) given by the inequalities

0 0,

=

I

8 dynl

faYn

2

2


(8 \aY) dyn. n

Integrating over Y, we obtain

fshI2nnn)12dS' l

Ayn f f.

ayn) dyn ds'

(12.19)

12

< Ayn f cl'

I

aY

d Q'

<_ AYnIIII;,,I,,. 2

Taking yn -* 0 and replacing Ayn by P, we obtain

f ,K(YI,Y2,...,yn_I,p)12dS<-Pi IIvw. Integrating in p between the limits Ah and Bh (A < B), we find Bh

I Bh f (YI,Y2,...,Yn_I,p)I2dSdp
1e12dcl
(12.20)

where the constant K does not depend on h . We now prove the boundedness of (12.18). Let ho > 0 be a sufficiently small number. We decompose the domain Cl - Q. into a finite number of intersecting domains V. (i = 1 , 2, ... , k) such that each piece is "based"

upon some piece S. of the surface S. For each V,, by virtue of our assumptions about the surface S, we can find a one-to-one continuously differentiable transformation of V. onto a cylindrical set Q, with base Si such that the transformation has bounded first derivatives. For this domain V (S2h -1Z3h) (the intersection of V, with Ph - i23h)) is mapped into some domain lying in the strip A;h < y,, < B,h.

90

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

Applying (12.20), we conclude the boundedness of (12.18) for each V and therefore for the whole domain f2h - f23h . Thus for every v E W21) and E W21) (HIS = 0) we have

D(v, C) = li mD(v,'Xh) and the lemma is proved. t PROOF OF THE THEOREM. Let us assume that in addition to v0 there exists

another function u E W21)(9) such that Au = 0. For such a function D(u) > d. If D(u) = d, then u could be interspersed infinitely often in a minimizing sequence converging to vo , and then we should obtain the conclusion that vo = u since the resulting minimizing sequence must converge and have the same limit as each of its subsequences. Thus if we demand that u be different from vo, it is necessary that D(u) > d. We shall now show that this is not possible. If U E W21) and in addition Au = 0, and is the function from the lemma which we have just proved, then

This equation is obtained by an obvious limiting process from D(u,x*) = 0 by virtue of the lemma. Furthermore, D(u + A4) = D(u) + 2AD(u, 4) + A2D(F, )

= D(u) +A 2

>D(u)>d by virtue of the condition D(u) > d. On the other hand, putting A = 1 and

4=vo-u (41, = 0), we find D(vo) > d, which contradicts the fact shown earlier that D(vo) = d. The proof of the theorem is complete. 5. Hadamard's example. In conclusion, we present an example due to Hadamard showing that summability and even continuity of a function given on the surface is still insufficient to ensure that the function can be a boundary value for a function from W21) .

Let Q be the circle x2+Y2 < 1 in the (x, y)-plane and (p, 0) polar coordinates in this plane. I Formula (12.20), and with it the whole lemma, may be proved not only for functions C having continuous derivatives in the interior of the domain but for arbitrary functions in W21j vanishing on the boundary. For this purpose it suffices to take the limit in (12.19), replacing by its averaged function 4h .

§2.1. THE DIRICHLET PROBLEM

91

Let 0C

w(B)=E

cos n48 2

n

n-1

Obviously, 9(8) is a continuous function and the function 00 cosn40

u1(P,0)=E

2

R=1

P

n

is a harmonic function in the open circle x2 + y2 < 1, continuous on its closure, and coinciding with to(8) for p = 1 . Furthermore, we have (,,U,)21

ff"

ax fpo =

dxdy

Oy

[I 2n a {() + n4p2n'-I dp E

p

()2} do] dp

2

JJ

00

= 2n fo °0

n=1

00

=nypa" ~oo forpo n=1

from which it follows that u1 V W21) in S2. If 9(0) were a permissible function in the sense of item 1, then, solving the Dirichlet problem by the variational method, we would find a harmonic function u2(x, y) E W(1 such that 2x

Ju2(p, B) - p(B)l 2 d8 - 0 for p -» 1,

f and therefore

f2x Ju2(p,

forp - 1.

o

The same condition holds for u1 since u1(p , 0) is uniformly continuous. Therefore (12.21)

for p- 1. Let po < p < I. Then by the Poisson formula for harmonic functions on the disk, we have u2(Po, Bo) - u1(Po, Bo) I

f

2n o

2n

P2

- Po

po-2ppocos(B-Bo)+p2(u2(p, 0)-u1(P, 0)]dO.

(12.22)

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

92

For fixed po and p -. 1 , the function p2-p02

PO-2popcos(0-0o)+p2 remains bounded and therefore, on the basis of (12.21), the right side of (12.22) converges to zero for p - 1 . Since the left side of (12.22) does not depend on p, it must equal zero and by virtue of the arbitrariness of po and 60, we have uI u2 . This is impossible, since uI f W") . As a result, 'F cannot be a permissible function. (47) §2.2. The Neumann problem

1. Formulation of the problem. We have considered for the Laplace equation the simplest problem, the Dirichlet problem. Let us now pass to the study of another basic problem, the Neumann problem.

Let n be a domain of n-dimensional space bounded by a sufficiently smooth surface S. Let u E WO) . Consider the functional H(u) = D(u) + 2(p, u),

(13.1)

where

5 )2 dl) D(u)= fE ( 0xi i-I

and (p, u) is a linear functional on W('). In all the following, we shall assume that

(p, 1) = 0.

(13.2)

THEOREM. If (p, 1) = 0, then H(u) is bounded from below.

PROOF. If u-v = const, then (p, u) = (p, v) , i.e., (p, u) has a constant value on each class lv E L2' . By virtue of the linearity of the functional p, we have

1(p, u)15 Mllullwv

Since for every class w E LZ'l, (p, u) has a constant value, while for one of the functions uI of that class IIuI IIW 1 = IIUIIIL=! , it follows that

I(p, u)I =1(p, uI)I 5 MIIu1IIL'm =MIIuIIL:1, = M D(u). Thus we have

H(u) = D(u) + 2(p, u) > D(u) - 21(p, u)I

> D(u) - 2M D(u) _( i.e.,

D(u) - M)2

- M2 ? -M2

H(u) > -M2.

,

§2.2. THE NEUMANN PROBLEM

93

The theorem is proved. As a result, there exists a greatest lower bound for H(u) which we denote

by -d: inf H(u) = -d.

(13.3)

2. Solution of the variational problem.

THEOREM. There exists u E W0) such that H(u) = -d. In addition, for an arbitrary E W2' , we have the equality

D(p, ) + (p, ) = 0.

(13.4)

PROOF. Let {vk} be a minimizing sequence, i.e., Vk E W2(l) and

lim H(vk) = -d.

k-oo

Then it is obvious that D (Vk 2vm) = [D(vk)+2(p, Vk)]+

2

J

\

vk 2vm/1 \\

= ZH(Vk)+ IH(vm)-H(vk?vm) If we choose k and m sufficiently large so that H(vk) < -d+e and H(vm) < -d + a, and take into account the fact that -H((vk + vm)/2) < d, we have vk - vm -d + e -d + e D( 2 < 2 + 2 i.e.,

D(vk-Vm)<4e. Thus {vk} converges in L2('). By virtue of the fact that H(u) has a constant value on each class v E L(') while for one of the functions u1 of that class II u i IIW, ,, = II u 1 II L= > , for a minimizing sequence we may always pick a sequence

for which Ilvkllw=-> = IlvkllL"

For such a choice {vk} converges in W2"'. Let U E J4'

be the limit

function. Proceeding as we did above in the consideration of the Dirichlet problem, we obtain I H(u) - H(vk)I = ID(u) - D(vk) + 2(p, u - vk)I < lD(u) - D(vk)I + 2l(p, u - Vk)l <- (Ilulll2, + IlvkllL())IIu - VkIIL2> + 2Mllu - VkIIW=o ,

therefore,

lH(u)-H(vk)I-.0 ask -oo,

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

94

from which it follows that H(u) = limk.oo H(vk) = -d. Let

E WZ 1) .

Then

H(u + 9) = H(u) + 21[D(u, ) + (p, )] + A

= 0, and by Fermat's theorem we have

D(u, )+(p, ) =0. The theorem is proved. Each function u E W(1) is summable on S to any power q' where

q
_ 2(n - 1)

n-2

We denote by L. (S) the set of functions v defined on S and summable to the q'th power. Let (ps, v) be a linear functional on L4. (S) . Then by the imbedding theorem (ps, uls) is linear on WZ I) Indeed if u E W(I) then I(Ps, uls)I S K1IIUIIL,-(s)
where KI , M, and K are various constants. The inequality so obtained establishes our assertion. 3. Solution of the Neumann problem.

THEOREM 1. If (ps, 1) = 0, then there exists a function u E WZ 1) such that (1) In f2, u has continuous derivatives of all orders and satisfies the equation

Au=O.

(13.5)

(2) Let f1k be an arbitrary increasing sequence of domains having sufficiently smooth boundaries Sk contained on Q and converging to Cl. Then

8avdSk u

(13.6) = -(Ps> ), kin` sk is an arbitraryfunction. where v is the exterior normal to Sk and E W(1) 2

The problem of finding such a function u we will call the Neumann problem.

PROOF. On the basis of Theorem 2 there exists a function u E that for anyi;E W21).

W(1) such

(13.7)

0 on some strip around the boundary S, then (ps, i;) = 0 and we obtain D(u, i;) = 0. From this it follows by a literal repetition of the argument of the preceding paragraph that u is infinitely continuously differentiable and satisfies the equation Au = 0. The first condition is established. If we choose

§2.3. POLYHARMONIC EQUATIONS

Let

95

be an arbitrary function from W.(') . We then have

r au a D(u, )=kim-oc J0,Eax;axidi2k = k'm Ifk kloc

TV

J

dsk -

J}

ctiudS2J

avdsk.

Sk

Taking (13.7) into consideration, we obtain (13.6). THEOREM 2. The solution of the Neumann problem is unique up to a constant term.

PROOF. Suppose that there exist two functions u1 and u2 E W2W satisfying the equation Au = 0 and the same condition (13.6). Their difference w = u1 - u2 is a harmonic function and satisfies the condition kim

J avdsk=0. Sk

Putting

= V, we find

lim f W av dsk = kim f (grad wl2dQk = D(W) = 0, k

nk

from which it follows that w = const. REMARK. If (ps, u) = fs Vu dS, then the "boundary" condition (1) takes the form

f p ds+ lim fk

avdsk=o.

The functional (ps, ) represents a generalized boundary condition for the function u, and the quantity su/av converges to (in the weak sense. Such a formulation is completely natural in the given situation since a limit value for au/av may not exist if u E W21) . (4s) §2.3. Polyharmonic equations

1. The behaviour of functions from W(") on boundary manifolds of various dimensions. The variational methods considered above may be carried over to boundary value problems for polyharmonic equations. We pass now to the study of the basic boundary value problems for such equations. Suppose that the bounded domain Cl of n-dimensional space has a simple boundary S (see Chapter 1, § 10, item 2). The equation

m(14.1) ()mu=O

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

96

or

a2mu

m!

Eb=maIla2!... and aX bj8X2a2... axn2n

is called a polyharmonic equation. The equation (14.1) is the Euler equation for the functional D(u)

amu

m!

=ln Ee=m

aI!o2l...anl ax;1ax2=...axn.

Al.

(14.2)

Obviously D( = IluH D(u)

" .

Let u E W (m) . If n - 2m < 0, it follows from the imbedding theorems that u(xl , x2 , ... , xn) is continuous and has limit values on manifolds of arbitrary dimension. If n - 2m > 0 and n - s > n - 2m, then u(xl , ... , xn ) is summable on manifolds of (n - s) dimensions to any power q' < q = 2(n - s)/(n - 2m). Since q > 2, we may take q' = 2. Therefore if n - s >

n - 2m > 0, then u E L2.,-, on manifolds of (n - s) dimensions, which we shall denote by Sn_s Thus, u E L2 n_3 on S,,-, if

s < 2m.

(14.3)

Moreover, from the complete continuity of the imbedding operator it follows that u(xl , ... ) xn) is continuous in the sense of L,,,-, and therefore has limiting values in the sense of L2,,-, on portions of the boundary of

S having dimension n - s, where s < 2m. As was shown in 2 of §10, u(xl , ... , xn) has all generalized derivatives of order less than m. We consider the derivatives of kth order, where k < m. If n < 2m, the derivatives of kth order belong to L2 on manifolds Sn_3 of dimension (n - s), where

s < 2(m - k).

(14.4)

From the complete continuity of the imbedding operator, it follows that the

derivatives of kth order are continuous in the sense of L2,,-, where s satisfies (14.4) and therefore have limiting values on portions Sn_, of the boundary in the sense of L2 n_, . From the inequality (14.4) it follows that

k<m-121 -1,

(14.5)

i.e., on manifolds of dimension (n-s) there exist limiting values in the sense

of L 2

_,

for derivatives up to order m - (s/2) - I.

§2.3. POLYHARMONIC EQUATIONS

97

for functions and all derivatives up to order (inclusive):

On the manifolds of the boundary:

there exist limiting values in the sense of:

Sn-1

L2,n-1

Sn-2

L2 n-2

Sn_3

L2 , n-3

m-1 m-2 m-2

Sn_4

L2 n-4

m-3

Sn-2k

L2, n-2k

S.-2k-1

L2,n-2k-1

m-k-1 m-k-1

Sn-2m+2

L2,n-2m+2

0

Sn-2m+1

L2,n-2m+t

0

We form these results into a table.

This table is formed under the assumption that n - 2m + I > 0. If n - 2m + 1 <0, then for some k either n - 2k =0 or n - 2k -1 =0. Then we have m - k - 1 < m - (n/2), and as a result, the function itself and all its derivatives up to order m - k - 1 are continuous. The table ends in the following now: on So there exist limiting values of the function and all derivatives up to order m - k - 1 . On the manifolds S1, S2, ... , Sn_1 , all the derivatives up to the (m - k - 1)st order are also continuous.

Obviously from the continuity of a function there follows its continuity in the sense of L2 ,,_ . Therefore in the following estimates ordinary continuity will not be mentioned and all arguments will be carried out under the assumption of continuity in the sense of L2,,-, 2. Formulation of the basic boundary value problem. Suppose that the boundary of the domain consists of the manifolds Sn_1, Sn-2 1 ... , Sn-2m+1

if n-2m+1 > 0, or of the manifolds Sn_1, Sn-21 ... , So if n-2m+1 <0. Some of these manifolds may be absent. Suppose now that on each of the manifolds Sn_, we are given functions

an-S)

where 0<Ea;<m

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

98

If there exists u E W2 ') such that °, ..... (, ,

ax°, ... ax" I

(14.6)

then we shall say that the system of boundary values

is permissible. (49)

For any permissible boundary values, it is obvious that if n - 2m + 1 > 0, then all the goo"-' E L2 and if n - 2m + 1 < 0, then all the functions for which 0 < E a, < m - [n/2] - I are continuous, and the rest belong to

L2.n-s'

For the polyharmonic equations, in contrast to the Laplace equation, we may not only give boundary values on z boundary surface of dimension n -1 but also give permissible boundary values on a surface of lower dimension, as a simple example shows. We consider the equation A 2 u = 0 in a three-dimensional space. We have

n = 3, m = 2, n - 2m + 1 = 0. In the role of boundary manifolds we necessarily have S2, and perhaps SI and So. Since m - [n/2] - I = 0, the function u will be continuous. On the manifold S2, we prescribe uls continuous and au/axes= E L2 , which must be permissible. On SI and So, we prescribe the function u. Let f2 be the ball of unit radius with its center omitted: 0 < x1 +x2 +x3 < 1 . In our example SI is missing. We consider the solution satisfying the conditions

u(0, 0, 0) = 1 (given o n So), u I

..,

=

au ar

= 0 (given o n S2). ,

I

The data given on S2 is obviously equivalent to the following: u s,

=

au 8x1 s,

= au

axe s,

= au

0. 8x3 s, =

The function u =(I - r) 2 is a solution of the problem. Indeed, it satisfies the equation with the boundary conditions. At the point r = 0, the derivatives do not exist. The second derivatives are square summable on fl, i.e., u E W22) As we shall show later, there are no other solutions in W2'). If we omit So, then the only solution is u = 0. 3. Solution of the variational problem. We proceed to the study of the basic boundary value problem for the polyharmonic equation in general form.

§2.3. POLYHARMONIC EQUATIONS

THEOREM. If the system {rp("-')

}

99

is permissible, then there exists a

unique function u E W("') satisfying the conditions (14.6) and giving a minimum to the integral D(u) among all such functions.

a^} the set of functions v E W(m)

PROOF. We denote by

is permissible, the set W(m) x

satisfying (14.6). Since the system rpQ"-S

is nonempty. For each function v in this set we have 0 < D(v) < cc. There must therefore exist a greatest lower bound for the values D(v), which we denote by d :

d = infD(v), From the set W

rpQ"-S)

Q^

v E WZ'"){sp("-J) o^}.

} we may choose a minimizing sequence {vk }

such that

lim D(vk) = d.

(14.7)

k-oo

We shall show that {vk} converges in W(m) In order that one may define some natural norm on the space W("') 2 we have seen, it suffices to give a system of linear functionals

as

bounded in one of the natural norms and such that for an arbitrary linear combination

p = E Ap p=,....p^pp,.s.....p^ Eu,<m-1

with not all coefficients zero one can find at least one monomial xI' x° for which this combination does not vanish. Consider the surface Sn_ . By assumption, this surface consists of a finite number of smooth pieces. Put pp,,p=.....p^v = Js^_,

ax,' ...axn

dS"-1,

(14.8)

where v is an arbitrary element of W(") , E=1 f, < m - 1 . It is not difficult to see that all the functionals pp, p, p^ are linear on WZ'") if the norm is defined on the latter by means of a spherical projection operator. This follows immediately from the imbedding theorems. Now let p = E Ap, P. pp, p^ be an arbitrary linear combination of the functionals Let p+o, be one of its non-null coefficients for which the sum p,(°) + fl2) +

+ f(°) = a(01 has its least value among all o,

the sums fl1 + P2 +

+ P,, = a. Then

rot

x2 2

ro)

xR' 0 0 .

100

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

Indeed, for (91

(fl(°) ,

f,I(°)) we have that

,

... xa

I

o1

=0

1

8xr'

Bxp.

fas

because '3k > fl (o) for at least one k, since pI + Therefore, we get that ^o,

roi

px ... xn =

W2 m)

by the formula

_ E(p,,,v)2+D(v),

Ilvllw,

+ pt,°j .

dSn_I f j°jI ... fro 1 96 0,

Ad

required to prove. Thus, we can define the norm in

+ p >- p(°) +

(14.9)

N

where p denotes P, with Fn., ft1 < m -1. Then for two functions Vk and v, of the minimizing sequence we get that pµvk = p,,v,, and thus llvk - vrll W;.' = D(vk - v,).

We choose k and l so large that D(vk) < d + e and D(v,) < d + e, as is possible according to (14.7). Obviously, Vk - V1 2

E W(m){rp1" 2

a)

}

) > d. From the preceding equality

and therefore D((

v,l D(vk2 we find D

l

ZD(vk)+2D(v,)-D(vk+v,l

=

(Vk - v,< d+e+d+e-d=e, 2

2

2

i.e., D(vk - v,) < 4c, from which by the arbitrariness of a it follows that

k,I-»oo.

(14.10)

Since the space W2 m) is complete, we conclude that there exists a limit function u E W 21m) for which Il u - Vk I I W=", -' 0. Just as in the Dirichlet problem, we show that 1)

u E W2 m) { rq.....

(14.11)

(14.12) D(u) = d. There do not exist two different functions satisfying (14.11) and (14.12). Indeed if u1 and u2 were two such functions, then the sequence u1, u2 , U1 , u2 , u1, ... would be minimizing and therefore would converge, which would only be possible if u1 =U 2. The theorem is proved. 2)

§2.3. POLYHARMONIC EQUATIONS

101

4. Solution of the basic boundary value problem. THEOREM. Thefunction u giving a minimum for D(v) in W2(m){9(."-s) °.) has continuous derivatives of all orders in the interior of Cl and satisfies equation (14.1). E W(m)(0) . Then

PROOF. Let

u+) E

.1= const,

and hence D(u +. ) = D(u) + 22D(u, 4) + A2D(4) > d for all A and has a minimum equal to d for A = 0. From this it follows that d(D(u + 1. ))/dJ.jA.0 = 0, which gives

D(u, ) = 0.

(14.13)

We consider the elementary solution of the polyharmonic equation

g(r)

_ (r2m-" Sl

if 2m - n < O or n is odd, r2m-" In r if 2m - n > 0 and n is even.

It is easy to verify that Amg(r) = 0 for r # 0. Let 6 > 0 be a sufficiently small number. We consider the domain i26 and form the function = g(r)[w(r/h1) - w(r/h2)], where 0 < h1 < h2 < b and w is the averaging function considered earlier (see gl2, item 3). From the properties of this function we conclude that

= 0 for r < h,/2, r > h2 ,

and that all the derivatives of C are continuous. If the point from which r is calculated lies within i26 , then and all its derivatives vanish on the boundary of Cl. Therefore (14.13) holds for 4. Since has continuous derivatives of all orders and is zero outside of f26 , by the definition of the generalized derivatives 9mu/ax*' ... ax.*- we have

f

n OX0

8m

8 mu

u

=(-1)m

8x ax; axRa aXI dQ

8 2m4 d ...axn°.

Hence equation (14.13) gives 1a2m.

1

D(u,

(-1)m

n

u

E°a,"

2m

On! ax2°a... ax2°. Al = 0 n

I

or

Inm dC=0.

(14.14)

But

Amy = Am [g(r)w ()J

- Am

[g(r)w (h)]

and since w(r/h,) = 1 for r < h1/2 and Amg(r) = 0,

Am [g(r)ey (i-)] = 0 for r < 2'

(i = 1, 2).

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

102

Hence, equation (14.14) may be put in the form

fuA" [g(r)W (rhI)] d2= JUAm [g(r)W (r )J d1, t

(14.15)

h2

where both sides of the equation make sense.

Let n > 2. Consider the function

w(r, h)

r [$(r)W 2)AQ"Am

(n -

11

(h/1 '

where a,, is the surface area of the unit sphere in n-dimensional space and

A is the constant from the equation Am-Ig(r) = A/r"-2. It is obvious that w(r/h) has continuous derivatives of all orders and is equal to zero for r < h/2 or r > h, since the same properties hold for Am{g(r)W(r/h)} . Furthermore, fn w(r'

h)d

(n-2)Aa,

f

Am

[g(r)W (jr)] )dig

ds2 (n - 2)Aon f 0A lAm-I [g(r)W (h)] J

= (n - 2)Ao" L=h

d

JA--' [g(r)y,

(h)]

dS

{Am-I [g(r)W (h)] } dS

(n - 2)Aon ,.. dr h n-l = (n - 2)o"A dr 1Am-I [g(r)v, (h)] },=n (Am-I

(n - 2)v"A dr

[g(r)W (h)} },.}

.

Since W(r/h) and all its derivatives vanish for r = h, the first term must be zero. The second term differs from zero((, since W

)I,=4

wtkl lh)Ir,

1,

-0 (k> 1).

As a result, we obtain

()n-I d r[Am-18(r))W (r) t r w(r, h) A) = (n-2)Adr h

d

A

}I

,=a

r

l (n-2)Adr [r"=2W (h)JI.=4

--( h "-I (n-2),(r)+ n-2 I-rn-

Ihr""2 I

h

Thus fn w(r, h) dig = 1 , which shows that w(r, h) is an averaging kernel (cf. Chapter 1, §2, item 4). We multiply (14.15) by 1/[(n-2)Ao,), rewriting

12.4. UNIQUENESS OF THE SOLUTION

103

it in the form

fuw(r. hl) dig =

fuw(r , h2) dil.

This equation shows that the averaged functions for u do not change on i2a for a change in the averaging radius h (h < d) and therefore on fl,', u is equal to its averaged functions. Since the averaged functions have con-

tinuous derivatives of all orders, the same must be true for u. In view of the arbitrariness of 6, we conclude that u has continuous derivatives of all

orders at all interior points of it for n > 2. The case n = 2 is handled similarly.

Let C have continuous derivatives up to mth order in fi and vanish outside some closed domain entirely contained in Q. For such a function (14.13) must hold, since obviously E WZm){O} . Integrating by parts in (14.13), we find fo Emu Al = 0, from which, in view of the arbitrariness of , there follows Emu = 0. We have shown that u is a solution of equation (14.1) with the conditions (14.6). We shall call the problem just considered the basic boundary value problem for the polyharmonic equation. §2.4. Uniqueness of the solution of the basic boundary value problem for the polyharmonic equation 1. Formulation of the problem. THEOREM. The solution of the basic boundary value problem (14.1), (14.6) is unique in WZ m)

PROOF. If we assume that there exists in WZ') still another solution w of equation (14.1) with the conditions (14.6), then we must have D(w) > d, since in the contrary case we may construct a minimizing sequence by interspersing with repetitions of w a sequence converging to u. From the convergence of this sequence, we would then arrive at the equality: u = w. We shall show that for every solution w E W(/) of the equation (14.1) under the conditions (14.6), it is impossible to have D(w) > d. (15.1) Let r; E W(m){O} and suppose that

has continuous derivatives up to mth

order in the interior of fl. If we can show that

D(w, ) = 0,

(15.2)

then, repeating word for word the corresponding arguments for the proof of uniqueness of the Dirichlet problem, we would arrive at the result that

D(u) > d, contradicting (14.12). Therefore for the proof of the impossibility of (15.1) and to establish uniqueness in the basic boundary value problem, it suffices to prove (15.2).

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

104

2. Lemma. It is convenient to proceed as we did in the proof of uniqueness in the Dirichlet problem: we introduce in 12 the function

-

y2h

(

I

t0

in 02h , outside S22h,

where h > 0. We form the averaged function for 'I 2h with respect to the kernel (Xh")- t {v(r/h) . This averaged function we denote by Xh. It is obvious that X. has

derivatives of arbitrary order, X. = I in n3h and vanishes outside of flh , and everywhere JXh1 < 1 . In addition, it is not difficult to show that k

ax-1, Xaxn°^ I

-

(15.3)

k.

The function Xh has continuous derivatives up to mth order in f2, vanishes outside of flh , and coincides with 4 on f23h . We shall prove the following basic lemma. LEMMA. For every v E W('') , we can choose a sequence {h,} (h, -, 0) such that

D(v, 4) = lim D(v, ,Xh r-.oo

(15.4)

PROOF. It is easy to see that

D(v,

f

m!

(

x

am( - Xh) 1 df2

a"`v

axial ... axn, ax0' ... axn ) amy

M! O_nM

aIt

f

... a".t axI°'

am(4 - 4Xh) df2 axnu,

u ", . ... ex8x1

amy

m!

am4 at! ... an! axal ... axn, axial ... axn df2

= n_nM

F

fl-n

amv

m!

Ck

>A

am-k4

X

axr'

akXh

ax!. ax"--P,

ax:.-P.

Al.

For h -, 0 the first integral tends to zero since v and 4 E W2(), while m(!;2 - f23h) -» 0. For the proof of (15.4) it suffices to show that the second integral tends to zero.

§2.4. UNIQUENESS OF THE SOLUTION

105

For this purpose it suffices to prove that Jk (h) ==

am-k

amtl

nh_0 19x0 , ... axn^ axa ... aXn akXh digI x axa '-Bl ... ax".-P-

(h

0

-» o )

(k=1,2,...,m) 0 since v, E W2"") and IX.1 :5 1).

(for k = 0 it is obvious that J0(h) By means of (15.3), we have

Jk(h)<

amy am-k dig ax; ' axO,. 04, axn^

h

amv axn^

axe'

am-k

2

dig

I

(15.5)

an.

I 094 ...19X19. n 1

Set h. =1/3° (ju = 1, 2, ...). Then 00

i - (i2hr - 3hM) + .U. I

and since v E

W2(-), the series 00

E p=1

amt,

llp-n,'p

I

aXn

.

19X1

2

do

converges. Hence we may find an infinite subsequence of the terms of this series less than the corresponding terms of the divergent series

E 00

p=2

l

µ In u'

In other words, for the infinite sequence {µ,} we have, setting h = h,: 2

amt! nh-n,A 19x1

...

do < axn

_

I

/L, In pL,

In 3

IInh,I [1nhinh,I -InIn31

K I1nh,I.lnllnh,I'

(15.6)

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

106

If we can show that am-k

ax",

2

df2 < Bh2klnJhI, . ax,6 n

(k=l,2,...,m)

(15.7)

then, obviously, we will have from (15.5) and (15.6) Jk(h,)

hk

lnh,l nllnh,l Bh,kIlnh,l

A BK Consequently, Jk (h,) -. 0 for r oo , from which (15.4) follows. Thus for the proof of (15.4) and the basic lemma, it suffices to prove (15.7).

3. The structure of the domains Q. - 03k . We shall concern ourselves with a more detailed study of the structure of the domains 0h - S23h . We decompose the whole boundary of the domain 0 into a finite number of smooth pieces Sn_, of various dimensions; to the collection of these pieces we adjoin all boundaries between two smooth pieces of the same dimension and all singular manifolds of the type of conical points or conical lines. The boundaries between pieces of dimension I will be, generally speaking, of dimension I -1 . For example, if the domain Cl is a cube, then the manifolds S2 will be all the faces of the cube, S, all its edges, and the manifolds So all its vertices. If the domain is a right circular cone, then we have to consider two manifolds Sz , the lateral surface and the base, S, will be the boundary of the base, and SS will be the vertex of this cone. We construct for each of these manifolds the domain (nh - f 3h)n-, consisting of all points of the domain Cl whose distance from the manifold S;,_s is less than 3h but more than h. The domain Cl,, - !Q3h is covered by the sum of the domains (L) - r3h)n-,

We extract from each of the domains

(lh -

(Cl,,

- 03h)n-, , a portion

which, by the aid of a nonsingular coordinate transformation with continuous bounded derivatives, may be transformed into a cylinder of radius h , the "axis" of which is a hyperplane of dimension n - s and is the image of Sn-, , and we do this in such a way that the domains (Cl,, -Q3h)n-, completely cover the whole domain 12h E13h An intuitive figure (Figure 6) shows how this decomposition is to be carried out if the domain Cl is a nonconvex hexagon. Here the domain (Ch-!Q3h)n-, is shown by shading. To prove the correctness of (15.7), it suffices to show its validity for each of the domains (Cl,, - C13h)n-,

-

§2.4. UNIQUENESS OF THE SOLUTION

107

FIGURE 6

To obtain the corresponding estimate, we may assume from the beginning are points or pieces of a hyperplane of the correthat all the manifolds sponding dimension. Obviously, we may always reduce the problem to this by a change of coordinates. We will assume from the beginning that the manifold S,;_5 is a domain

= xs = 0. By virtue of the assumption

in the hyperplane x1 = x2 =

about the simplicity of the boundary S, the general case may be reduced to we introduce cylindrical this. In the space (x1 , x2 , ... , xs , xs+1 , ... , coordinates with "axis" SS,_5 , i.e., put x1 = p cos 91; x2 = p sin f01 coS V2;

x3 = psinp1 sing2cosc3;

xs_ I = p sing I sin (02 ... sin 9s_2 COS 9,-1 xs = psin r9I sin 92 ... sin g,-2 sin 97,-, Xa+1 = Xs+I; ...............

X = X,,.

Then the manifold xI = x2 =

= xs = 0 goes over into the manifold p = 0

(Figure 7).

The proofs for the domains tending to S,,_s for the cases s even and odd

FIGURE 7

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

108

differ slightly. We shall carry through the proof in a unified form, indicating the differences where they are unavoidable.

Let s = 2t or s = 21 + I. Then it will be necessary for us to consider separately: Case I when k = 1, 2, ... , t, i.e., to estimate the integral (15.7) for derivatives, the values of which are not defined on S,;_s (cf. § 14, item 1), Case II when k=t+l, and finally Case III: k = t + 2, . .. , m , i.e., the case when the integrated derivatives in (15.7) are defined on S,,_, .

4. Proof of the lemma for k < [s12]. t. Set

We consider first Case I: k =

1,2,...

k=1,2,...,t i= 1,2,...,s-1

q1 , xj)

Z(P, 9`' x)

axft ...ax.n

j=s+I__ n

Let po > 0 be some number. We let 0 < p < p0. Since i has continuous derivatives up to mth order, Z will have continuous derivatives up to order k everywhere except on the manifold p = 0. Hence, applying Taylor's formula, we find

l! a z (P, vi, x)=Z(Po, co,,x.)+ P-P0a?I P +

(p-p0)k-Iak-IZ k (k - 1)! 0P-I

+... v=vo

1v(p-p')k_I ak7

+

(k - 1)!

vo

o=vo

a pk

dp

''

(15.8)

Set

IIZ(P, c,,

1/2

f

IZ12dx.,+1...dxn

v=const

ll

e,-const

Then equation (15.8) gives

IIZ(P, c, xj)L2

,_, <_ C

j11z(P0,9j,Xj)11,L2.._,

+ (P -IPo)2 (1.)2

(p -

+,

ZZ

II

+ .. .

Po

p0)2k-2

19

II2

L2.e-,

k-t

apo

l

2

II 12

f r° (P - PI)k-1 akZ d p, dxs+l... dx. (k - 1)! aP + ° =000S °, (15.9)

§2.4. UNIQUENESS OF THE SOLUTION

109

Multiplying (15.9) by p"- I du, and integrating over the unit sphere c o, and denoting by S. the (n - 1)-dimensional manifold p = const # 0, we obtain

fS'0 IZ12dSp

IS-1

c I Pa P

1410 IZ I2 dSpo

(p-o)2 f

+

(1!)2

is , o

,a I

ps- Id

+C

(15.10)

apo-I

w p=const

()

I,=con$t

f (p-p,)

k-I kZ a

(k- 1)! aPt

O

dSpo+...

ak-Iz

L

[(k -1)!j2

f

po

PO)2k-2

+(p -

x

IaZ I2

2

dp,

dx,+1...dxn.

E W2(-), by the imbedding theorems all the derivatives of i up to order m - I belong to L2,n-, and all of them figuring in the right-hand side of (15.10) may be estimated by Since

Aljfljw2(_).

Consider the last term on the right side of (15.10). Noting that p < po we obtain

f

m

p

s-I

*-1 akZ 12 dp, J dxs+, ...dxn dw fp«o>ut fILf p (p(k- -PI) 1)! E=const

< p' f do) f

o =eonst

rv

a

ap;

po

k

z

ap

f

p

p,

PI) 2k-2

(P -

((k

dp1

2

PI

dp1 dx:+, ... dxn

(continues)

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

110

p0

,

Jpp°

p

s-I a

fdwfp=c0nf PI ,

k

z

api

const

2

dp1 dxs+I ...dxn

(P - pl )2k-z

[(k - 1)!]2ps1

dpi

2

akz x _po

dig

ap

(P - p1)2k-2 x

<

dp'

v [(k - 1)!]2pt

11 CPSI IIIIWJpo(p-pI)

p1)2k-2/p"-'

We estimate f p°((p -

2k-2

dpi.

d p1 .

For this purpose, set p1 = px,

PO=Py

J

y p2k-2(I -X) 2k-2

pl)2k-2

!p0 (P PJ-1

di1 = f, X

ps-Ixs-, ry (1 xs_I

J

pdX=P 2k-s-1

x)2k-2

dx.

If 2k < s, then the integral converges as y - oo (i.e., as p - 0). If 2k = s, the integral grows like In y, i.e., like Iln pl. Hence, if 2k < s, we have from (15.10) and (15.11) IZI2dsp <_ PS-IIIIIW=. [CI +C2p2k-SIJnpI]

Integrating in p from Ah to Bh, we obtain IZl2dci <

llnhl] < K3h2kllnhl.

fnAh-nah

Thus (15.7) is proved for k=1,2,...,: (s=2t or s=2t+l).

§2.4. UNIQUENESS OF THE SOLUTION

III

5. Proof of the lemma for k = [s/2J + 1 . We pass to the proof of (15.7) for Case II: k = t+ I . In this case we already know that the derivative under the integral sign tends to zero on the boundary. However, the estimates which we obtained above in § 11 turn out to be insufficient and we give some slightly strengthened forms.

Consider some domain its in the hyperplane xs+I = const, ... , x. _ const and denote the distances from the points xI = x2 = . = xs = 0 and (Axi , Axe , ... , Axs) (E Ax? = IAPI2) of this hyperplane to an arbitrary point of the same hyperplane by r and rI , respectively. Set (am-`-'/axe axn^) = q(xI , ... , xs , xx+I , ... , x.). Derivatives of order t + 1 with respect to xi , ... , xs of tP are derivatives of mth order of 4. Hence, applying the formula (7.12) for estimating ...,X11)

4

=rp(AP,x/)-c(0,x), we find that, as in the proof of (11.14), I(P(A P , x/) - 1P(0, x/)I

IA J L

IAPI

(rr + r

I

Jn rs-t-I rs-t-! digs+B

"rr I

x

(r + rI

aM axe ... 8x°

2

)2s-2t-4

fn r2s-2t-2,,2s-2t-2

d.,

I

2

digs + B

n,

digs

We consider the cases s = 2t. Then using the representation which we encountered earlier in Chapter 1, § 11, we obtain

(r + r,) 2j-2t-4 1n

digs

)21-4

(r + rt

In, r2(t-1)1'2(1-I) IAPI

' J p

M,

(p + pIpzr-z )2t-4 digs pzr-z

R/inPi dp

z

=CI+C2IlnIAPII

P

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

112

Thus we find from (15.11) 19(A P , xj) - to (O , x1)12 < IA P I2

[KI + Kz 11nIA P III

am4 =m

fA,

I

axe ... ax

Integrating over S,_s (XI = x2 = Js;_,

Idil,

+ K Imls 2 dig

= xJ = 0), we see that

Io(AP , x) - 4,(0, x,)I2 dQR_s

[KI +K211nIAPII]

< IA-F12

12

f1 n

am,

n, m

(15.12)

f

dR

df2 + K3 n

axi' ... axo^

2

< A P 12{[KI + K2I1nI0PII]IRII2i .) + K411CII2 ,-

<

}

1211nIAYIj.

This estimate is more precise than that which follows immediately from

formula (11.15), since by using it we would get

IA P 12-`

instead of

IlnIA P MA P I2 in (15.12). In deducing (15.12), we used the fact that

f

am-1-1 .

InIcI2

2

dQ_n,ex

digNIItfIIW=-,.

Using the fact that 9(0, x) = 0 (in the sense of L2,,,-,)' we give (15.12) the form

fax x,=Ax.

I

axn

1

dx:+l ...dxn 5 KSII IIW==,IA

I2I1nIAPII. (15.13)

I2

Introducing cylindrical coordinates with axis replacing Io P I by p in (15.13), multiplying by p21-1 dw, and integrating over the unit sphere w, we obtain m-r-1 2 L9 4 (15.14) in p1. d So < JS 0x°' ... ax°^ n K611CIIWi., pzr+l

(

1

1

I

Integrating in p from Ah to Bh, we find am-r-1

IAA-n,h

axI

.. axn.

2

dig < Kh21+2Iln hI.

(15.15)

§2.4. UNIQUENESS OF THE SOLUTION

113

Analogous arguments for the case s = 2t + 1 give the estimate (15.14) with a right-hand side of order pet+1 and (15.15) with a right-hand side of h2t+2 . Thus (15.7) is proved for k = t + 1 . order

6. Proof of the lemma for [s/2] + 2 < k < m. We pass to the proof of (15.7) for Case III: k = t + 2, t + 3, ... , m. Set am-kC(P,

Z(P, i';, xj

q1;,

xj)

Z (p , tp, , x) has continuous derivatives up to kth order (k > t + 1) with respect to all arguments everywhere except on the manifold p = 0. Applying Taylor's formula, we find pP-po

l!

+(P-p0) k-,-2 fa k-t-2 Z1

(k-t-2)!

(TP_)v=va+...

+

apk_t_2/I

v-vo

0(P-PI) k-,-2 ak-t-1 Zdp1. (k-t-2)! app-1-1

fvo

(15.16)

From formula (15.16), we obtain

IIZ(P, q't, xj)IIL,..-, )IIL,.,-,

Iv-P I Ilrap

k-t-2 Cak_t-2Z pk-t-2 (k - t - 2)! a

I!

a

II

v=vo L2.,-,

+ IP - Pol

v-°o IIL2..-,

° (P - PI )k-t-2 ak-1-1Z

+

r - t - 2)! app -1-1 JS(k fo.

/2

2

d pI

dx,+I ... dxn

By virtue of the boundary conditions for C ( and all its derivatives up to (m - i - 1)st order are null in the sense of L2 _J on the manifold p = 0),

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

114

in taking the limit po

0, we obtain

IIZ(P, tv;, xj)Ilc:.,_, P (P

[

-

pI)k-t-2 ak-t-IZ

r<

k-t-2!

12

dxs+l ...dxn

k-r-1 dp1

ak-t- I Z z

A_ fs;

=

1 ap'- t-I

AIP2k-2r-3

P

dpI

(p_ pI) 2k-2t-4 dpI

ak-t-IZ 2 a p; -t-I dxs+1 ... dx dp1.

foP

I

s;_,

(15.17)

We consider the case s = 21. Multiplying (1S.17) by pet-1 dw and integrating over the unit sphere c o, we obtain L IZ(p, t0;, xj)12dSP 0

A zP

2k-4 / P

a m-1-I

I

pet-I fs,

o

i E °,=m-1-1

1

I

2

ax0,...axn.

dSP, d PI

from which, taking (15.14) into consideration, we find 2

2

I IZ(p, Pi, xj)I dSP < A311IIW"T p

2k-4 P p;t+llln p11 pi

Jo P

IIw,p2k-a

dPl

1

p2

fo

,Ilnp,Idpl

A4p 2k-I

IlnPl Integrating in p from Ah to Bh, we find IZ12 dQ < Ash2a`IInhI.

In the case s = 2t + 1 , the right-hand side is of order h2k . Thus (15.7) is

proven forall k= 1,2,...,m. From this the basic lemma immediately follows. t From the lemma an obvious method yields equation (15.2). Indeed for an arbitrary function w satisfying the equation A'"w = 0 we will have

D(w, Xt,) = 0.

(15.2)

Passing to the limit, we establish (15.2) and with it the uniqueness theorem. 1 Here, as in the proof of the lemma in 4 of §12 we may dispense with the assumption of the continuous differentiability of 4 up to order m . For this purpose one only needs to pass to the limit of the averaged functions in formulas (15.9), (15.12), and (1 S.17). The remainder of the proof proceeds without change.

§2.5. THE EIGENVALUE PROBLEM

115

7. Remarks on the formulation of the boundary conditions. We shall call a function in W(m)(0) if it is the limit in W(") of functions a function vanishing on a strip near the boundary. If functions u and yi in W('") are given, then we say that u coincides with yi on the boundary of the domain in the sense of W(m)(0) if u - yr E W("')(0) . In a series of publications on variational methods, there has been considered the problem of finding functions giving an extremum to the functional D(u) (for the case m = 1) and coinciding on the boundary with a given function yr in the sense of W2(m)(0) . Such problems obviously have a unique solution, as one can easily see immediately. However, without our results it was not clear what are the real boundary conditions for a solution for which u - y/ E W'ro)(0) (50 §2.5. The eigenvalue problem

1. Introduction. Let 0 be a bounded domain of n-dimensional space with a sufficiently smooth boundary S. The problem of the eigenvalues of the equation

Au+Au=0inf2

(16.1)

for the conditions on the boundary S

8u hul = 0 (v is the exterior normal), 8v is - is

(16.2)

where h is continuous on S, consists of the determination of all values A for which the equation (16.1) with the conditions (16.2) has a non-null solution.

We use the fact that there exist continuously differentiable functions a,, bounded with bounded derivatives on the closed domain D + S (1a11 < M ;

18a,/exl < M, M = const), such that lhl:5

ails cosna,,

(16.3)

where a; is the angle between v and the Ox,-axis. Since S is a suitably smooth surface, one may construct sufficiently smooth functions C,, reduc-

ing to cos nx, on the surface. Then it suffices to take a; = MC,, where

M=maxslhl

The variational method enables us to construct a solution of this problem, to find all its eigenfunctions, and to show the orthonormality and completeness in L2 of the system of such functions. We recall that an orthonormal

system of functions in L2 is called complete if one can approximate any function in L2 with linear combinations of these functions, with arbitrary precision.

116

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

2. Auxiliary inequalities. We turn to the solution of this problem. Consider the functionals on W21)

()2 dA- f hv2dS;

D(v) = f

(.!.)2 d12;

f

J(v)

H(v) =

s

i

n

.

Ivl2 dg = IIvIIfn

LEMMA. The following inequalities hold:

J(v) < LID(v)+L2H(v),

(16.4) (16.5)

D(v) < KI H(v) + K2J(v), where LI , L2 , KI , and K2 are certain positive constants. PROOF. We shall prove (16.5). We have, using (16.3):

fhv2dS

rv2a,cosnx;dS s

;=1

r E8(a'v2)di2 8x;

/n ;_I

=2 fn v ra 'v8x;dQ+ f v2 < M f 2IvI

aa` Al

,_, Ox,

n

r=,

I 87xav I dig+MnH(v).

Using the fact that 2

Kv2 + T 8XI

21vl i

where K > 0 is an arbitrary number, we obtain

Ifshv2 dSI < MnH(v) + KMnH(v) + K J(v) MJ(v). = Mn(1 + K)H(v) + From the definition of D(v) and J(v) we have

D(v) < J(v) +

I

jhv2dS

I

< Mn(1 + K)H(v) + I 1 +

=KIH(v)+K2J(v),

J J(v)

X/

12.5. THE EIGENVALUE PROBLEM

117

K1 = Mn(1 +K),

K2=1+M, and the inequality (16.5) is established.

On the other hand, from the definitions of D(v) and J(v) follows

J(v) = D(v) + f hv2 dS s

< D(v) + Mn(1 + K)H(v) +

K J(v),

from which

J(v) (1 - K) < D(v) + Mn(1 + K)K(v). If we choose K > 2M, we obtain (1 - (M/K)) > 1/2, and J(v) < 2D(v) + 2Mn(1 + K)H(v) = LID(v) + L2H(v), i.e., inequality (16.4) is proved and the lemma as well. From (16.4),

L2H(v) ? -L2H(v), D(v) >-Ll1 J(v) - Ll Ll and hence if H(v) = 1, then D(v) > -(L2/LI). It follows from the latter that there exists

inf D(v) =).

H(v)=I

Hence there exists a minimizing sequence {vk} : vk E W21) ,

H(vk) = 1, lim D(vk) = 21

koo

3. Minimizing sequences and the equation of variations. THEOREM. There exists a function ul E W21) such that

H(u1) = 1, D(u1) = A1. The function u1 has continuous derivatives of all orders in 0 and satisfies the equation

AuI +Alul = 0. PROOF. Let {vk} be a minimizing sequence. We shall show that I1vk11w2..

is bounded. Indeed, if we put (p, v) = favdfl, (p, 1) # 0, and use the boundedness of D(vk) , (16.4), and the condition H(vk) = I , we get that IIVkllw," _ {(p,

vk)2 + J(vk)} 1/2

{mi2H(vk) + L1 D(vk) + L2H(vk)}l12

<{(mil +L2)+LIA}"2,

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

118

where A = Supk ID(vk)l. Thus the boundedness of JIvkIIW7 is established. From the complete continuity of the imbedding operator it follows that {vk} is strongly compact in L2 on Cl. From {vk} we may choose a subsequence converging in L2. This subsequence will itself be a minimizing sequence. Hence there exists a minimizing sequence converging in L2. We may thus assume from the beginning that our original sequence has this property. Furthermore, 1Ivk - v11IL2 = H(vk - v,) < e as soon as k, I > N(a). From the obvious equality, (vk v1 ) H 2 = ZH(vk) + 2H(v,) - H (vk 2 vi) using\\H(vk)

= H(v,) = 1 and H((vk - v,)/2) < e//4, that

we obtain,

H (yk 2v1 J>1-4 The functionals D(v) and H(v) are homogeneous quadratic functionals and therefore their ratio D(v)/H(v) does not change under the passage from v to cv (c = const # 0, H(v) A 0), and hence

inf D(v) =

vEwm H(v)

inf D(v) =A1.

H(v)-1

vEWx(0

Therefore D(v) > A1H(v) for all v E WZ') and in particular

DI

yk2v1)>A1(1-4), k,l>N(e).

Then, taking k and 1 large enough that D(vk) <. +a and D(v1) <'l +a we obtain

D(vk2y1) =2D(vk)+ZD(v1)-Dtvk+VI) \\ <11+e+2 +a-.1 +\.lle 2

4

//2

-8 1+ 4'). From the inequality (16.4),

0 < J(vk - v,) < LID(vk - v1) + L2H(vk - v,), and therefore,

D(vk2v11 >-L24,

JLL2e+Lle(4+)1).

0< J(vktt-v,)

<

Consequently,

D(vk-v1)-O and J(vk-v,) 0, k,l--.oo.

§2.5. THE EIGENVALUE PROBLEM

119

But

llvk - V/Ilw2;;; _ (P, Vk - v!)2 + J(vk - VI)

< mQQH(vk - VI) + J(vk - VI) -. 0, i.e., II Vk - V/ II H= - 0. Hence {vk } converges in W21) and as a result of the

completeness of W21) there exists a limit function ul E W(l) such that IIu1 - VkIjHZCn -- 0, k -. oo. In addition, ID(vk) - D(u1)I = D(vk)
+

(u1) +I h(uI - vk) dSl is

J(vk) -

J(ul)I ( J(vk) +

J(ul))

J (ul - vk)2 dS 1 h2(u) + Vk)2 ds -- 0

for k -. oo, since J(ul)I =

J(vk) -

IllvkIIL2;

<- Iivk

[f(UI_vk)2ds]

- IIUIIIL=,I <- Ilvk - UIIIL=')

- u1II121 - 0,

1/2

< Nil u l - vk II W= -, # 0

while the terms V/T(i + \/T(u1 and fsh2(uI + Vk)2 dS are bounded. Therefore D(vk) -. D(ul) and, hence, D(u1) = )1.

(16.6)

H(u1) = 1.

(16.7)

Analogously one sees that

Suppose now that

is some function from W(). Consider the ratio D(u1) + 2,uD(ul ,

D(u1 +

H(ul +

H(u1) + 2pH(ul

,

) )

+'U2

D0)

+'U2

H(C)

where

au rl

aXl aXi

da-J$

H(u,1)=J u4drI.

n It is a continuously differentiable function of µ on some interval around the point µ = 0. This ratio has a minimum at µ = 0 equal to 2 and by the basic theorem of Fermat, we have

[D(uI +2D(u, )H( ul) - 2H(u1 H(u

1

v=o

H(2 ul)

,

= 0,

120

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

which by virtue of (16.6) and (16.7) gives

D(u1,4)-A,H(u1,C)=0.

(16.8)

Equation (16.8) holds for any * E W. We shall show that u1 has continuous derivatives of all orders and satisfies the equation Au1 +21u1 = 0. (16.9)

Indeed, let b be the distance between some point P E i2 and the boundary

S. For n>2 let

=Iw\hl)-w(r)]

(r)'

where h1 and h2 < b, w(r/h1) is the function introduced earlier in 3 of § 12, and X (r) is a solution of the equation AX + A1X = 0 and for r = 0 has a singularity of order

r-"+2, and X' a singularity of order r-"+' . Since

v(r/h,) = lv(r/h2) on the ball r < I min(h, , h2) , we have = 0 on this has no singularity for r = 0 and has continuous derivatives ball. Hence of all orders. Therefore, E W21) . In addition, = 0 outside the ball r > max(h, , h2) and since max(h, , h2) < 6, we have 0 on S. Therefore (16.8) for

takes the form

fn

8x18x

/ dig=0.

Since

f 8 8x df =- r u,a df

n n 8x; (by the definition of the generalized derivative 8u,/ax;), it follows that

fui(t+A1)dc=0.

(16.10)

We set

c(h)

lA[ (h) X(r)] +A,w(h)X(r)} =wh(r),

where c(h) is a constant to be defined later. It is obvious that wh(r) = 0 for r > h. In addition, cvh(r) = 0 for r < (h/2), since in this case w(r/h) = 1, and AX +A,X = 0. Hence, coh(r) has continuous derivatives of arbitrary order. Since

AC + A,C = c(h, )wh, (r) - c(h2)wh,(r) ,

equation (16.8) takes the form

c(hl)faulwh (r)di2=c(h2) fuu,wh2(r)df2.

(16.11)

§2.5. THE EIGENVALUE PROBLEM

We put

c(h) =

f fA

\h X(r), +.1, v

lw

121

X(r)} d[l.

Then

fwh(r)dQ= 1. It is not difficult to show that limb. ,0 c(h) = co exists and is nonzero. The function wh(r) may be considered as an averaging kernel. The equation

(16.11) then means that the averaged functions for u, on the domain Q. for various h < & differ only by a factor. We obtain uhz =

c(h1)

By virtue of the fact that u, is the limit of its averaged functions, it differs merely by a factor from any one of its averaged functions. But the latter are continuously differentiable arbitrarily often and hence the same is true for u, on S2.. Since & is arbitrary, u, is infinitely often differentiable at any interior point of 0. Suppose now that 4 is continuously differentiable and vanishes on a boundary strip. Then from (16.8) we obtain by integrating by parts

fa (eu1+A1u,)Al =0, from which by virtue of the arbitrariness of 4 , there follows (16.12)

Au1 +A1u1 = 0,

and the theorem is proved for n > 2. The case n = 2 is handled similarly. REMARKS. Consider a sequence of domains {i2'} lying in the interior

of it and converging to Q. Let the boundaries S' of these domains be piecewise continuously differentiable. The equation (16.8) for 4 E WZ 1" takes

the form

axi

1a

dig-I hu, dS=O.

ax'

But

184 '9U 8x, Oxj

)

gal

ninf'

dig

au,'n r=1

8x; 8xi -11u1

dig

= lim L-1 4(Au1 +A,u1)di2+ f

n-n

19

= n'

nJ

s'

n,

8v

dS.

f a dS]

(16.13)

122

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

Thus (16.13) takes the form

lim f haul dS -

O.

J

(16.14)

If in addition S' S in the sense that not only the points of S' converge to the points of S but also the normals at these points converge to the corresponding normals of rS , then

r

fs

lim

n,--.n JS,

if we assume that h is the value on S of some function given on Q. Then condition (16.14) takes the form

n' nJsl

(ovl -hul )4 dS=O.

(16.15)

Thus, u1 satisfies the condition (16.2) "in the weak sense." Therefore, an eigenfunction of the problem (16.1), (16.2) will be defined to be a function u(x) l- 0 satisfying equation (16.1) in ( for some A and the boundary condition (16.2) in the sense of relation (16.15). The number A is called the eigenvalue corresponding to the eigenfunction u(x). The theorem proved implies the existence of an eigenfunction ul corresponding to the eigenvalue Al in the sense indicated. We turn now to the search for the other eigenfunctions. 4. Existence of subsequent eigenfunctions. Let us assume that we have already found (m - 1) functions u. E W(t) and (m - 1) numbers A. (i = 1, 2, ... , m - 1) such that

D(uj, ) - A,H(u,, ) = 0;

H(u,)=1,

(16.16)

H(u,,uk)=0 (i#k) i,k=1,2,...,m-1, (16.17)

where (16.16) holds for an arbitrary satisfies the m - 1 conditions

E W(1) . Suppose that v E Wl l

H(v,u1)=0 (i= 1,2,...,m-1).

(16.18)

The family of such functions v E WZ 1) we denote by Wz 1)(u1, ... , um_1) )(u1, ... , u,,,_1) is a linear space, since every linear combi2 Obviously nation of its elements belongs to the set. We shall show that this set is closed. Indeed, suppose that a sequence {vk } converges to v in W(1 , i.e., Ilv - vkllw( 0. By the imbedding theorem, llv - VkIIL2 <- Cllr - vkIi,u ,

§2.5. THE EIGENVALUE PROBLEM

123

and hence Ilv - vkIIL, - 0. Thus IH(v, u,) - H(vk , u.)I = lI ur(v - vk) dill <

If U'2dil j iv - VkI dZ

=IIV - vkIIL= - 0

i.e., H(vk , u,) = limk_,_ H(vk , u,) , and if vk E W21)(u1, ... , um_1), then V E W21)(u1, ... , U,-,). Hence W(1)(u1, ... , um_1) is closed. Since W2(1) (u1

, ... um-1)C W2(1),

D(v) is bounded from below for v E W21)(u1, ... , UM-1) with H(v) = 1 . The greatest lower bound of D(v) for this family we denote by Am :

inf vE Wi "'(u, "',U.-,)

D(v)=A,,.

(16.19)

H(v)=I

THEOREM. There exists a function Um E W21)(u1 , ... , um- 1) with H(Um)

= 1, such that D(um , 4) - AmH(um, 4) = 0

for arbitrary i E W21) . In particular, for

(16.20)

= um , (16.20) takes the form (16.21)

D(um) = elm.

In addition, on 12, um has continuous derivatives of arbitrary order and satisfies the equation (16.22)

Au"' + Amum = 0

and the boundary condition (16.2) in the sense of the relation (16.15). PROOF. From the definition of the greatest lower bound, it follows that we

can find a minimizing sequence {vk} in W(1)(ul, ... , um-1): Vk E W21)(U1 , ... , um_1),

H(vk) = l ,

limD(vk) = Am

Repeating literally the proof of the theorem in item 3, we show that Ilvk II W:" is bounded and hence that {vk } is strongly compact in L2 . We may assume

that the same sequence {vk} converges in L2. Then H(vk - vt) < e for sufficiently large k and 1, and H(vk + vt)/2 > I - (z/4), as in the theorem in item 3.

For all vE W(1)(u1,...,Um_1),wehave D(v) > ArH(V),

124

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

and since (vk + v!)/2 E W(1)(us , ... , UM-I) simultaneously with vk and yr ,

D(vk2 v!) >AmH(vk2 vl ) >A.(I-4). From this, as in the theorem in item 3, we find that D(vk - v1) -- 0 as k, I -a oo, and further

1IVk-v,Iluu,-.0 as

k,/-.oo,

i.e., {vk} converges in WU) . As a consequence of completeness of W2I) , W21)(u1 there exists a limit function Um E Since vk E , ... , um-1) and WO) 2 (us , ... , um_1) is closed, it follows that um E W2(1) (U Um In addition, as in the theorem in item 3, we show that D(um) = An and H(um) = 1. Let 17E W11)(u1 , ... , um_1). Consider the ratio W21)

D(um + un) = D(um) + 2sD(um , g) + u2 D(7) H(um + fzq) H(um) + 2,uH(um , q) + p2H(q)

This ratio is a continuously differentiable function on some interval about the

point p = 0, and has for u = 0 a minimum equal to A. (since um + pq E W(I)(us , ... , um_1)) . From this, it follows by Fermat's theorem that D(um , q) - ArH(um , q) = 0.

(16.23)

We shall show that (16.23) holds for all q E W21) . Indeed, put = um in (16.16). Since um E W2 11(u1, ... , um_1), it follows that H(u,, um) = 0 (i = 1, 2, ... , m - 1) , and (16.16) takes the form D(u, , um) = 0. Therefore (16.23) holds for q = u, (i = 1, 2, ... , m - 1). Hence (16.23)

holds for an arbitrary linear combination of the functions u; and q E W2I)(u1 , ... , Um- 1) .

Let 4 E

be an arbitrary function. Then m-1

n = - E u;H(ur, ) E

W21) (u1 ,

... , um-1),

i=1

as is easy to verify. Thus

= q+E"

1

u,H(u,, 4), i.e., an arbitrary 4 E W2 I

is a linear combination of the u, and of q E W21j(us , ... , um_1) . Thus (16.23) holds for 4, i.e., the correctness of (16.20) has been established. In the proof of the fact that um has continuous derivatives of all orders in 12 and satisfies (16.22) there is a literal repetition of the corresponding part of the proof of the theorem in item 3. As for the sense in which the functions um satisfy the boundary condition (16.2), the same remark can be made as in the case of the function u1 in the theorem in item 3.

§2.5. THE EIGENVALUE PROBLEM

125

5. The infinite sequence of eigenvalues.

THEOREM 1. There exists a nondecreasing infinite sequence of numbers {Am} and a corresponding sequence of infinitely differentiable functions um on S2 such that Aum +Amum = 0,

H(u;, u) = aid

(16.24)

D(u,, u.) = bij.ii , where bit is the Kronecker symbol. Each um satisfies the boundary condition (16.2) in the sense of equation (16.15).

PROOF. The theorem in item 3 asserts the existence of a number 2I and a function u1 such that Aul +A1u1 = 0, H(u1) = 1, D(u1) = ill. Consider the subspace W. ') (u 1) . By the theorem in item 5, there exists a

number A2 and a function u2 such that Au2 + A2U2 = 0 , H(u2) = I

,

H(uI , u2) = 0, D(u2) ='12 , D(u1 , u2) = 0.

Thus AI, u1 and A2, u2 satisfy (16.24) for i, j = 1, 2. Since W(')(u1) C W (') , we have

D(v) > inf D(v),

inf vE W'"1(u,)

H(v) = 1.

vE W=1)

This means that A2 >-'11 . Considering the subspace W2(')(u1, u2) and using the theorem in item 5, we obtain '13 and u3 . Using the theorem in item 5, this process for obtaining Ai and u; can be continued infinitely. We obtain

thereby elm =infD(v) (V E W(')(ui , ... , um_I), H(v) = 1). Since W2 (uI , u2, ... , Um-1) D WZ')(u1 , ... , Um)

(a function v belonging to W(')(u1 , ... , um_1) satisfies m - 1 conditions, and at the same time v E W2(1)(u1, ... , um) satisfies the additional condition H(v, um) = 0), we have Am < Am+i The fulfillment of condition (16.24) follows from the theorem in item 5.

Since H(ui, u .) = bid, the sequence {um} forms an orthogonal and normed system of functions. THEOREM 2. For the sequence {Am} of eigenvalues

lim Am = +oo.

m-oo m

PROOF. Indeed Am is bounded from below and nondecreasing. Suppose that A. does not converge to 00 . Then .Im is bounded:

tArI
2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

126

From (16.24), it follows that

H(um) = I (m = 1, 2, ... ).

ID(um)I
On the basis of the inequality (16.4), we have

J(um) < LID(um) + L2H(um) < LIA + L2 = B,

i.e., the sequence {um} is uniformly bounded in the norm of LZ') by the number V B-.

We show then that the sequence {um} is bounded in the norm of W(') Indeed, if we define the norm on WZ') by the equation 2 2

2 (fva'cz) +llvlli,),

2

(fUmdf2) + Ilumll41)

llvllw=I) _ we find

2 2

11U m11 W

<milf u2 d1I+B=mil+B.

n As a consequence of the complete continuity of the imbedding operator, we would conclude that {um} is strongly compact in L2 on Q. In other words, from {um) we may choose a subsequence {um } (i = 1, 2, ...) such that

H(um -umk)-»0 asi,k - oo. The latter is impossible, since

H(um - um,) = H(um) - 2H(um , umk) + H(umk) = 2.

Thus, the assumption that limAm # oo leads to a contradiction, and the theorem is proved. 6. Completeness of the set of eigenfunctions. THEOREM. The system {um) is complete in L2 . i.e., for every p E L2 we have the equality co

l rp2 dil = n

m=1

(fn

2

taum dig)

PROOF. Since Am - oo, it follows that, starting from some integer j we

have that Am > 0 (m > j). Let k be some number, k > j. Let v E W2( be some function. Set k

Rk = v - E umH(v , um). m=1

It is obvious that Rk E WZ'>(u1 , ... , uk) , since

H(Rk,u;)=0 (i=1,2,...,k).

§2.5. THE EIGENVALUE PROBLEM

127

Hence D(Rk)/H(Rk) >_ )1+1 , from which it follows that 1)

D(Rk) > 0

2)

H(Rk) <

and

D(Rk) '1k+1

On the other hand,

0 < D(Rk) = D(v) - 2D (v ,

E urH(v ,

um)) + D

/

M=1

(E umH(V, um)) `m_1

k

= D(v) - 2 E H(v, um)D(v, um) M=1 k

k

+ E E H(v, um)H(v , ur)D(um , ul) m=I 1=1 k

k

= D(v) - 2

H(v , um )2mH(v, um) + F (H(v, um)]2D(um) M=1

m=1

k

= D(v) - E ).m[H(v ,

um)]2

m-1

l

r

= S D(v) - L )m[H(v, um))21 `


-

E A,[H(V, um))2 m=j+1

m=1

Am[H(v, um)]2. M-1

Thus D(Rk) is bounded, and from H(Rk) < D(Rk)/2k+l and 2k+1 00, it follows that H(Rk) 0, which means the completeness of {um} in the class of functions v E W(1) . Let 9 E L2 be an arbitrary function. For any e > 0, no matter how small, we can find v E W21) such that (16.25) II9-vliL2 S 2 Indeed, denoting by 12, the subset of points of 0 lying at distance more

than 8 from the boundary, we will have ell mJn qi2df2,

from which it follows that for given e > 0, we can find 6 > 0 such that

f

r

2

jo2dQ-f 41' Al < i6.

128

2. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

Introducing the function ry

in Q.

0

outside f?,

(IPa E L2),

we obtain

z

L

Ico-cal2di2< 16.

(16.26)

If we construct as 9;a the averaged function with kernel of radius < 6, we obtain a function v having continuous first derivatives on the closed domain S2. Consequently, V E W2') and choosing a sufficiently small averaging radius, we will have 2

Jo

196 - vl2 Al < 16.

(16.27)

From (16.26) and (16.27) we conclude the correctness of (16.25). For the function v , as was shown, we can find a linear combination of a finite number of the functions um such that k

H(v ->amumf

/

m-l

v=

G

amum

which, together with (16.25), gives IIc - Em=I amumllLi < c. If the am are replaced by H(9, um) , then, as follows from the general theory of orthogonal systems, the left side does not increase, and hence k

<E,

c->umH(co,um)II L2

mal

which, in view of the arbitrariness of z, leads to the equality

0 The theorem is proved. (51)

(fcumd)2.

CHAPTER 3

The Theory of Hyperbolic Partial Differential Equations In the present chapter we consider some problems arising from the theory of hyperbolic partial differential equations. We shall treat two topics:

1. The solution of the Cauchy problem for the wave equation. Dependence of the solutions upon initial data, and generalized solutions. 2. Hyperbolic equations with variable coefficients. These topics are united by a common method of investigation which consists of studying solutions in the spaces L,') and W(t) considered in the first chapter. However, en route we shall have to solve an auxiliary problem in the integration of hyperbolic equations with sufficiently smooth variable coefficients. It is necessary for us to show the existence of a solution of the Cauchy problem for such equations with sufficiently smooth initial conditions. For this purpose, we employ the theory of characteristics in 2k-dimensional space and the method of descent in the space of 2k + 1 dimensions. (52) §3.1. Solution of the Cauchy problem for the wave equation with smooth initial conditions

1. Derivation of the basic inequality. Consider the wave operator 2

Ou = Du - ate

(17.1)

on the domain n in the space of n + 1 dimensions with coordinates x1 , x2 , x,,, t, bounded by a smooth surface S. Let u(x1, x2 , x,,, t) and v (x1 , x2 , ... , xn , t) be twice differentiable in 0 with their first derivatives continuous up to the surface S. Let

Ou=j,

Ov=9. 129

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

130

Consider the integral

s

avau

Ua-U u 8v

"

J(u, v)

cos«;

x; at + ax at

,=1

au av 1 cosao l dS,

Cau av +

at at +

ax; 8x;

where v is the interior normal to S, a, is the angle between v and the Ox;-axis, and ao is the angle between v and the Ot-axis. A simple transformation yields

J( u, v) =

a

f

=

axax)-in

ar

a22

at [at

au av + av au

au 8v

au av +

fn{at(at

i=1

8x;

alZ + 8v 82Z _ at [ar

x,

(ax;at a2 2 ;=1

ax;

at)

dS2

ax;

f{ov+ou}dcz.

(17.2)

Replacing Ou and Ov by their values, we get

f. MU f + -V f) We have the equality

;=I

8x.,

cosa0 - 8l Cosa.

(ax CosaO - of Cosa, I i

" au av 2 2 +cos aoEa;ax; at at E(cosa,) i=1 i=1 (OuOv + -avau - COs a; COs a0

_ au av "

at ax;

=cosa0 l`

at ar

atax;

cos«0+iax. az Cosa0 i=1

i

au av

av au

-E arax;+atax;)cosai au av

cost a, - cos ta0 + at at 1 `i L 1

d1Z

§3.1. SOLUTION OF THE CAUCHY PROBLEM

131

FIGURE 8

Let S = S' +ksse S" , with cos ao 0 0 on S' and c\os/ao = 0 on S" . Then

J(u, v) _

f

o

(aX; cos«o - 8 Cosa; I ax; cos«o - a cos«;

au av

l - 2 cost ao 1

at at

cosao "

+f E

J

au av

dS

av au (8x;at+ex;8[)cos;

I dS

.=1

= fs O dS,

(17.3)

where by 4) we denote the whole integrand of the integral J(u, v). It is useful to note that if u = v, then at all points of the surface where Icosaol 2! 11,42-, the integrand has the same sign as - cosao : sgn 0 = sgn cosao .

(17.4)

Suppose that u is a solution of the wave equation 17u = 0

(17.5)

in the half-plane t > 0. Taking v = u , we apply the formula (17.2) derived above to the function u, taking for the domain Cl a truncated cone whose generators make an angle of n/4 with the 01-axis (Figure 8). Then

-I on S1, cosao =

-1

on S2,

+1 on S31 where S2 is the lower base, S3 the upper base, and S1 is the lateral surface of the truncated cone. Suppose, for the sake of definiteness, the quantity t on S3 is equal to to . By means of (17.2) and (17.5), we obtain

J(u, u)= f OdS+J 4dS+ / OdS=0. ,

s,

s,

(17.6)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

132

By (17.4), fs' 4) dS > 0. Then

-f 'bdS=f 'DdS+f s,

,

ddSS

f

OdS.

s,

,

From this, since on S2 and 53 : cos a1, r= 0 for i

0 , we have the estimate

dS< f fS3

dS. (17.7)

j.1

2. Estimates for the growth of the solution and its derivatives. For our purposes, it is also necessary to estimate the integral

fu2dS. ,

From (17.7) it follows that fs,( )2dS is bounded. Denoting by y(t) the quantity y(r) = fu2dS,

where ET is the section of the cylinder with base S3 and axis Ot by the plane t = T. Then y'(t) = 2 r u(t) au dS. Applying the Cauchy-Bunjakovsky inequality, we obtain

()2 dS

2 [f

1/2 l

and by virtue of the inequality (17.7)

J

I I ,u2dS

l!

J

IY'(t)I 5 2A[y(t)]1 /z ,

where

1/2

A

_ [L2{t()2 +

()2}

dSJ

.

This implies that 71,j < A. Integrating this inequality from 0 to 1, we get that y(t) < y(0) + At . Setting y(0) = fs7 u2dS = B2 , we have that

y(t) <(B+At)2, 0
()2 ()2}

f,{

+

dS

{l ()2 + ()2} dS. (17.8)

§3.1. SOLUTION OF THE CAUCHY PROBLEM

133

Integrating in t from 0 to to , we obtain

(5)2 + ()2}

. {7n

dV

(8u)2+(ett)2


=I

87

8t

}

dS
(17.9)

In a way analogous to the way the estimate was proved for y(t), we get

fu2ds<(B+At)2.

(17.10)

,

Integrating in t from 0 to to, we obtain

fu2dv < 3(3B2t0+3ABto+A213).

(17.11)

The inequality (17.1) has a number of important corollaries.

COROLLARY 1. Suppose that the initial values of u and 8u/8t vanish in

the interior of S2. Then u = 0 in V. PROOF. Since u =

= 0 on S2, it follows that A = B = 0, and u e 0

on V by (17.11). COROLLARY 2. The value of the function u. a solution of the given equation,

at a point x0 , x2 , ... , x, , to is determined by the values of the initial data of

u and 8u/8t on the ball (Z' I (xi - x0)2)1/2 < t0, which is the intersection of the characteristic cone with vertex at the given point with the plane t = 0.

PROOF. Indeed, if for any two solutions of the wave equation the initial data of u and 8u/8t coincide on this domain, their difference will vanish on this domain and by Corollary 1, the difference will be null at the vertex of the cone. Hence, at the vertex of the cone, i.e., at the point in question, the two solutions will coincide, as was to be shown. 3. Solutions for special initial data. THEOREM. Let the function u be a solution of the homogeneous wave equa-

tion. If the initial values ul,=0 and 8u/8ti,_0 are infinitely differentiable on the whole of the space of the xI , ... , x,, , then the function u itself has all derivatives of all orders.

PROOF. To begin with, we shall establish this theorem in a more special situation. We prove a lemma. LEMMA. Let the function u satisfy the equation

Au-a u=0 8t2

(17.5)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

134

on the domain -a < x, < a (i = 1, 2, ... , n), where a is some constant, and suppose in addition that

i=1,...,n,

ujX=to=0,

(17.12)

i.e., the function u vanishes on the boundary of this domain. Suppose in addition that for t = 0 ult.o = go, (17.13)

at = where the functions go and rp( have continuous derivatives of all orders and 11-0

vanish on the boundary of the domain together with all their derivatives. Then the function u has continuous derivatives of all orders. PROOF. Indeed, in this case, the solution can be written in an explicit form with the aid of Fourier series. We expand the functions go and 9I in Fourier series: 90

=

0 91 = L

j

(XI + a)n

b

s in

gj. , .......

sin jI

2a

l

(x1 + a)n

2a

(xn + a)n x .. . x si njn 2a

x ... X Sin in

(x + a)n 2a

(17.14)

These Fourier series converge uniformly together with all their derivatives of arbitrary order. Consider the partial sums of these series. Set rp(N)

D(N) 1

=Eb =

(xn + a)n (XI + a)n ... sin in

sin.

2a

g .gt=...,j- sin j1 j,.....jua1

2a

(XI + a)n (xn + a)n 2a ... sin in 2a

If in the initial conditions we replace go and 91 by go) and p (IN), we obtain a solution u(N) of the wave equation satisfying these initial conditions: N

u (N) =

)bj

, j, , ... , j.

2a

ii + j2 +... + jht n

2a

zsin2a

j1+j2+...+jn

+ x sin

cos

z

1

z

(XI + a)n 2a

sin in j

jl+j2+ Fjnt 2

(xn + a)n 2a

2

2

§3.1. SOLUTION OF THE CAUCHY PROBLEM

135

FIGURE 9

This solution, obviously, is infinitely differentiable. We shall show that with increasing N the sequence u(N) converges in every space WZ') , where 1 is an arbitrary number, to some function u. It follows already from this

that the limit function u is a solution of the wave equation satisfying the initial conditions (17.13) and is infinitely differentiable. Since this solution is unique, our lemma will follow. It remains for us to prove the convergence of the sequence u(N). Applying the integral equality (17.6) to the parallelepiped whose lower base S2:

1xi 5 a lies in the plane t = 0 and whose upper base S3 lies in the plane t = to (Figure 9). If we use the fact that on the lateral surface S, of this parallelepiped we have the equality u(N) = au(N)/at = 0, we obtain

f

ou(N) aulN)

S

at

an

dS = 0,

and hence

(au(N)12 s,

r=1

8x7

+

JJ

dS

at

E (OU(N))'+

= /

J

S

1=1

ax

((N))21 dS.

at

(17.15)

We consider also the functions (N)

a*U(N)

= axI-...axn.

These functions in their turn satisfy the wave equation. t We note also that will on the faces of the parallelepiped for Ix I = a, the function v((N) satisfy either the condition voN) = 0, if a is even or the condition t If the solution u of the wave equation has continuous derivatives up to (1 + 2)nd order on the whole space, then obviously an arbitrary derivative of u up to order I will also be a solution of the wave equation.

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

136

aV(') °"/an = 0, if ai is odd. Applying formula (17.6) to this function, we obtain in this way 2

" fs=

av(A)

av(N)

2

ax

f

(91

I

av(N)

"

2

aU(N)

+

8x;

2

dS.

at

(17.16)

On the initial plane S2 all the integrals for given a, , a2 , ... , a" are bounded by a number which does not depend on N. We consider also the functions

- U(r)

= v(k)

w(k .r)

For these functions we obtain as before 2 aw(k ,) I " aw(k ,) °,.....°" + °,....°" , ax; at

2

JEI

/

aw(k

L, E

r)

eX;

dS 2

aw(k r)

+

)21

dS. (17.17)

It is not hard to establish that for sufficiently large k and r, the integral on the right-hand side of the last equation will be as small as one pleases. This follows immediately from the convergence with all derivatives of the Fourier series for go and g, . From this it follows that the quantity on the left side will be arbitrarily small: " S,

E

aw(k

,)

2

aw(k

,)

2

of

ax;

Integrating this last inequality in the variable I between the limits 0 and T, we will have 2 aw(k r) z " aw(k r) ...°" dig < Te, + of 8x L

L

where, by i2 , we denote the domain 0:-5 t:_5 T, fix; I < a. Using the same arguments as we employed to derive (17.11), we show that (k, r)

(w°

°_)2 dig < Me,

where the constant M is independent of k and r.

§3.2. GENERALIZED CAUCHY PROBLEM FOR THE WAVE EQUATION

137

By virtue of the completeness of the space W(°) , we conclude that the sequence v(N)

, which satisfies the Cauchy convergence criterion, must

converge in this space. The convergence of all the derivatives of u(N) in W(°) implies the uniform convergence of all the derivatives of this function by virtue of the imbedding theorem. The lemma is proved. If we use the lemma, it is easy to prove our theorem. Indeed, the values of the unknown function in the interior of the cone

0
(17.18)

depend only on the values of the functions q0 and q1 within the ball lxi < a12. We construct the functions TTn77

90)-9901

(1aiJ)'

fJ -I (Ia'I)' is a function equal to one for < 11 and zero for > 1, and infinitely differentiable. We will seek a solution u(O) of the wave equation satisfying the conditions where

U (a)

(a)

Ir=0=(00 19U

(a) (a)

I

=V,

09t

r=0

On the basis of the lemma, we see that u() is infinitely differentiable. But by what was shown earlier, within the cone (17.18) this solution coincides

with u. Hence u in turn will be infinitely differentiable, as was to be proved. §3.2. The generalized Cauchy problem for the wave equation

1. Twice differentiable solutions. We pose the following problem: to find a solution of the wave equation z

Ou=Au-a Uu=0 all

(17.5)

in the whole space satisfying the initial conditions ulr=0= U0,

au ar i_0

_u

(17.13)

It has been shown that if u0 and u 1 have derivatives of every order, the solution of the problem exists and is infinitely differentiable. But of

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

138

course, there is no necessity for infinite differentiability of the data to obtain solutions, especially since the equation involves only derivatives of second order. We consider first of all the following problem. PROBLEM 1. To determine what conditions imposed upon uo and u1 will assure the existence of twice continuously differentiable solutions. In the previous chapter we became acquainted with the generalized derivatives, and we now introduce the generalized wave operator according to the same scheme. Let u(x1, ... , xn , t) be a summable function on any closed bounded subdomain of the domain i2 of the (n + 1)-dimensional space. If there exists a function f (x1 , ... , x" , t) summable on any closed bounded subdomain of

) such that

fuDwdv =J Wfdv n

for any twice continuously differentiable function yr(x1, ... , xn , 1) vanishing outside of some closed bounded subdomain of i2, then f is called the result of applying the generalized wave operator 0 to u and we shall write

Ou= f. If Ou = 0, where 0 is the generalized wave operator, then the function u will be called a generalized solution of the wave equation. It is natural to pose the Cauchy problem for generalized solutions.

PROBLEM 2. Find conditions on uo and u1 ensuring the existence of a generalized solution. We shall consider both these problems. THEOREM 1. If u0 has generalized derivatives up to order [n/2]+3 squareintegrable on every bounded domain and u1 has similar generalized derivatives up to order (n/2] + 2, then the equation (17.5) has a twice continuously differentiable solution satisfying the conditions (17.13). PROOF. We construct the families of averaged functions {uph) and {u1h) . By the theorem of § 17, item 4, there exists a solution uh of equation (17.5) satisfying the initial conditions u1r=o = UOh

8ul

(18.1)

8t 1=0 = u1h

and having derivatives of arbitrary order. Consider the function: vD 9 = vh - Vh ; vp. is a solution of equation

§3.2. GENERALIZED CAUCHY PROBLEM FOR THE WAVE EQUATION

139

FIGURE 10

(17.5) with the initial conditions avn.v VD.Qit=O - UOho - UOh'

at

t=0

Ulhp - 141h,

=

By inequality (17.7) we have (Figure 10)

n (OV,,,)2 +

f,

(,I)2] dS<

o

and analogously for any derivative of v, " ",

8°v

a

r=1

,

E (ax,

ax;)

(

a°yp.9

a a t axial 2

axa^) +

221) at

dS

J

. 8°v

2

a.v + a x; 8x;1... 19x*. n )

f

f

1 l

2

v.v

... ax:n a

dS

)

a°VD.4

(at ax", Jaxe.)

dS.

2

(18.2 )

Moreover, by (17.10),

f (vv.a)2dS <_

[f,VP2.4dS

[f, +

1/2

)\ 4)2dS+f ax;

Si

2

2dS1/2

of

r

/

(18.3)

B y the imbedding theorem in 2 of § 10, °, sup

F 1a2PQ

r)

I

M

v + II

at Ilw

(18.4)

z

where the norm of vo a is taken in and that of 8vD c/8t in W(1"/21+2) over S2 , and M does not depend on I E (0, to).

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

140

By a property of averaged functions (§5, item 2), u0h uo in WZIn/213) u1 in Therefore, the right-hand side of (18.4) can and uIh be made as small as desired for sufficiently small hD and hq, and then the left-hand side is also as small as desired, i.e., the sequence {uh } converges uniformly with respect to the truncated cone V, and hence the limit function u is in C2 . By analogous estimates we prove that 8u/8t E CI . Since Auhr W(xn/21+2).

82Uhr/812 , it follows that the sequence {82uhr/812} converges uniformly on

V, and thus 82u/8t2 E C. Thus, the function u is twice continuously differentiable in (n + I)dimensional space and is a solution of the wave equation. 2. Example. We consider the wave equation in a 5-dimensional space. We saw above that it sufficed to demand by way of the initial conditions the existence of square-integrable fifth derivatives of uo and fourth derivatives of u1 . We shall show that if only derivatives of lower order are square-integrable (for uo derivatives of the fourth order, for u1 derivatives of the third order), then the solutions may not have continuous derivatives of second order. Let r = (Es , x01/2 ; then if f(y) is continuously differentiable m times (m >_ 5),

u=

f(t-r)-f(t+r)+ f(t-r)+f (t+r) r3

(18.5)

r2

will be a solution of the wave equation with initial data uo =

ul =

- 2-f3r r) + 2fzr) r -

2/''3r) + 2j2(r)

r

r

where

,

,

2f1(r) =f(-r) -f(r),

where 2f2(r) = f(r) + f(-r).

We have 82u

_

["(t- r) - f(t+r)]+r[f"(t-r)+f"(t+ r)] r3

8l2

If m > 5, it is not difficult to verify that 02u/8t2 and 82u/8r2 are continuous, and an application of l'HBpital's rule gives 82u

8t2

= lim

r) - f'(t + r)] + r[f"(t - r) + f"(t + r)] = 2 f(V)(t) r3

r-0

Let

0,

f(Y)_ CY-42,

Y<1, Y > 1.

S3.2. GENERAUZED CAUCHY PROBLEM FOR THE WAVE EQUATION

141

Then

uo -

(0,

- (0, u

1

r<1,

it='L + a '5--r

r> 1

r< 1,

- a ' - + a (a -

r> 1 ,

and it is not difficult to see that u(, E W(') , u1 E W23) but uo ul g WM if 9/2 < a < 11/2. Thereby we obtain 2

a u [_0

0,

t A(t- 1)°-S,

ate

W(s)

t < 1,

t> 1

(A=const#0),

and the function u(r, t) given by (18.5) turns out not to be twice continuously differentiable if 9/2 < a < 5. As is easy to see, u(r, t) is a generalized solution of the wave equation for the data uo, ul and moreover the unique solution (see the theorem in item 5). If 5 < a < 11/2, then although uo ¢ W(S) and u1 0 WM , the corresponding solution is twice continuously differentiable, and, as a result, the conditions u0 E and u1 E W[",'2)+2 are not necessary. Analogous examples may easily be constructed for (2k + 1)-dimensional spaces. (53) 3. Generalized solutions. We saw above that a solution of the wave equation having continuous derivatives possesses the property that its norm in an arbitrary space W.') (if we consider it as an element of this space) remains uniformly bounded. This norm may be estimated if the initial values of the function are known, more precisely, if we know the norm of u0 in the same space W (r) and the norm of u 1 in W(-')

We shall say that the function u is a generalized solution of the wave if Du = 0 , where 0 is the generalized wave operator, and

equation in WZ

1y

if the function lies in W(1) on every bounded domain. Obviously, every generalized solution of the wave equation in WIM is the limit of a sequence of functions uk converging in W(l) on every bounded domain and themselves twice continuously differentiable solutions of the wave equation: Duk = 0 . Indeed, by what was shown above, u will be the limit in Wl') on bounded sets of the sequence of averaged functions on any bounded domain: uhk hk . Furthermore, each averaged function obviously satisfies the wave equation, since the wave operator can be shown to commute with the operation of averaging. This is proved by the same means which we used earlier to prove that the derivatives of averaged functions are the averages of their derivatives (see Chapter 1, §5, item 2). We shall prove the converse theorem.

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

142

THEOREM. If the function u is the limit of a sequence uk of twice continuously differentiable solutions of the wave equation having uniformly bounded W(I) norms on any bounded domain with the sequence converging weakly to

u in LI , then u is a generalized solution in

W(1)

PROOF. On the basis of the theorem on the existence of generalized derivatives, we conclude that u belongs to WZ 1 ) on any bounded domain. The fact that u is a generalized solution of the wave equation follows from

f

kim /

where yr is an arbitrary function having all continuous derivatives and vanishing outside some bounded domain V,, . Since

f we obtain

0,

JuDwdv=0 ,(18.6)

as was to be shown.

4. Existence of initial data. THEOREM. Every generalized solution in W2I) on any bounded domain G has on the plane t = 0 limiting values uo E WZ 1) and uI E L2 , i.e., for any bounded domain G li m 11u(t) - uOIIWzI) = 0, im II watt) 1-0

- uI

(18.7) 11

'L2

= 0,

where u(t) = u(xI , ... xn, 1).

The plane t = 0, of course, was chosen completely arbitrarily. Our theorem leads to the conclusion that a solution of the wave equation in ly() can have the plane t = coast only in the role of a removable singularity. We recall that from the imbedding theorem there follows in this case a somewhat weaker result, namely that every solution u of the wave equation in WZ I) can have the plane t = const only as a removable singularity if we consider the value of u itself in L2 (without derivatives). We proceed to the proof of our theorem. PROOF. We consider in the plane t = 0 an arbitrary n-dimensional ball S2 and construct through its boundary the reverse characteristic cone (Figure 11).

Denote by E(t) the ball obtained from the intersection of this cone and the plane t = const. The value of E(t) for t = T will be denoted by S3 .

§3.2. GENERALIZED CAUCHY PROBLEM FOR THE WAVE EQUATION

143

FIGURE I I

Let V (t) be the domain bounded by S3, 1(t), and the lateral surface of the cone, V(0) = V . Suppose that v is some twice continuously differentiable solution of the wave equation. For it we have the inequality

401:1-1 (aX;)2+

(at)2) dS> f (<)

[f

(ax;)2+(ar)2j dS

if t > tl , which is obtained from (17.7) by replacing t by T - I. Integrating in t from t1 to T, we obtain

{n()2

()2 (y)2] t1)L()

()Jds.

2

(18.8)

Let vh = uh(t + k) - uh(t), where Uh is an averaged function for u and

k>0.

Applying (18.8) to vh for a fixed value of k and substituting its value for Vh , we will have

T tax,

(auh(r+k)

1

JV(t,)

E(t,>

- aUh(r))2+ (auh(r+k) - auh(1))2J dv ax;

at

(Ou(1+k) - auh(r)2+ (auh(t+k) ax; ax, ) at

at

- auh(t)1dS. at J (18.9)

In formula (18.9) we may pass to the limit as h - 0, since the convergence of Uh to u takes place in W211. The L2-convergence of the derivatives j:xA and

on 1(11) follows from the estimates for Uh of the form (17.8).

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

144

F,

Therefore,

n

a

2

(u(t + k) - u(t))] +

8 + k) - u(t))] [(u(t

2

dS

I

+I_ (u(t+k)-u(t))]2ydV.

(18.10)

The functions 8u/8x; and 8u/8t `belong to L2 on V, and, as a result, are continuous in the large on the domain V. Let t1 < T/2. We may then choose 8(e) so small that for k < d(e) the right side of the inequality (18.10) will be less than a independently of t1 . Hence, l ll £(t,,

{[a(u(t,+k)-u(t))]+ [a(u(t,+k)-u(t))]2} dS<e.

=1

i

111J11

We put t1 + k = t2 and obtain Ilu(t2) - u(1I)IIL(" < e, u

Ft (`2) - ar (ti)ll

<e

(where the norms are taken over E(t1)) for t1 and t2 sufficiently close and in particular for sufficiently small t1 and t2 . By virtue of the completeness of the spaces L2 and LZ1) , there follows

from this the existence of a limiting value as t - 0 in L2 for 8u/8t and a limiting value as t - 0 in LZ1) for u, on any bounded domain in ndimensional space. If we note also that a limiting value for u in L2 exists by virtue of the imbedding theorem, we arrive at the conclusion that the

values u(0) and (8u/8t)(0) exist and are in Wit) and L2, respectively. Therefore, for any bounded domain in n-space Ilu(k) - u(0)IIw2n) -0,

u

at (k) at (0)II LI

0

as k - 0. The theorem is proved. 5. Solutions of the generalized Cauchy problem. THEOREM. For any functions u0(x1, x2,.. - x,) ,E WZ 1) and u1(x1 , ... , xn)

E L2 on any bounded domain there exists a unique generalized solution u of the wave equation in W21) satisfying the conditions "11-0

U0,

8u = u1, at it-0

(17.13)

§3.2. GENERALIZED CAUCHY PROBLEM FOR THE WAVE EQUATION

145

where for t -+ 0. u(t) converges to uo in W21 ) while 8ulat converges to u1 in L2 on any bounded domain. PROOF. Consider the averaged functions for the initial data: uOh and ulh . If we put them in place of uo and u 1 as initial data, we may find a solution of the Cauchy problem, since these new initial data are infinitely differentiable. Denote this solution of the wave equation by uh .

converges in Wlll on It is easy to show that as ho - 0 the sequence ukP any bounded domain in the (n + 1)-dimensional space of x1 , x2 , ... , x , t and on any bounded domain in the n-dimensional space of x1, ... , x,, for an arbitrary fixed value of t. Indeed, put uho - uh. = up 9 . The function vp a is a solution of the wave equation, for which the quantities A and B in (17.9) and (17.11) will be as small as we please for sufficiently small hp and by . If we use these inequalities, we see that for sufficiently small ho and hQ we obtain

JvqdV<e,

f [E it

ax.Q)2+ ("VP

from which it follows directly that the sequence

uho

2J dV <e,

converges by virtue of

the completeness of W21) . The limit function u by virtue of the theorem in item 3 will be a generalized solution of the wave equation. Furthermore, if we employ the inequalities (17.8) and (17.10), we will have {t()2

J a,

+

(aa

)2]dS
(Vp.e)2dS <

These inequalities mean that the sequence uh converges on an arbitrary v

bounded part of the plane t = const in the sense of WWWW , with the conver-

gence uniform with respect to t on any interval [0, T] . One easily concludes from this that the limit function u, which was shown

in the theorem in item 4 to have limit values for u and au/at respectively in W(1) and L2 as t - 0 on any bounded domain, assumes on the initial plane the given values uo and u1 . In fact, IIu(t) - u0IIW=I

< puh(t) - uh(0)IIW2(') + Iluh(0) - u(0)IIW=>> + Iluh(t) - u(t)IIN=i1

For sufficiently small h we have by the estimates (18.11), which are uniform

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

146

in t, that Iluh(t)

- u(l)II w( 5 3 ,

Iluh(0) - uolly=" = Iluok - uoll.=o 5

Choosing t sufficiently small for fixed h, we will have Iluh(l) - uh(0)II W=o <

because uh(t) is smooth. Consequently, llu(t) - u(0)IIWzo < C.

Analogously, it is easily shown that II

et (t) -

u1IIL2

< C.

Thus Theorem 3 is proved. §3.3.

Linear equations of normal hyperbolic type with variable coefficients (basic properties)

1. Characteristics and bicharacteristics. In the present section we shall show the existence of solutions of the Cauchy problem for linear equations of hyperbolic type with sufficiently smooth coefficients for sufficiently smooth initial data. We consider the equation 2k+I 2k+12k+I a2u 8u (19.1) Lu + E B.ax + Cu = F, E A, 8 8x i r J i=o j-0 1=0 where A,, (A1J = Aj) , Bi , C, and F are functions of the variables xo , x1,

... , x2k+1 continuous together with their derivatives up to order K + 1 ,

where K is a sufficiently large number. We assume that for each point of the space, the quadratic form 2k+I 2k+I

A(p) = E E Aijppj

(19.2)

i=o !=o

may be brought into the form 2k+1

q? + qo

A(p)

(19.3)

r=I

with the aid of a linear change of the variables p,. The equation (19.1) is called in that case an equation of normal hyperbolic type.

A characteristic surface or characteristic for equation (19.1) is a surface G(xo, x1 I ... , x2k+I) = 0

(19.4)

§3.3. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE

147

for which, on this surface,

A(p) = 0, 2k+I

(19.5)

2

ME )

(19.6)

> 0,

where p = (p0, p2k+1), Pi = Tx' [294]. An equation of normal hyperbolic type has characteristic surfaces with a conical point at an arbitrary given point of the space xo , xI , xz , ... , X2k+1 . These surfaces are called characteristic conoids. In the case of an equation with constant coefficients the characteristic conoids reduce to characteristic cones.

We recall some of the simplest properties of characteristic conoids and their construction. Everything below takes place in a sufficiently small neighborhood of the particular point xo0 , ... , X2k+I

We set

aG aXi

=P;.

(19.7)

As is known from the theory of partial differential equations of first order (2711, a surface (19.4) satisfying the equation (19.5) is obtained as a manifold

built up out of bicharacteristics, i.e., solutions of the system of ordinary differential equations:

dx, _ _dp, = ds. (19.8) 1OA/ap, - 11 aA/ax, More precisely: a parametrized equation for the surface (19.4) is obtained in the form (0) (0) xi=ci(s,X0(0),X1(0),...,X2k+l,PO ,PI(0) ,..., Pro) 2k+1),

(19.9)

where xo) , x1°) pik+1 are functions of 2k inde, Xik+l , Po) I Pi°) I pendent parameters vl , v2 , ... , v2k . For these: a) the functions (19.9) together with >

p.=3R;(s,x0 ,...2X2k+I+Po ,PI ,...,P2k+l) 1

(0)

(0)

(o)

(0)

(0)

(19.10)

should represent the general solution of the system (19.8) depending on 4k+4 arbitrary constants, where in (19.9) and (19.10) we set (19.11) p.3_0=p,°) (i=0,...,2k+1); X,j.,,o=x;0), b) the functions x(°)(vl , ... , v2k) and p(°)(v1, ... , v2k) should satisfy the conditions (19.12)

A(p%I o=0, 2k+1

Ep

(0)

(0) axe

8v

=0

(

j=0 , I

,

2

,..., 2k ),

( 19. 13 )

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

148

with the quantities x; and p, in the condition (19.13) understood as expressed in s by formulas (19.9) and (19.10): c) the equation (19.9) should give a parametric representation for a manifold of dimension 2k + 1 (and not lower). We shall show how to construct by means of this theory the characteristic conoid with vertex at the point X(0) xl°) , X2k+I Assume that we have constructed the solutions (19.9) and (19.10) of the system (19.8). In them we shall consider p(6°) , p(o) Pik+I as independent parameters, upon which we shall impose two conditions: the equation (19.12) and the normative condition 2k+I

p(o)2=1

(19.14)

iso

The quantities x6 °) x1°) , ... , xzk+I will be taken as constants, not depending on p(0 , p(°) , ... , P2k)+1 It is not difficult to verify that equations (19.9) give a parametric equation of a surface satisfying the equation (19.5). Condition (19.12) is satisfied by the choice of the p;°) . The condition (19.13) is also satisfied, since 0)

0. As to the fact that in this case we will have from (19.9) the parametric equation of a manifold of dimension 2k + I, we shall give a brief proof of it later. We note first of all some important properties of the equations (19.9) and (19.10). Set

SPi°) = y; , spI = n; .

(19.15)

We show that the functions r;, and n, depend on s and on p(,°) only through

y,, i.e., that (S' X°(0),X1(0) , 71

(0) .,x2k+I,

Yo S

S ,..., y2k+1 S )

Y1 ,

(S' X°(°) ,XI(°) ,. .,X2k+I+ (°) Yo, Y >..., S $

,

(19.16)

y2k+1 S

do not depend on s for given y, and x(o) . Indeed, in place of s and p, consider the new variables s1 and p0) , setting (I)

OSI'

Pi

(19.17)

Putting these new variables into the system (19.8), we obtain a system of equations for p;1) and xa with independent variable s1 . This system

§3.3. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE

149

turns out to coincide with the system (19.8) and may be obtained from it by

a simple renaming of the independent variable and of the functions p. It follows from this that the functions (0)

(0) (0) (0) X, _ ;(asl , XO , ... , X2k+1 , p0 > ... , P2k+1)

(0) (o) (o) = apf = S 7r,(asl , X0(0) , xl(0) , ... > x2k+I , PO , ... , P2k+I)

(1)

1

Pi

(19.18)

I

also satisfy the system (19.8). Set (out)

(I)

Pi

(19.19)

where p(O)(1) = ap(0) or P(0) . = p(o)(1)/a r r r i We have obviously (0)

(19.20)

xr1J,=0=x; . The equation (19.18) may therefore be rewritten as (0)

X, _ i asp , X0

, .

(1)

(o)

(

Pi

=

1

,X

6_

(0)(u

(0)(1)

(0) ,

a

Em I

,

a

1

(0)(1)

(0)(1)

Pp

(o)

asl , x0 , ... , x2k+1

7t, S

.

,

(19.21)

P2k+1

a

a

1

On the other hand, for the same functions x1 and p as solutions of the system (19.8) satisfying conditions (19.19) and (19.11) we will have on the basis of the uniqueness theorem (0)

X.

p,1)

X0

=

1 7r (sl s r

,

, .

X(0), 0

(0)(1)

(0)

(0)(1)

, X2k+I , Po

... ,

X(0) 2k+1

,

P2k+1

p(0)(I), 0

... ,

P(0)(1) 2k+1

(19.22)

The right-hand sides of (19.21) and (19.22) are identical for any c f. Setting

sl = I and a = s, we obtain our assertion. Set (0) (0) (0) ,(s,x0(o)I...,X2k+1,P0 ,...,P2k+1) (0)

= X,(X0(0) , ... , x2k+1 , Yo, YI > ... I Y2k+I) (o)

71,(s, XO

(0)

(0)

x2k+1 , po

,'

(0)

, P2k+I)

n;(x0(0) , ... ,

(0)

x2k+I , Y0, YI , ... > Y2k+1) .

We shall show that the equations (0) X,=Xi(x0(0) , ,X2k+I,YO,...,Y2k+I)

(19.23)

express a change of coordinates from x; to Y, in our space which carries the point y, = 0 into x, = has close to this point a Jacobian different

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

150

from zero, and is continuous together with derivatives up to order K, where K is a sufficiently large number. By known theorems of the theory of ordinary differential equations, the functions n, and , have continuous derivatives with respect to and up to order K + 1 .

Let us consider the functions x, and p1 as functions of the variable (o) (0) (o) p(0) p(0) p(0) s and the parameters x(0) o ' x1 I X2 'x2k+1 o I '. 2k+1 and consider the derivatives of xi and p3 with respect to s. We shall show that lims_0(d°x1/ds°) is a homogeneous polynomial of degree a and iim(d°p;/ds is one of degree a + I in the p(o) For the proof we employ once more the system of equations (19.8). Differentiating the equations of the system with respect to s successively and eliminating each time from the right-hand side the first derivatives, we may

express d°x,/ds° and d°p1/ds° in terms of the quantities xi and p.. We show that ,

X, °) (x0, x1 , ... , x2k+1 , PO , P1 , ... , P2k+l) ,

ds°

d p;

(19.24) __

O

P(Q)( XO,

XI , ... , x2k+1 , PO, P1 , ... P2k+i)

where X,(°) and P;(°) are polynomials of degree a or a + 1 in p, . In order to prove this, we apply the principle of mathematical induction. For a = 1 , our assertion is obvious. Suppose that for some a our assertion has been proved. Differentiating (19.24) with respect to s, we obtain d ds

2k+1

d°x _

( ds°)

-

ds

F,

axe

ds +

2k+1 aX(°) 2k+1

ax!

i=0

2k+1 8X(°) dp.

aX,°) dxx

F,

app

2k+1 ax(a) A11P1 +

1=0

i=o aPi

ds 2kk++I 2k+I

2 1=0 M=0

aA 1 mPiPm' 8x-

from which it is clear that our assertion is still valid for a + 1 Passing to the limit for s -» 0, we obtain the desired assertion. Analogously we prove the assertion for p, . Since the derivatives in s exist up to order K + I, we obtain by applying Taylor's formula 2k+1

A;°)pj°)+Fs

x, =x;°j+s !=0

P,

=P(0)+

l

K

K

°=2

X;°)(p!o))+RK)

0=2

°+If'°+l)( O))+RK'>

( 19 . 25 )

§3.3.

LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE

151

rI!°+1)

are polynomials of degree a and a + 1 in pil°) . From this it follow that for any r < K - I , we may express x; and p, in terms of where X'°) , y j by

x-

X;0) = 2k+1 A'0)yj

E j=0

r+I

R+,

+` n=2

r+1

P1 =

Y,+ Erlin)(Y1)+Rr+1 n=2

where the ath-order derivatives of R(')1 and R;+') with respect to p vanish for s = 0 like . Simple calculations show that thereby the derivatives of R;+1 and R;+') in yr up to order r+ I vanish at the origin of coordinates. As a result, the derivatives of x, with respect to yj up to order r + 1 exist and are everywhere continuous. It is obvious that the Jacobian sk+2-°

D(x0, x1 , ...

> X2k+l )

D(yo,y1,...,Y2k+1)

s=000,

and on the basis of the implicit function theorem there exists a domain with center at xo) , ... , xik+1 on which an inverse is defined: yo, yl , , Y2k+1 are one-valued functions of the variables xo , x1, ... , 'x2k+1 On this domain, we obtain

yj-=

2k+1

r+I

j +R, j- x(0)) j + Y(")(xj- x(0))

H(o)(X ii j=0

(19.26)

n=2

where H f?) is the inverse matrix to A(°) and Y(") is a polynomial of degree

n in The equation

j=0,...,2k+1. 0 in the variables y, may be written in the form A(y) = 0

(19.27)

and consequently represents the equation of a cone, i.e., a manifold of 2k + I dimensions. Thus we have proved that conditions a), b), and c) are satisfied as given above, and consequently (19.27) is the equation of the characteristic conoid.

By our assumption that the equation is of normal hyperbolic type, the characteristic cone will divide the whole space into three parts: the exterior, the upper interior, and the lower interior of the cone. Any direction at an arbitrary point of the space will either point into the interior of the cone and will by analogy with the ordinary case be called time-like or it will point into the exterior of the cone and will in that case be called space-like. In the analytic definition, those directions I will be space-like for which A(l') < 0 and those time-like for which A(1) > 0.

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

152

In a neighborhood of the given point (x (0) , ... , x2k+1) we may make a linear transformation of coordinates carrying the matrix IIA;°)II into canonical form at (xo ) , ... , x2k+1) . Suppose that the transformation reducing (19.1) to canonical form at (xu , ... , x2k+l) has the form

xi-XI

j

j

where aij and yi j are functions continuously differentiable k + I times in the variables x(0)' X(0) ' x2k+1 for which the determinant satisfies Iaij I > h, where h is a positive number not depending on xo , x(1°) , ... , x2k+l , and K is a sufficiently large number (see, for example, [243]). >

Then under the substitution pj = Z "I yjigi we have 2k+1 2k+I

2k+1 2k+1

2k-+1 2k+1

1=0 m=0

i=0 j=0

Aijpipjlxo xo

Aijyilyjm

i_0 j=0

g/gm,

and, as a result, by the assumption about the transformation being canonical we must have

0, 1 # m,

Zk+12k+1

E E Aijy,lyjm = i=° j=°

-1, 1,

1 = m # 0'

/=M=O.

We shall assume that such a transformation has already been carried through, and that yi is the corresponding local system of coordinates, giving (19.27) the form 2k+I 2

Y°- EY, =0

(19.28)

i=1

or if we set p =

E?ki 1 y? , then

yo-p2=0,

(19.29)

i.e., in the coordinates yi the characteristic conoid turns into a right circular cone. Such a transformation can be made in some neighborhood of each point of the space. In the case of variable coefficients it is necessary, however, to note the following important circumstance. In the solution of the Cauchy problem for the wave equation, the bicharacteristics were straight lines and therefore our solutions of the characteristic equation could be extended to arbitrary time t. In the case of an equation with variable coefficients, the field of bicharacteristics may have some kind of singularity (for example, a focus) and therefore we can construct a conoid only in some neighborhood of its vertex. The size of this neighborhood may be estimated from the coefficients of the derivatives of second order.

§3.3. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE

153

2. The characteristic conoid. We transform the equation (19.1) to the coordinates y, . In the new coordinates it takes the form a2u

2k+12k+1

2k+1

au ayiayj +EB-+Cu=F. EA.

j=o j=o

i=o

(19.30)

' ayi

For this, the cone with the equation G(Yo, Y 1

, ... , Y2k+1) = Yo + P = 0,

(19.31)

2k+1 2 where p = F;=1 yi , is the characteristic cone, and the lines y, - a;yo, where E2k+l a? = 1 , are the bicharacteristics. Let us consider in greater i-O

detail what consequences follow from this fact. On the characteristic cone

q,=aG=y'-yi=-ai (i#0), Yo a'i p

aG

(19.32)

qo=ayo=1.

We substitute these solutions into the system of equations

=ds (i=0,...,2k+1),

dyi

laA'/aqi

where A = E;ko' Ejso' Aijgigj . Since W = aia; (i = 1, ... , 2k + 1), we can write this system using the notation

s- = V(s)

(19.33)

in the form _

amp(s)

Aoi - -j=1 Aijaj A00 2k+1

90(s) 2k+1

_

=1

(19.34)

- Ej=1 Ajoai

Thus, A00

-

2k+I

Aioa, = 9(s),

(19.35)

2k+ l

Aoi

- E A'ijaj = a,9(s) -

(19.36)

j=I

Multiplying the second equality by a, and adding over i from I to 2k + I we will have 2k+I

E Aoiai i.I

2kk++1 2k+I

L j-1

2k+I

E Aijalaj = E ai07(s) = Se(s).

i=1

i=1

(19.37)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

154

From this it follows that on the surface of the characteristic cone 2k+I2k+I

A00 - E E A,ja,aj = 29;(s) i=I j:I

(19.38)

or 2k+1 2k+1

E E A,ja,aj = Aoo - 2So(s).

(19.39)

j=I

r=1

3. Equations in canonical coordinates. We pass to the study of the Cauchy problem for the equation (19.1). Let it be required to find solutions of that equation which satisfy the conditions , x2k+I),

uIxo=o = uo(x1, auIX0=0

( 19.40)

8xo

We make another important assumption. We assume that at each point of the part of space being considered, Aoo > m > 0, A,, < -m < 0 (i # 0) in addition to (19.2)-(19.3). We now introduce a new variable, setting t = yo + p. For convenience in the arguments, we shall in the following designate by 8/8y, the partial derivative with respect to y, taken on the surface yo = coast, i.e., for constant yo, and by D/Dy, the partial derivative with respect to y, taken on the surface t = const. Then 8

8

Y, D

D Dy,

ay,

p Dt ' a2

__

a2

ay,ayj

_

Dt'

D2

y, D2

Dy,D: +

ay,0ya

yyj D

D2

Dy,Dyj

D2 a2 ayo = Dt2

D

8yo

PT Dt2

Y,

p3 Dt

D2

p Dy,Dt 2

2

+ a2 _ D2 a y? Dy? +

1

(p

y? p3

2

D

2y,

D2

(i#.1) y2 D2

/ DI + p Dy, Dt + p2 D12

§3.3. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE

155

Setting these values in our equation, we will have 2k+I2k+1 A1.,

2k+1 2k+1

p Dye Dt

2

Ao,

D2u

2k+1

A oDy, Dt

P2 D

2k+1

D2u

Y

1

D2u

Y )'

J=I

+2

+2

Dy, DyJ

+ =1

2k+I2k+I

D2u

D2u 2 + A,0-

Yj D

,_

+-A, 3J 2k+I2k+1

2k+1

Y y

_

2k+1

Y

+

1+

Du

+ Bo

Dt

2k+1

+EB;Dy +Cu=F ,=1 or

D2

2k+I2k+l

2k+1

DY, DY; 2k+1

D +E-+C u

2k+1

+ 2E

D

FA,Jy'+.9',0

,=1

1

Dy,

P

J=1

2k+1

2k+I2k+l _ Y y

2k+1

Aj ''-3 +E h, Y,+Bo

+ P 2k+I2k+1

=I

P

J=I

P

P

11

Du Dt

D2u Ao Y, + Aoo

2 J=1

1=1

2k+I

Y y

+ =1

DY;

=l

,=t

P

2 = F, Dt

(19.41)

which we rewrite more briefly in the form JD-2

L1Olu + M1o1Du +

Dt

u

Dt2

= F,

(19.42)

where by L10j and M10) we denote the operators appearing, respectively, in the first and second curly brackets on the left side of (19.41), and by J the third curly bracket in (19.41). As is easily seen by virtue of (19.32), (19.35), and (19.39), on the characteristic cone i = 0, we have Jj,_0 = 0, and (19.42) takes the form Du = F. L(O)u + M(O Dt We carry out one more change of independent variables, introducing polar coordinates instead of y1 , y2' ... ` y2k+l

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

156

4. The basic operators M(° and L(° in polar coordinates. We will define the position of a point by the coordinates p, where 2k+I

P =

Y? 1=1

and the unit vector Y. It is necessary for us to investigate more closely differential operators in these variables. In order to calculate any differential operator on the unit sphere, we choose some coordinate system on the surface of this sphere. For us the choice of the system is indifferent, and therefore we may, for example, choose for each point of the sphere its own coordinate system regular in a neighborhood of this point. We may, for example, consider for this purpose polar coordinates. The system of polar coordinates on the unit sphere is given by the equations Z2k+I = PCOSt92k, 22k = p sin t92k COS t92k- I ................................................

(19.43)

z3 = p sin t92k sin t92k -I "' sin $3 cos 192, sin 193 sin 62 cos i91 ,

Z2 = p sin t92k sin 102k_ I '

z I = p sin a2k sin t92k _I .

Sin 193 sin a2 sin 19, ,

.

where -n < a1 < n , 0 < 19; < X, i = 2, 3, ..., 2k, and z, is an arbitrary Cartesian coordinate system obtained from the yj by a rotation of axes, i.e., by an orthogonal transformation y; = 1:2k I y; J z It is useful to note the formula

D(Y1,Y2,...,Y2k+I)

D(z1,22,...,z2k+I)

D(p, 6 1 , t92.... 102k)

D(p, 01, 82, ... 1 62k)

=P

2k

sin2k-202,_,

sin2k-1

2k

sine 633 sin $2 (19.44)

From this, there follows immediately d!Z = p2k

sin2k-2 t92k-

sin2k- I

$2k

I

'

Sing $3 sin 62 dp dal

.

d 62k . (19.45)

If we take into consideration that the surfaces p = const are spheres, the lines 01 = const, ... , $2k = const are radii of these spheres and that

ap/av = I , where v is the exterior normal to the sphere, we conclude that the surface element of the sphere is dS = p2k

sin2k-2

sin2k-1

82k

02k- I

, ... , sin d2 dt91 d$2 , ... , dd2k . (19.46)

§3.3. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE

157

The system of equations (19.43), in the form when they are solved for p and the 6,, will be 2k+I

P=

ZZ i=1

01 = (sgn z1) arccos Z2 , P2

02 = arccos

z3

, ... ,

(19.47)

P3 elk-1 = arccos

z2k

,

P2k

elk = arccos z k+I , where

2

2

ZI + Z2 = P2 ,

2k Eil Zi

2

Pzk

In the following, for convenience in calculating any operator on a function at the point A0 on the sphere, we shall choose the axes zt , Z2 , ... , z2k+l in

such a fashion that the point X. under consideration falls on the z1 axis, i.e., that we have at this point z 1 =P, z2 = Z3 = = Z2k = 0. A function `I' given on the sphere is said to be s times continuously differentiable at the point 1° if it has at that point continuous derivatives up to order s with respect to all the coordinates t91 , t92 , ... , 62k for the choice of coordinate system as given above. A function which is continuously differentiable s times at every point of the sphere is said to be s times continuously differentiable on the sphere.

A linear differential operator of order s on the sphere is said to be continuous at the point 1° of the sphere if it consists of a linear combination of derivatives up to order s with respect to the variables 61, d2 , ... , 02k with coefficients continuous on a neighborhood of 1°. If it is continuous on a neighborhood of each point, it is said to be continuous on the sphere. An operator of order s may be applied to any s-times continuously differentiable function. We will call a differential operator L defined on the unit sphere r-times differentiable if its coefficients at any point are r-times continuously differentiable. If an operator of order 1 which is r-times differentiable is applied to a function which is m-times differentiable, where I < m < 1 + r, then the result is a function which is (m - 1)-times differentiable. After these remarks, we pass to the calculation of the operators M(0) and L10 in polar coordinates. To begin, we calculate the form of the operator M(0). It is easy to see that 2k+I Dv Dv (19.48) yi J.1 / .

`e,,

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

158

In turn, 2k+1

Dv

Dzj

Dv DO Dv Dp Dfls Dz + lip Dzi

Zj av

av

p 8p

8t9i-1

2k+1

Pjj=1

Pj

$=j-1

8v Zs+1 + zj

86s psp +I

Replacing all the z's and p by their expressions and taking into consideration that at the point concerned all the pk equal p, we may give M(0) the form G(O)8v + 1 M(O)v = A(0)v aP

(19.49)

,

P

where G(° is a function of the variables t, p, $1,

$2k defined on the whole t9-sphere and for all p in the interval 0 < p < M and continuously differentiable with respect to all its variables, while A(0) is a differential operator of first order on the unit sphere which is sufficiently often differentiable.

We also examine the value of the operator M(0) for t = 0, i.e., on the surface of the characteristic cone. For this, we have y j / p = -a , and by virtue of the equations (19.36): 2k+1

2k+1 _ y

i=1

j=1

DU

2k+1

2F, FA;j-'+AiO D P y;

Dv

y

i=1

2k+1 yi

Dv - 2o,(s) E p Dyl ice[

2to(s)a-, and by (19.39) 2k+1

i-I

yty

2k+12k+I

I

Aij

Aii

p

1-1

j=i

p

2k+1

+E

hi

i-1

[E_+2s] 2k+1

)

I

y

` + B° p 2k+I

+

Biyp + B° .

G

Therefore,

M°vj,.0 = -2rp(s)

8v + (1 (App

lop

P l\

-

2k+1

s) A;i

29p(s)

- i) J

2k+1

2r0(s)

y

B; p)

§3.3. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE

159

We show now that 2k+1

A00 -

1

1: Ai,J pc0 = (2k + 2)10(0).

We consider for this purpose the values AiiI equalities hold:

P=0.

(19.50)

Obviously, the following

2k+1

aiA10lp=o =90);

A001P=0 2k+1

Aoi'p=0 -

=I

aiAijIp=0 = airp(0),

where a, are completely arbitrary numbers such that E k;1 a2 = 1 . If we set al = ±1 , ai = 0, i # 1, we see that A®I,_0 = *AO,IP=0 + ?(0), from which it follows that A001p=o = 0(0),

Ao1iP=0 = 0.

Further, putting al = sin w am = ± cos to,

where / # m,

i#1, i#m,

ai=0, we will have

sin co[A,, I p=0 + V(0)) ± cos wAlm I p.o = 0,

from which we obtain, using the arbitrariness of w,

Al10.0 = 0 (196 m),

A11ipao = -q'(0)

From these equalities, formula (19.50) follows. Furthermore, from the existence of continuous derivatives of the coefficients Aid with respect to the variables yo , y1, ... , y2k- I it follows that (1)

Aid = Aid Ip=0 + pA1 J +

2

2

(2)

Aid

+

+

Pk

kl

(k) Aid + Rid(A')

where RAN) (the remainder) vanishes together with its derivatives up to order

N for p = 0. Analogously, V

9(s)

= ArjP=0 + q0(0)

N

p mel

"A-tm)+P(N)

'

i

Considering this decomposition, we finally obtain M(0)vlr=0

= -2,p(s)

f

8v aP

+

Ck P

+ G/ vl J

Lo

(19.51)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

160

where the function G(;, p) is bounded and has bounded derivatives of sufficiently high order. We shall now express the operator L(0) in the same coordinates. We will have after an obvious computation L(0)v = Q(o) O22 +

S(°)v

(19.52)

P R(0) aP + P2

where do) is a function sufficiently often continuously differentiable on the sphere, R(0j is an operator of first order on the sphere, and S(0) is an operator of the second order on the sphere, all is differentiable sufficiently many times. It is of interest to calculate the value of the operator L(0)v for t = 0. For this purpose, we represent the operator L(0)vl,,, in the form L2k+12k+1

2k+1 D2v L(O)VI 1=0

=

D2 r=t

y

D2v

+ PE EG+1Dy1Dyj ,=t J=1

2k+1 1: g. Dv +

1-1

Cv

l

(19.53)

9D(s) DY; +

to

Such a representation obviously will be regular, where A1j will be functions continuously differentiable in p and t9. Obviously,

2k+1 D2v

2k+1 a2V

=I DY;

t=1 aP

2k 8v i+--+ TAV, 1

P aP

P

where a is tthe surface Laplace operator. We have L(°)vI,_0

= S ap2[l +PQI(P, X)l + p[2k+PQ2(P,

)]aP

+ p[o+PQ3(P, ))lv} p(s)

.

(19.54)

1=0

5. The system of basic relations on the cone. Let A(0) be an operator of second order in the variables yt , ... , y2k+I , or, equivalently, in the variables p , 191, ... , $2k , the coefficients of which depend on the variable t : 2k+12k+1 D2v 2k+1 _' Dv A(0)v

= E E A.. t=1

)=I

+E ,=I

Cv . Dyt

We shall denote by the symbol d'A/dt' = A(t) the operator of the form D2v + 2k+1 DAB Dv + D'Cv D 1A' Av= (19.55) () Dt' DY1 DY, Dt)1 DY; Dtt >

§3.3. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE

161

the coefficients of which are the derivatives of order I of the coefficients of the operator A(O)

We form the operators M(1) and L(1) by the rule written above and construct their expressions in polar coordinates. By calculations analogous to those already made, we obtain m(l) = L(1)

+ -'A(l)v;

(')-'9v

aP

P

(1)49 2v

1

=Q Opt+pR

(1) 8v

(19.56) 1

8P+p2S(1) V.

The coefficients G(1) and Q(1) are functions continuously differentiable on the sphere, A(1) and R(1) are operators of first order on the sphere, and S(1) is an operator of second order on the sphere, differentiable sufficiently many times.

We return now to equation (19.42). Differentiating it 1-times with respect to r and setting t = 0, we obtain (if we take into account that Jl r=o = 0) : II

E r! (1 - r)! r=o

L(1-r) D'u +

Dt'

M(t-r) Dr+I u

I!

"

(1- r)!

rso r!

Dtr+1

+ E r! ( 1!- r) r=O

D1-'JD r+2U

1

1

r

L.11

i. D'F

= Dt

r-r Dt

(19 . 57)

Introducing the notation Dol)v

= L(1)v,

D(,l)v

= IL('-1)v + M(1)V'

v = M(o)v + Dt v 11

D(1) v =

r! (1-r)!

L

(1-r )

I!

19.58) 11

v+

MU-r+),U

(r-1)!(1-r+l)! DI-r+2 J 2
+ ( (11 Dt we rewrite equation (19.57) i n the form 1++11

r=o

We now set

Du

(1)D'u

L D'

Dt,

__ `Nk1

u,

Dt .

D'F ... Dl'1 ul = Dt' Dtl+1 1

D'ul

= ur . Dt r.o In this notation equation (19.59) takes the form

Mk_1(uo,u1,...,u1+1)ED;)u,= F r=0

19.59)

19.60)

(19.61)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

162

On the right-hand side of (19.61), there obviously appear known quantities. §3.4. The Cauchy problem for linear equations with smooth coefficients

1. The operators adjoint to the operators of the basic system. Given some

system of m linear operators on I functions in the (2k + 1)-dimensional space of p, 61 .... , $2k with the volume element dig= p2kdpdS= p2kxdpd6I *-d62k

(20.1)

where dS is the surface element on the unit sphere and IC is a variable factor equal to sin2k-I

K=

132k

(20.2)

sin2k-2 62k-I ... sin2193 sin 02.

The operator MJ(S) may be displayed in the form of a rectangular matrix

M having m rows and I columns.

rMM t111j , 2

M=

... , , ... , M(2) 2 M(2),

M(s) 2

...................

L.

MM 2

,

...............

m

m

`

MV)

M(s)

M11.1)

M(S)*

Lei

denote the operator adjoint to the operator MSS) , i.e., such that

KP2k(wiM (s)vs - vsMi(S) wi) _ where

2k BPS)

Bp 3)

8t9 +

p

,

(20.3)

denotes certain functions. t

We shall define the system adjoint to the system Mi as the system Ns consisting of 1 linear operators on m functions w1, W2, ... , wm of the form

m

m Ns(w1,w2,...,wm)=EMIS)'wiENs')wi.

i=1

1=1

The matrix of the operators N(j) will have the form N(t) t N21),

N(2),

... , N22),..., t

N(j), ... , 1

N2

...,

N(m) t

N2

N s

S

s

s

Nlm)

t A simple calculation shows that the operator Mi,1- is defined uniquely in this way.

§3.4. DUCHY PROBLEM FOR LINEAR EQUATIONS

163

This matrix is obtained from the matrix of the operator Mss) if in the latter one replaces rows by columns and takes the adjoint of each operator (N(j) = M(s)*)

The system of adjoint operators satisfies the following identity: m

xp2k

f

WjMj(vl,...,VI)-F-vSNs(wl,w2,...,tum) i(JI

$=1

= E -" +-'0, i=1

where P; = E

p

1

(20.4)

j-1 S-1

We construct the system of operators N (j = 0, 1, ... , k) on k functions 01 , a2, ... , ak adjoint to the system (19.61). The matrix M for the system (19.61) has the form D(k-1)

D(k-1) 0

D(k-i)

D(k-1)

k3

1

D` k-2)

D(k-2) D(k-3)

,

D(k-3)

0

(

D11-2),

Dkk

D(k-3)

D(k-3)

D(k-1) Dkk

k2

D(k-3)

k

0 0

3')'

0

0

0

0

0 0

0

0

0

2

D(, 3),

D(2)o,

D( 2)

D(3), D (2)

t

2

0 0

D(1), DO('),

D(1), 1

D(1). 2

0

0 0

D(00),

Duo),

0

0

0,...,

D(k-1)

,

0

k

Do"),

M=

)

0

3

DZk )

(,k

k-I ' 0

k

2

D(k-1) Dkk 22J ,

Dik-4).

Dok 4

.

2)

0

The corresponding matrix N will be D(k-2)'

D ( k-1)'

D2 k1) 3

D( k-1)'

D(k-2) 3

D2k-3) D(k-3) 3

D(k-2)'

D(k-3)'

k-3 Dkk 22)

k -3

DI t D( k-1)'

'

D(k-2)

-1

,

I

D ( k-1).

0,

D (31'

0

0 D'3) .

D'k-O)

DZk-2).

D( k-1)'

k

Do

0

D(k-3)'

D(k-2)

D(I k-1)

N=

D (k-3)'

0

0

k-3

,

Dkk 23)

,

'

D(2 )' 0

D2k-k)

D23).

D(k-,) 3

D(3) 3

D(2 ) 3

D(k-°)* k-3

D22 )

,

D 1)' 0

D',')

'

D(0) Do'0) .

DZI)

0

0,

0

0,

0,

0

,

0

0, ...,

0,

0,

0,

0

'...,

0.

0,

0

.

0

0,

0,

0,

0

0,

0

0,

0.

2. The construction of the functions a; . We now construct the system of functions a1 , a2, ... , ak which are solutions of the system of equations D(k-`)*a = 0; k I

- U; D(k-2) o2 + D(k-1) v1k k l 1

Dk

+Dk

D(0)1*ak+D(t`)*ak_1+DI

+D(ik-21)'x1

(20.5)

= 0;

Ik-`) a1-=0. I

164

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

This system is recursive, the functions al , a2 , ... , ok being determined from it successively, one after another. Further, let

N0(1 a ,...,o)=D(0).o +D(I)*o k_1 +D(2)'o k_2 k k 0 0 0

0

(20.6 )

1

Obviously there holds the identity obtained from (20.4) and (20.5): Kp

2k

k-l

a1F

E ak_1at<-u0N0(a1,...,ak)J

=Kp2kk-I ak_1 1=0

aIf -Mk-,(u0,...,u1+1) all

k-1 1+1

+ Kp2kE E{ak_1D, (1)ur - urDr(1)' ok_1} 1=0 r=0

= Kp

2k

k-1

E ak_1 1=0

2k

+

a1F

atl

- Mk-1(uo , ... , u1+1)

8P0 a19m+ ap aPm

(20.7)

k-I 1+1 (r) where PM = E1=0 _r=0Pm.k-1' From the expression for J(:, p, 1) in (19.42) it is easy to see that J(1, p,

A) and its derivatives in t up to order k + 1 are continuous for p = 0. Therefore, taking into account formulas (19.58) and (19.56) for L(1) and M(1) for t = 0, we find D1+ 1v

Dr(1)v

= M(°)v +

DJ

v = _29(s) {8p +

= L(0)v + 1M(1)v +

[-P

+ G(p, X)J v}

1! D2J 21(!-2)! ?'

2

_ -1v(s) lap 2 11 + pQ1 ] + -[2k + pQ2]L + z

D;1)=cp(s) A2p2+IBOv+-2Cv

,

[Z + pQ3]v }

,

0
where G, Q1 , and A are functions, Q2 and B are operators of first order, and a , Q3 , and C are operators of second order, sufficiently often differentiable.

§3.4. CAUCHY PROBLEM FOR LINEAR EQUATIONS

165

We pass to the calculation of the adjoint operators. We note first of all for this purpose that the operators B' and Qz adjoint to the operators of first order B and QZ defined on the unit sphere are in turn operators of first order defined on the unit sphere. The operators A' , Q3 , and C' adjoint to the corresponding operators of second order on the sphere are operators of second order on the sphere. The operator a' will coincide with the operator A, since the Laplace operator on the sphere is a self-adjoint operator. Further, the following formulas hold: ((

aP - (

PI

2k

l aPP2k/

CQ a 1' 09P /!

-

(

-

2k 1

aP

2k

z

Y

= LPl

)

aPz (P

,

(20.9)

a P2k Q2-

1

P2k ap

Similarly P2k

a 2k a 2k au au 2k +up op(P v) = (P v) a- +uaP(P v) vaP 1

_ y (uvp2k),

p

2k

[v- - u- -119Pa2(P 2k v) 19 2u

92 a2u (p v)- u- (P 2k

1

P

=

= ap [(v2kv)a-

P2k

(V62-

(

- + u-2 l Pp k62) vJ }

[p2kV62

=

2k

v)

5 P7

- u8-(P2kv)

\aP/ - Q2(P2kv)8

+ [Q2(P2kv)8u 2 2 aP + u aP (Q2P2ku), 2k

= ao[ a u(Q2P2kv)I +

aP;

Ea-6-

where Pi are some expressions given on the surface of the unit sphere.

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

166

If we use the fact that the adjoint operator is unique, we see that our assertion is established. By virtue of formulas (20.8), we obtain 2k9(s)a 2 8 - G(p, I)`21a(s)a] [(s)pzka] +I =

7T-(

D(1)-o

-

P

29(s) 1Tp + I (k +.oH(p, 2))01 ,

(20.10)

where H is a continuously differentiable, bounded function. Carrying through analogous calculations for D(1) and D(1) where r < I - I , we have finally

D()a=29(s)+p[k+pH]a(20.11) lop 18z

DI a= -r9(s) + DrU)'

2[I+pTTr)]+ p[2k+pT2 ]y-

I [a + pTY) ]a }

(20.12)

,

(2aP2 + I M,(') 80 Or a=rp(s){'1 ap+ -1p2N,(,

q>

1

a

r< (20.13)

3. Investigation of the properties of the functions a, . We pass to the study of the character of the functions o; defined by the system (20.5). We show that with the appropriate choice each of them can be represented in the form 21r(2k - I - 1)r(k) 1

'-k1 = r(I + 1)r(2k -1)r(k - I)

p2k_,-1

[1 + pP,J,

(20.14)

where `P, is a bounded function of the variables p and 8, , differentiable sufficiently many times. First of all, we remark that a, is a solution of the equation

a p

and consequently

+ p[k + pH]al = 0,

al = X(191 ,192, ... , t92k)e-fi = X (191 , 192 , ... , 192k)

dpi-fo H(pi8)dpi

e- fo H(pi 6) dp, k

We set X(t911

62,...1 82k)=

Our assertion for a, is established.

(20.15)

2k-Ir(k)

r(2k-1)

(20.16)

§3.4. CAUCHY PROBLEM FOR LINEAR EQUATIONS

167

The properties for the function ak_, for arbitrary I are proved by induction.

The equation for ak_1 has the form 2

[La p ' + p (k + pH)ak_/] = Xk_!(p , 191, ... , d,) ,

where 1

Xk-1

(1+1)'

(S)D1+1

1

ak-1-1

- (P(S)D1+1 (

ak-1-2 D(j+11)-a1.

(20.17)

Substituting in the right-hand side of the last equality the values of a1, az , , which we assume to be known and to satisfy the condition asak_I-1

serted above, we see that the main term of the right-hand side is obtained from substituting the function ak-1_1 in the operator D1+1 and has the form +1

2

az 2k r(2k - I - 2)F(k) ll r'(1 + 2)r(2k - 1)I'(k -1- 1) apz pzk-1-2 + P pzk-l-z I"(2k - l - 2)I-(k) = 21+1 2) [(2k - I - 1) - 2k] r'(l + 2)1'(2k - I)r'(k - I - 1) (2k r'(2k - l - 1)I-(k) _ -21+1 r'(l + 1)r'(2k - 1)r'(k -1- 1) pzk-1 1

1

1

1

A partial solution of equation (20.17) is easy to obtain by standard methods. We will have ak-1 = dp1 . (20.18)

2

Iv

From this the value of the principal term in the decomposition of ak_1 is obtained by elementary means. Our formula for ak_1 is proved. We consider one further question, namely the estimate of the principal

term in the operator N0(al , az, ... , ak) for p -y 0. This principal term is obtained by substituting the function ak in the operator DO (O)' , and is clearly equal to

10z

2k 0

p zk-1

=0.

Thus, N0(al , ... , or.) begins with terms of order order p-(2k+1) vanish.

p-2k

apz+

P

gp+p2o

, since the terms of

4. Derivation of the basic integral identity Bu = SF. We now return to the identity (20.7). We consider for t = 0 the domain c2 in the space of y1, ... , yzk+I containing the origin of coordinates in its interior. In the variables x1, xz , ... ,

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

168

x2k+I , it will be a domain in the characteristic cone containing the vertex of

the cone. We remove from this domain a ball p < e around the origin of coordinates and denote the remaining domain by 0'. The boundary of Sl' consists of the exterior bounding surface S and the surface p = e. From (20.7) we obtain 2k

1R'

'

xp (,ok_r I -Mk-1(u0,u1, ... ,ur+1) + , 00 + 8 dpd0 l Zk k-1 8r J m-I 8Pm 8 . 1-0 n,

K P 2k

J

k-1

E 0k-1

/^ 0F_uN(o 0tr

0

o

1 , ... ,

ok) dpd0,

(20.19)

1=0

where dO = d01 dO2k . Obviously [(0rF/0tr)-Mk-1(uo u , ... , uI+I)1= 0 by equation (19.61). The left side may be transformed into an integral over

the boundary of i1'. We obtain

f rcp

2k

k-1 !_0

01F

ar

-u0N0(o1,...,ok) dpd0 =

f 3dS'+ f PodO,

(20.20)

p

S

where E is defined by the equality 2k

.rdS'

_ lPm M=1

dpd01dO2...d02k +Pod01...dO2k. d0^

We now take formula (20.20) to the limit as a

(20.21)

0. On the left side we

will have a convergent integral since No has a singularity of order not greater

than p-u` and each of the functions or has a singularity of order not higher than p-2k+I. We consider the limit of f Po dS as a - 0. We show that v=, this limit will equal Cu01P=0 , where C is a nonzero constant. Indeed, in this case there can be nonzero limits only for that one among the terms making up f,, P0 dS that corresponds to the term in Po not containing the factor p. Such a term can be obtained from no other ok_f than ok , since even , and the derivative of p2kok-I with ok_1 has a singularity of order respect to p must vanish for p = 0. It is also obvious that such a term cannot be obtained from terms containing u1 , since u1 and ak appear in equation (20.7) only in the term p-(2k-2)

okDIO)u1 1

- u1D(O)1* ok,

but D(10) is an operator of first order, and as a result the term P(l) contains the product irk u 1 p2Ar , which is equal to zero for p = 0.

§3.4. CAUCHY PROBLEM FOR LINEAR EQUATIONS

169

Thus, it remains to compute the principal part Pokl , which has the form a kD(o) 0 u

a (02k aako

l

ak- 0u P 2kap 0- u0D(0), 0 1

8 1u

= P2k aP 1

p (2k auo k 2kap lP ap J

aP 2kaak

0p

a

= 7 8P u0P

aP

2kt9U0

a kP

Op

2kOak

eP

From this, there follows

limf 4-0

Pod9 = -C2u0Io_0,

where C is some nonzero constant. Using this, we finally obtain after substituting this result in (20.20) and dividing by the corresponding constant that (S4) uo1v=0

Ifnu0No(al,...,ak)da k-i r=o fn

ak_I

a'F Al f s_T dS J . all

(20.22)

5. The inverse integral operator B-1 and the method of successive approximations. The relations we have obtained yield the possibility of constructing the solution of the Cauchy problem for equations with sufficiently smooth coefficients in the case when such a solution exists. The existence of such a solution we shall establish later. We consider the problem of finding the solution of equation (19.1) for the conditions (19.40). Suppose that we need to find the value of the unknown function u at the point P0(x0, x1 , ... , x2k+1) . It will be assumed that all the bicharacteristics emanating from the point P0 intersect the plane x0 = 0. We construct the characteristic conoid with vertex at P0. This conoid in its lower part will intersect the plane x0 = 0 in a manifold S bounding a part

fl of the surface of this conoid which contains the point P. in its interior.

The quantity ' dS' is known on the manifold S. Indeed, this manifold lies in the plane x0 = 0, where all the derivatives of the function u are obtainable from the equation (19.1) and the conditions (19.40). All derivatives containing at most one differentiation with respect to x0 are given directly by the conditions (19.40) differentiated with respect to x1, ... , x2k+1 , the derivative 82u/8xoJxos0 is determined from the equation (19.1), and after it all the derivatives having not more than two differentiations with respect to x0 are found. Differentiating (19.1) with respect to x0 , we obtain equations

3. HYPERBOLIC PARTIAL DIFFERENTIAL. EQUATIONS

170

for finding a3u/axo3 , a°u/axo4 , etc. An arbitrary derivative atu/axo is determined by differentiating equation (19.1) and substituting the derivatives already known. The remaining derivatives are obtained by differentiation with respect to x , ,. . . , x2k+1 of the derivatives of the form asu/axo . We introduce some notation. Set ,

1

u(xo,xt,...,xzk+1)C

u(xo,...,x2k+1)N.di2Bu.

(20.23)

The operator B carries the function u defined on the domain x0 > 0 into a new function Bu defined on the same domain, and can be applied to an arbitrary continuous function. Let also k-I asp. 1

fks 5=o

at

(20.24)

The operator S can be applied to an arbitrary function having continuous derivatives of order k and defined for x0 > 0, and carries it into a function defined on the same domain. The equality (20.20) can be written in the form of the equation

Bu = SF + ft,

(20.25)

where F and f, are known functions. If we set SF = fz , fl + f2 = f , we will have

Bu=f.

(20.26)

The equation (20.26) is an integral equation of Volterra type. A little later, we shall establish its solvability and find its solution. We investigate a particular form of the equation (20.26). We suppose that the function F vanishes together with all its derivatives

of order up to k for x0 < 0, and that the function u also vanishes for x0 < 0. Then in the equality (20.22) the integral

jdS' vanishes, since it depends only on the initial values of the function u and its derivatives of order not higher than k, and these initial values are all zero, as follows easily from the way of computing them indicated above. We obtain in this case

Bu = SF, Bu = SLu,

(20.27) (20.28)

where by L we mean the hyperbolic differential operator standing on the left side of the equation (19.1).

We study the properties of the operators B and S.

§3.4. CAUCHY PROBLEM FOR LINEAR EQUATIONS

171

THEOREM 1. Suppose that the characteristic conoid E has been constructed

for the point P0(xo) ,

,

xix+1) and 0 < T1 < T2 :5-x6(0). Denote by Q1

and f22 the domains cut out from the characteristic conoid by the hyperplanes

x0 = T, and x0 = T2 (obviously, t21 D

'12).

Further, suppose that Q is a

bounded closed set such that every bicharacteristic going out from points of Q intersects the hyperplanes x0 = T, and x0 = T2. Then 1

/'

SCIJn,-n,INo1dig
where K is a constant independent of the point P0 E 12 . PROOF. This estimate is most easily proved by replacing the independent variables p, t91 , ... , 62k in the expression dS2 by the variables x0 , 191, ... , 02k Further, D(p,61,...,t92k) _ 8p

ax° But p = -y0 in view of our construction. Therefore, a p/8x0 < M0, where M0 > 0 is a constant independent of P. E Q . Further, I N01 < K, p-2'1r , and D(X0, 191,..., 192k)

hence

f _n

IN0IdcJ
=KKI r

Kp2kp-2'dpdtI...d$2k dpd$1...d$2k

< KMOK1 l

n,-n,

dxo d8I ... dt92k

.

From this, our inequality follows immediately. THEOREM 2. The operator B has an inverse.

PROOF. We shall establish that the equation Bu = f is solvable by the method of successive approximations. From this the existence of an inverse operator will follow. We set

u(°)

= f;

(0) 1 ... , X(°) u(")(x(°), No dl+ f( X(°) 0 2k+1)= C fro, u("-1)N o ...,x2k+1)

We show that the sequence u(") converges uniformly. From this, it will follow obviously that its limit will be a solution of the integral equation. As usual, it suffices to consider u (n+1) - u (n) = v(n),where the v (n) is related to v("-1) by the homogeneous identity v(n)

= f v("-I)Nodc2. n

Suppose that jv(0)1 < M.

(20.29)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

172

We show that then (n)

(0)

(0)

n (XO

(20.30) , ... , xik+I)I < MK n! where K is the constant in Theorem 1. The proof proceeds by induction. Suppose this inequality has been established for some n ; we show that it remains valid for n + I. By Theorem Iv

(xo

)

1, Iv(n+1)N(0)

x(0)

_ 1 IV(n)I INoId!Q I

<-L fMK"'°-N0Idfl f <_

ICI

(0,

1 f a MK"

(xpd Idx(l) o)> n!

0

x(o> U

MKn-1

< MK n+1

INoldc

dxo

(1) n

(x0 )

o0

x(1)

0-o

n1

dx(1 0 )

(X00))n+1

(n+1)!

as was to be shown.

Having constructed the solution of the equation Bu = f, we find the solution of the Cauchy problem for the given equation of hyperbolic type if that solution exists. It remains for us to show the existence of such a solution. As a preliminary, we shall prove some further theorems. 6. The adjoint integral operator B' . Let v(xo, x1, ... , x2k+1) be a func-

tion of the variables x0, ... , x2k+1 vanishing for x0 > To > 0 and for (x, I > To, where To is some constant. The function u will be assumed to vanish for x0 < 0. We form the integral f J v(x0, x1, ... , x2k+1)Bu(x0, x1 , ... ,

x2k+1)dxo...dx2k+1

(20.31)

over the whole space. This integral may be transformed into the integral

f u(x0, x1 , ... , x2k+I)B'v(xo, x1 , ... , x2k+I)dxo, dx1 ...dx2k+1

The operator B' is called the adjoint operator of the operator B. THEOREM. Both the operators B and B' can be applied to arbitrary continuous functions u and v satisfying the conditions indicated above, and Bu and B'v are also continuous functions. The operators B and B' have inverse operators. In other words, each of the equations Bu = f and B'v = w has a unique continuous solution when the right-hand sides are continuous.

§3.4. CAUCHY PROBLEM FOR LINEAR EQUATIONS

173

PROOF. The existence of an inverse operator for B was shown above. In order to prove the theorem, it is necessary for us first of all to construct the

operator B' in an explicit form. Let us transform the integral (20.31). We have that

f v(x0,

I f u(xo, ... , X2+1)

... , X2k+1) [u(x0, ... , X2k+I) - C

x No (p , 191 , ... , 192k , XO , ... , X2k+ I )

x p2kKdpdt91 - d02k) dxo...dx2k+1

=

fv(xo, ... , X2k+I)u(xo, ... , I

X2k+1)dxo... dx2k+1

fnv(x0,...,X2k+1)NO(P,61,...,192k,xo,...,X2k+1)

X u(XO(XO

... , X2k+I I PI1 1:9

620 2k

X2k+I(XO, ... , X2k+1 , P, t91 , ... , 02k))P K

x d p dt91... d62k dxQ ...dx2k+1.

(20.32)

In the last integral the function N. depends upon the coordinates of the vertex x0, x1, ... , x2k+l and polar coordinates on the cone. On these same coordinates depend the variables xo , ... , xzk+1 calculated on the surface of the cone. We carry through a change of independent variables, taking as the new variables the coordinates X0'' ... , , 02k P, $1 f Thereby, the second integral in the equality (20.32) can be rewritten in the form ANO(P,191,...,192kXO(XO,...rX2k+1IPr191,...,192k)

, X2k+1(x0, ... , X2k+I , P, , X2k+I (... ))u(xO, ... , X2k+1)

xdxo.

191

, ... ,

t92k))v(XO(...

D(xo,...,x2k+l,P,$I,...,02k) D(XO,...,X2k+1I P,t91,...,02k)

dx2k+IP2kxdpd191... dd2k

= J u(XO,...,X2k+1)

jf.N0(P161,...,t92k,X0(X0,

P, 01 , ... '92k), ... I X2k+1(XO, ... , X2k+1 X2k+1(... )) P, 61, ... , t92k))v(XO(... ), ... ,

X2k+1

xID(Xo',...,X2k+lIP,191,...,0 )I x pcdpdt91 2k . . d92k } dxo ...dx2k+1

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

174

From this it is clear that

B v( xox2k+1)v(xo,X2k+1)

fo NO(P,191i92kIXO(XOX2k+I, P, 16162k) X2k+1(XO , ... , X2k+1

xID(xo'

x2k+1IP61''.

D(x0'

xv(xo('. ),

P, 191 , ... , $2k)) $2k)

x2k+I ' p' d1 ' ' ' 62k) ... , X2k+1(... ))p2kxdpd191 ... d132k .

(20.33)

The resemblance between the operators B and B' extends not just to the external form of their representations but also to many properties. We investigate first of all the character of the surface in (2k+2)-dimensional space given by the parametric equations

x0_x0(x0'X1'

'X2k+1' P, 61,...,t92k),

................................................... x2k+1 = x2k+I(xo,

X'11

(20.34)

.. , X2k+1 > P, t91 , ... , 192k)

We shall show that the surface (20.34) is itself again a characteristic cone with vertex at the point xo, x1 , ... , x2k+1 but extending in the direction of increasing values of x0', i.e., the upper portion of the complete characteristic cone. Indeed, the point x0, x1, ... , X2k+1 and the point xo, x' , ... , X2k+1 by construction must lie on a common bicharacteristic. Consequently, the set of all points x0, x1 , ... , X2k+1 for which x0', x'1, ... , x2k+1 is on the surface of the characteristic cone directed downward coincides with the set of points of all bicharacteristics passing through xo, xI , ... , X2k+I and this set is the upper part of the characteristic cone, as was to be shown. We set

IND(xo,x1,...,x2k+l,p,d1,...,192k)_N. C

No

D(xo,x1,...,x2+1P,61,...t92k)

o

and estimate the size of the Jacobian

D(XO,xI,...,X2k+1P,...,192k) D

D(XO,x'l ,...,x2k+1P,...,192k)

This determinant will be bounded. Indeed, we may for its calculation first of all change coordinates from the variables x,, x' , ... , x2k+1 to the variables y0' y1 ' . . . > y2k+I and calculate the functions yO(XO , X1 , ... , X2k+I ' XO, X1 , ... , X2k+l) , y2k+I (xO , xI , ... , X2k+1 > XO , XI , .. , X2k+1) ,

S3.4. CAUCHY PROBLEM FOR LINEAR EQUATIONS

x 175

where x0, x1 , ... , X2k+1 are the coordinates of the vertex of the cone in which the substitution is carried out, and xu, x' , ... , X22k+1 are the running coordinates related to y0 , y1 , ... , Y2k+I by the formulas (19.23) investigated above. It is not difficult to see from general theorems on ordinary differential equa... , tions that for fixed s, p(,0) , .. , p2k)+1 in the equations (19.9), xo , Xk+1 will be continuous and sufficiently often differentiable functions of the

initial data x0, x1 , ... , x2k+1 with a functional determinant different from zero (the initial value of this determinant being equal to 1). As a result xo (x0 , XI ,

.

, X2k+1, Yo , ... , Y2k+I) ,

(20.35) X2k+I (XO , X1 ,

.

, X2k+1 , Yo ,

, Y2k+1)

will be continuous functions of x0 , x1, ... , X2k+1 with functional determinant different from zero. But the functions xx (x0 , ... ,x2k* ) , p , t91, ... , d2k) are the same functions (20.35), where we set YO = P = (E2k+1 ;=1 y , 2 ) 2 . The boundedness of the determinant D(x0, xl , ... , x2k+1) D(x0 , x; , ... , X2k+1)

is proved.

It follows from this without difficulty that the function No = NOD satisfies the inequality INo I

<

Mp-2k .

We shall show that the function No, just like No, satisfies the integral inequality (see Theorem I in item 5)

JN(p2kdpdS K(T2-T1), x0<
(20.36)

=-n,

For thi s, in distinction from the previous case, T1 > x0, since the cone on

which Na is defined extends in the direction of increasing x0. From this, as earlier, will follow the existence of an inverse operator B' on the space of functions vanishing for x0 > T, where T is a constant. (In the case of B, we had to do with the space of functions vanishing for x0 < 0). In order to establish (20.36), we need as before to replace p under the integral sign by x0 for fixed x0, d2k , X2k+1 , d1 7. The adjoint integral operator S* . Just as we constructed the operator

B' adjoint to B , we may construct the operator S' adjoint to S. We show that such an operator may be defined for all sufficiently smooth functions vanishing for x0 > To and Ix, I > TD , where T. is any constant.

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

176

In order to construct the operator S' , we use the previous method. We have

/ Su = _ 1 fn

.

f

k-I

C

k-I a

n s=o

zu(xo, k-s

... > z+l) zk P

dPdS

ayo

,

8P U

ak-s

, Po

IP;<s axo axt

(sl ,P

.. ax2k+I

,

2k

wPo .P,.....P:k.,P

dpdS. (20.37)

Here yr of , is the coefficient in the representation of the derivative asu/ayo in terms of derivatives with respect to xo, xi , ... , X2k+I , and the integral is over the part of the characteristic cone surface lying above the plane x0 = 0, as before. It may be considered as extended over the whole surface of the cone, since u =- 0 for x0 < 0. We consider the integral

J = f v(xo, ... , x2k+l)SudxodxI ..

dx2k+I

Transforming it as above, we obtain

J-

C

f

f

k-I

a Pu(xo, ,00

S-0 1116,<$

, xzk, l) /07k.1

L9

,

x n ak-s(XO(X0 , ... , X2k+l , P , 61 , ... X2kt1(xo, ... , X2k+1

61

, ... ,

$2k)

t92k))Wf',....P:k.l

x v(xo(...) , ... , X2k+l (... ))P2kKIDI dP d 0l ... dt92k xdxo

(20.38)

dx2k+1

Since u vanishes for x0 < 0, while v vanishes for x0 > T as well as for Ix11 > To, i = 1, ... , 2k + 1 , integration by parts in (20.38) gives us that

J=-

k-I

u(XO,...,X2k+I)E E -1) S-0 tEP, <s

aP

x

axo .OX2k+I

,

ak-s(xo(xO.... , X2k+1 , P,

X2k+ I (... )) W ), .... P:k. i v (XO

191 , ...

,

192k),

(...) , ... , X2k+l (...) )

x PzkKIDl d p dt91... d62k } dxo ... dx2k+1

§3.4. CAUCHY PROBLEM FOR LINEAR EQUATIONS

177

From this it is clear that the operator S' has the form

S v(x0, . , X2k+I) k-I 1

_ -C E E (-I)B aXO S=O Yft <s

,du«i

'

fk_s(XO(XO'

' ' aX2k+1

(s)

X2k+1 , P, 191, ... , 192k) , ...

X2k+1(...

xv(xO(...),...,X2k+I(...))p2kKIDIdpd$I'.. dt92k

.

(20.39)

From this very form of the operator St , it is evident that it is defined for functions having continuous derivatives up to order k -1 . We may now proceed to the proof of the existence of the solution of our Cauchy problem. 8. Solution of the Cauchy problem for an even number of variables. First of all, we reduce our problem to a simpler one. We calculate the value of all

derivatives with respect to x0 up to order k + 1 of the function u, and we set

u=w+uIx=O+XO

ak+Iu

Xk+0I

a uI

+

}

(20.40)

k+I

I

xo=0

For the new unknown function we will now have a homogeneous Cauchy problem and an equation with another free term F1 . It is obvious that F1 together with all its derivatives up to order k will vanish for xo = 0. We consider an arbitrary function to(x0, XI , ... , x2k+I) which is sufficiently smooth and differs from zero only in some bounded domain V' contained in a band 0 < x0 5 To. We construct the differential operator L' of second order adjoint to the operator L. The operator L' will have the form 2k+I a 2k+I2k+I a2

L'v= >2 >2

i=0 j=0

axax(Aijv)- >2 /-(B,v)+Cv. ,

i=0

J

i

We set

L*9=v. For the function rp , we may pose the Cauchy problem in the domain x0 < To. The initial data for this will be the conditions

c=0,

x0> To.

The theory which we have developed above enables us to write for the function (p the integral identity

BI c=S1t,

(20.41)

from which the solution of the Cauchy problem may be obtained in the form

=BI S1y. 1

178

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

The operators B1 and Ss are analogous to the operators B and S and differ from them merely in that the role of the direct and reverse cones have been interchanged. If one considers that the variable x0 denotes time, then the ordinary Cauchy problem consists in seeking a solution for later moments

of time for given initial conditions at the moment t = 0, and the adjoint problem in seeking a solution at earlier moments than t = To, for which the initial data are known. Multiplying both parts of the equality (20.41) by a sufficiently smooth function 4> which vanishes for xo < 0 and integrating over the whole space, we obtain f DB1 p dL2 = f OS1 yr Al. Using the concept of adjoint operators, we rewrite this equation in the form f vB; Ddf2 = WS;4?di2. (20.42)

J

The right side may be transformed in the following way. We substitute in it +v = L' . We will have, integrating by parts:

JL S;43d = I cLS14>dS2,

(20.43)

since the function (p vanishes outside a bounded domain. Bringing both integrals in (20.42) to the left side and combining them, we obtain, using (20.43):

r [B;0 -LS;0JdC2=0. This last equality holds for arbitrary q under consideration. From this, there follows the identity

B;4>-LS;4V=0. We form now the integral equation

B;4>=F,

(20.44)

where F is the right-hand side of equation (19.1). The operator B, has an inverse. Therefore, this equation is always solvable, where b = B;- ' F. Replacing 0 in equation (20.44) by its value, we will have

F = LS;B,-'F.

(20.45)

This equality says that the function S; B, -' F satisfies the equation Lu = F, if the function F has continuous derivatives up to order k and vanishes for x0 < 0. By means of the change of unknown function as we described earlier, we may reduce the general Cauchy problem for sufficiently smooth F and with sufficiently smooth initial data to this problem. Thus, we have shown the existence of a solution for a linear normal hyperbolic partial differential equation with sufficiently smooth coefficients and sufficiently smooth initial conditions in the case when the number of independent variables is even, in a sufficiently small neighborhood of a particular

point of the hyperplane x0 = 0.

§3.4. CAUCHY PROBLEM FOR LINEAR EQUATIONS

179

9. The Cauchy problem for an odd number of variables. The case in which the number of independent variables is odd, as is well known, can be reduced to the preceding case. Suppose for example that we need to find the solution of the equation 2k

2k

2k

a2u

aU

E> Ajjax.ax.EBiax +Cu=F ; i=o i=0 j=0

(20.46)

r

under the conditions u0(x1 , x2, ... , x2), au I aX0 xo=0

=u1(x1,x2,....X2k)

(20.47)

We assume that at each point of the space the quadratic form A(p) _ 2 i=0 j=0App ij i can j be reduced to the form 2k

2k

A(p) _ -

(20.48)

q2 + qo i=1

with the help of a linear change of the variables pi , A00 > 0 , Aii < 0 , i # 0 . We introduce one more new independent variable x2k+1 and consider the equation 2k

2k

a2u Ajj ax axj.0

i=0

82u

ax 22k+1

2k au + Cu = F . + E Bj ax

(20.49)

;

i=o

The equation (20.49) will be a normal hyperbolic equation with an even number of independent variables, while A,j , B; , C , and F do not depend on the variable x2k+I . By what has been shown, this equation has a solution satisfying the conditions (20.47), which may be interpreted as conditions in the space with

(2k + 2) variables where the right-hand sides in this case do not depend upon x2k+l

.

The solution of this problem gives us a function u which, as is not hard to see, in turn does not depend upon x2k+I . Indeed, the equation which is satisfied by the function 8u/8x2k+I coincides with the equation for u when F = 0, and the initial conditions for this function are 8u0 aX2k+ l

= 0,

8u,

= 0.

9x2k+ l

Therefore, au/aX2k+, = 0. However, this solution is unique, as we showed

earlier. Therefore, the solution u which we obtained has everywhere 8u/8x2k+1 = 0 and does not depend on x2k+I . But in this case the function u , as a function of the 2k+ 1 variables x0 , x1, ... , x2k+1 , satisfies equation (20.46) with initial conditions (20.47), as was to be shown. (ss)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

180

§3.5.

Investigation of linear hyperbolic equations with variable coefficients.

1. Simplification of the equation. In the preceding section we established the existence of a unique solution of a normal hyperbolic linear partial differential equation with variable sufficiently smooth coefficients for sufficiently smooth initial data. The methods developed by us in the first chapters permit us to make a significantly more precise estimate of the order of smoothness of the coefficients of the equation and of the initial data which is needed for the existence of a solution. Suppose we are given the linear differential equation

+EB, i=o J=O

i=o

+Cu=F,

(21.1)

f

where A,; = A;, , B. are continuous functions of x0, x1, ... , x,, while A00 i4 O.

In the following, there will be presented additional conditions imposed upon the coefficients A,; , B1, C and the free term F. This equation may be simplified with the aid of a change of independent variables. We set x0 = t and construct a vector field l with the help of the equations

1;=A1=1,2,...,n.

(21.2)

ao

We consider the system of ordinary differential equations dts=15

(s=1,2,...,n),

(21.3)

and let C,(t, x1 , ... , xn) (i = 1, 2, ..., n) be first integrals of this system. The system (21.3) implies that the lines

C,(t,x1, ..,xn)=const (i=1,2,...,n) are transversals in relation to the planes x0 = const. We set

yi=C,(t,x1,...,xn) (i=1,2,...,n).

(21.4)

We can pass to the variables 1 , y1 , y2 , ... , yn in a neighborhood of the particular point x6(0), .. , x,(,°) in equation (21.1). The equation (21.1) takes an especially simple form. In it the coefficients A0; of the mixed derivatives

a2u/ay;at vanish. We shall prove this. As is well known, the coefficients Aoj of the mixed derivatives in an equation of hyperbolic type after the transformation will be A01

Aol

ac; ac; ax + Aoo at

§3.5. LINEAR HYPERBOLIC EQUATIONS

Since

161

ac; _ ax, dt + at 0'

n aC, dx,

the equations (21.3) give us that

n ac

ac-

a;li+ etc =0. Taking (21.2) into consideration, we see that A0, = 0, j = 1, ... , n. Dividing the equation then by ADO = AOO , we arrive at a new equation of

the form

a zZ

at

n

n

n

2

i-I j-1

"U -rBiau+ha-Cu=F. ay,ayj at ayi i=,

We also set

u=e-ifohdi,V Then ail

jahdr,+v

+hat = at2e

(8t2 +hFt l e-

Iohdt,

After this substitution, the term containing 8u/8t also is eliminated from the equation. In the following for the investigation of the general linear equation, we shall always assume from the very beginning that terms involving

82u/8x;at and au/at do not appear in the equation. We consider the equation

LueEEAij88 is I j.1

1

at

+

n

i-,

Bi. +Cu=F,

(21.5)

i

where A,; , B, , C, and F are given functions of the point x1, x2 , ... , xn , t . Suppose that at every point of space and for every moment of time n

n

n

2 Ai;=Aji, A(p)=>EAijpipj>cEp;,

i:1 j.1

(21.6)

i-I

where c > 0 is some constant. (56) We seek the solution of this equation satisfying the conditions ult_0

=u0(x1,x2,...,xn),

= u1(x,,x2,...,xn).

au

at

(21.7)

11=0

We shall consider two different formulations of this problem. 2. Formulation of the Cauchy problem for generalized solutions. Let fl be a domain of the (n + 1)-dimensional space of t, x1, ... , xn . The function u(t, x1, ... , xn) defined on C and summable on every bounded domain

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

182

S2' with 12' C 12 is said to be a generalized solution of equation (21.5) if for every twice continuously differentiable function v on the whole space which vanishes outside some bounded domain i21 we have the equality

fuLvdcz = f vFdfl, n

where L' is the adjoint operator defined by n

L'v=r i=1

n

2

'

(Av)axiaxi 1

n

2v

aC

i=I

-(Bv)+Cv. axi

The first formulation of the Cauchy problem consists of the following. To find a generalized solution u of equation (21.5) which, on any section

of 0 by a hyperplane t = const, is an element of the space WZ 1 f , while au/at is an element of the space L2 = W2. The trajectory in W21) and L2 defined by the pair of functions u and au/at in this pair of spaces should be continuous in t and satisfy the initial conditions (21.7). We impose upon A;, , Bi , C, and F restrictions which we call conditions o), or conditions for the existence of a solution in the generalized Cauchy problem. These conditions are the following.

1) In the domain Q : the coefficients Aj are continuous, have first derivatives, and satisfy the inequalities

A,>m>0, IAI,I
0A..

aA. lax',I
k

I

at+
(21.8)

(i, j, k = 1, 2, ... , n, m and A constants) ; 2) the coefficients B; are continuous and satisfy the inequalities JBij <

A (i=1,

,n);

3) the first generalized derivatives of the functions B; exist and satisfy the inequalities

1 a Bi

ax;

+

Ia

101

dx

< A(t) < A (1=1,2..... n);

dxn

(21.9)

4) the coefficient C satisfies the inequality

140)


(21.10)

JJ

5) the first generalized derivatives of C exist and satisfy for some e > 0 the inequality 1

n

[I_I+

IaCIf'dX1...dX}"
(21.11)

13.5. LINEAR HYPERBOLIC EQUATIONS

183

where vl = (n + e)/2 for n>4 and v1 =2 for n= 1,2,3; 6) the free term F satisfies the inequality

dx"]

Ifn(l) IFI2dx1

(21.12)

J

where F is a constant; 7) the generalized first derivatives of F exist and satisfy the inequality 2

fa

(l)

I aFI+IaFJI E ax; 7 J dx l ...dx"

1/2

F(t)<

(21.13)

8) in Q(0),

u0E WZ`

(21.14)

u1 E W211.

(21.15)

9) in C1(0),

To the question of the solution of the Cauchy problem in the first formulation, the following theorem gives the answer. THEOREM 1. The fulfillment of the conditions o) is sufficient for the existence of the solution of the Cauchy problem (21.5), (21.7) in some neighborhood G of the set 92(0) in the first formulation. The solution depends continuously upon the initial data in the following sense: for any e > 0 there is a 3 > 0 such that if Iluollwo, < 0 and IIu111Lj < b on fl(0), then e

and IIIIIL,
3. Basic inequalities. Let w be an arbitrary function of the variables x1 , x2 , ... , x,, t which is continuous together with its derivatives of second order. We introduce some inequalities analogous to the inequality (17.7). Consider a domain V in the space (x1 , x2 , ... , x,,, t) with piecewise smooth

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

184

boundary surface S. Then from the Gauss-Ostrogradsky formula: "

Ow2

aw aw

"

fstuj(T)jcosao

-2 n

n

i-1 j=, 1

aw 19W Ajax./ at cosa

f{Ow[ -2 at

A

i=1 j=1 n

n

+EE

(OAj) aw aw

dS

a2w

-

axiaxj a

at axiaxj taxi

a2w 01

aw aw l dV' Ox. at 1

where a; is the angle between the outer normal v to S and the Ox;-axis, and cao is the angle between v and the Ot-axis. If on some part of the surface S n n E Aij cos a, COS a j - Cost a0 < 0, i=1 j=l

then on it, the sign of the quadratic form: W- 2w.

i-1 j=1

Aijax i ax. +

aw12

cosao-2F, FA;jax; itI j-1

COAij (coSao_ TCOSai) COS2 a

COSaO -

O-E "_"_ A cos a COs a

aw

at

cosao

aw cosa+

cosa1

2

)

coincides with the sign of cos ao .

Consider a closed ball S2 contained in Q(0), and let (S2, [0, Tj) be the cylinder constructed on S2 and bounded by the hyperplanes t = 0 and t = T > 0. Let cl = ni j , maxis= to fl1 I Ai j I . We draw through the boundary of S2 a conical surface S1 (Figure 12) such that cos 2 ao > c1 /(I+ c) on it. Denote by LY this truncated cone lying in (S2, [0, T)), and denote by S3 the upper base of the cone lying in the hyperplane t = TI < T. Then on the surface SI the quadratic form is positive, since on it cos ao > 0 and n

n

n

A,j cos a, cos a j - cost a0 < c1 i=1 j=1

cos2 al

- Cost a0

i=1

= c1(1 - Cost a0)

- Cost a0

=c1-(I+cl)Cos2 a0<0.

LINEAR HYPERBOLIC EQUATIONS

§3.5.

185

FIGURE 12

This yields an inequality analogous to the inequality (17.7) derived earlier. f,

S

" r"

aw aw

u Aii ax; axi +

`=i

C. "= ii=I j=II n aAii aw Ow <

n

(awl,

aw aw (ow Z ll ds A'' axi ai + (ar) 1

n

n

aAi) 19W aw

E E at ax iax -2 i=1EE ax ax at i=1 i=I j=I

+1fY

i

J

where

n

LIw =

ds

at J

=1

n

a2w

a2w

atz

Aiiax;ax.

i=1 i=1

aw

-tat LItt1

Denote by f2' (T) the section of the cone i1' by the plane t = T. Suppose that for the function w

dx" = N(t) < N.

(L1w)2 dx1

(21.19)

Then it is possible to estimate the integral over 91' in (21.18) as follows:

n aAi'. aw aw

n

" 8Ai aw aw

"

flEEaax; ax.-2F, E ax; ax at -2 aw aw 1=' ai i=1 i=1

i=1 j=l

1

aw atL1w

dS2

1

Ti

<

o

L.1'

i=

Jo

aw

+2nE

axi

i=1

< C1 0T

Let

1

l

Iw i=1

l

law

+A

ax

dx1 ... dx" IL1w1

+\ (

84t

dt

/l2+2l agt l ILlz'1I1 dx1 ... dx" } dt. J

/ \1 f " (,gW)2++aw ax at j dx1.dx"=K1(t1w).

JJJ

(21.20)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

186

Obviously, by (21.6) and the condition IA..I < c,/n,

cK1(Itw) < J

n

z

=1 ;.)

(t)

8wew

< CZK,(tlw), where cz = min{c, 1 ) and C z = maz{c, It is easy to see that

I0W ILIwIdxl...

1n (r)

8t

(8wlz

ax, ax .. + t 8t J

dX ... dxn (21.21)

, 1}.

dxn

\ 2fa.M

1/2

1/

z

dx, ... dxn

(L.i-i

i

(

J

IL,wlz dx1 ...

dxn)

I

< N(t)'/zK,(tlw)'/z Setting T, = I in (21.18), and hence S3 = S2'(1) , we get that c2K1(tlw) 5 C2K1(01w) +C1 oI[K,(t,Iw)+2K1(t,Iw)1/2N(t,)1/2)dt,.

(21.22)

0

For our purposes it is necessary to derive one more inequality. Let 2

ffl(z)

w dx,

dx, = Ko(tlw).

(21.23)

Obviously, Ko(tI 1w) < rn,(r) w2(11, x1, ... , xn) dx, ... dxn ,

if t, > t. Therefore, dK0(tIw) = K0(:, 1w) - K0(tlw) lim

dt

t,-r

t1 -t

rf

1

< lim

ricer t, - t

w2(t,x1

f d

dT

(f

wz(T, x1.... , x,,) dx, ... dx,,) .(1)

Lt

2n w 8-w dx1 ... dxn (,) 81 1/2

<2wzdx,...dxn(_)2 J < 2[K0(t, Iw)1'[K1(tIw)]'!z /z

1!2

§3.5. LINEAR HYPERBOLIC EQUATIONS

187

Then

K0(tlw) = fl d l Ko(t,Iw) dt, + K0(O1w) < K0(O1w) + 2lo,[Ko(t' lw)]'/2[K1(t, 1w)],

12

dil.

(21.24)

The inequalities (21.22) and (21.24) are basic for what follows. 4. A lemma on estimates of approximate solutions. We now prove a basic lemma. Consider an equation (21.5) satisfying the conditions o).

In it we replace all the functions A,, , B. , C, and F by their averaged functions Aiih , B,h , C,, and Fh with respect to x1, ... , xh , 1, and the initial data uo and ul by their averaged functions uOh and ulh with respect to x1, ... , x,,. The new equation Lhuh = Fh will be called the averaged equation. This equation, with the new initial conditions, has a solution uh in some neighborhood G of the set f2(O) independent of h > 0. (58) Let 91

be a truncated cone contained in

the integrals Ko(tlu),), KI(tluh), and ,,

KZ(tluh)

(

G.E i=1 j=I

a2uh

ax ax

lz J \z

///

+2

,U4

E ` ax;at J I

I

/

+

dX, ...

LEMMA. For the functions Ko(tluh), KI(fluh), and K2(tluh), IK,(tluh)I < yi(t),

i = 0, 1, 2,

(21.25)

where the y; are the solutions of the system of equations d(yo/2)

_

My1j2

dt llz )

(21.26)

d(dt d(y2/2) = M(;'1 2+y11/2+y1/2+F(t))

dt

under the conditions y01,=0 = MK0(OIu),

y,Ir=O=MKI(01u), y211=0 =

(21.27)

MK2(01u),

where M is some positive constant independent of h. (Thus, Ko(tluh), K2(tluh), and K2(tIuh) have an estimate independent of h.)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

188

PROOF. Let us return to the inequality (21.22) and consider for uh =- W the equation

B,hax

LI,w = Fh t=1

r

- Chw,

(21.28)

where, by properties of averaged functions, the functions B;h are bounded, (21.29)

IBMJ
the function Ch satisfies the inequality '/u,

Jo

dxn1

ijIChl"' dx1

< A(t) < A,

(21.30)

F(t)
(21.31)

+ IIChwIIL, .

(21.32)

1

and, moreover, 1/2

I In this case in fY(t) aw

IILIhwIIL, <- IIFhpL2 + i=1

B'h axi

Obviously,

1:

IIFhIIL, = F(t) < F;

r

<- AIIwIIL='.

IIB;haxIl

1

! L2

To estimate IIChwIIL2 we use the Holder inequality and the imbedding theorems.

If n > 2, then uI = n + e > 2, and by the Holder inequality and the imbedding theorem in 2 of §8 in Chapter 1, 1/2

d xl ...

(/

dx(fchwI2

llq* Lhln+' dx1

... dxn)

(Jo. Iwlq dx1... dxn/

R (r)

(1)

B,A(t)Ilwllµ.o> < B,A[K1(tIw)1 /2 + Ko(tlw)1/2].

where 1

_

1

1

2n

n+e)
is the constant in the imbedding theorem.

(21.33)

LINEAR HYPERBOLIC EQUATIONS

§3.5.

189

If n = I, then pI = 2, and according to the imbedding theorem in 1 of §8 in Chapter 1, 11/2

IChwI2 dx, ... dx

<_

)

(

IChl2dx, ...

1/2

dx.)

\Jn'(')

IIIIC

/J

< B2A(t)Ilwllkz , < B3A[KI (tlw)112 + K(,(tIw))1/2 ,

B2 , B3 = const. Thus, by (21.19), (21.34)

[N(t)l" /2 < CI(KI(tlw)1/2 +Ko(tlw)1/2

where C, is a constant independent of w, t, and h. Substituting this estimate in the inequality (21.22), we get K, (11w) < C2K, (01w) + C3

o

x (K0(t, 1w)

1/2

+ K,(tl lw)1/2 + F(tI)] dt, . (21.35)

Together with (21.24) we get a system of inequalities from which it is easy to estimate the functions K0(tlw) and K, (11w). Let yo and y, be functions satisfying the system of equations

2[yo/2+yI12+lidt,,

I

y1(t)=C2K,(Olw)+2C3 fo yo(t) = K0(Olw) + 2 f 1

y

(21.36)

/2yl/2

A.

It is not hard to show that yo(t) > K0(tlw) and y,(t) > K,(tlw). Indeed, y and y, can be found as the limits of the increasing sequences yo) and y(m) defined by the formulas y(m)

= C2K, (0Iw) + 2C3 f

'[y(m_`)l1/2([yp -I)11/2 + (y m

)]1/2

+ F) dt,

0

o= K0(Olw) + 2 f `(yIm-1)]I/2(yo -1)]1/2 dt 0

if the first terms of these sequences are taken equal to K0(tIw) and K,(tlw) . Our estimate is proved.

The functions yo and y, are the solutions of the system of differential equations 2 1/z y,1/2

yo = yo

y, =

2C 1/2(Y01/1 +y1/2 3y,

,

+F)

(21.37)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

190

Taking y1/2 = zo and yi/2 = zl as the new variables, we get z; = C3(z0 + z1 + F) .

zo = z1,

(21.38)

It remains to estimate K2(tlw). Differentiating the averaged equation (21.28) with respect to x, and with respect to t, we get ague anti _aF, _ L Ih ax, ax, ;_1 ,=1 axr ax;ax;

aBih auh

auh ax, ax; - Ch ax,

"

auh _aFh Llh

at -

at

"

a2uh

'h ax;ax,

;_1

- Uh

h ax,

aA;;h a2uk

a2uh

at ax ;ax;

E-- aB`hauh

Chauh

at ax;

uhOCh

at

(21.39)

at

We can easily estimate the quantity K2(tlw) by using (21.22). Applying it to each derivative and noting that

)+KI (181) =K2(tlw), FKI (tl' Yxi we find that K2(tlw) < C4K2(OIw) + C,

K2(tllw) + K2(tl lw)1/2

`

I

J

X [N,(tl)jlt2 +

dt1,

(21.40)

JJJ

where [N,(t)]1 2 = IILI ax r

and [N0(t)]1'2 = L,

aUh

L1at

We get [Nr(t)]1/2

ax il-il

rL,+AE

ax, ax

EIIaxZax;IIL,+A

i=1 j=1

11

L2

+I!Chax,

IIaxrax;II L:

+Ilax, C uhllL

(/=1,...,n).

Estimating ll(aB;h/axt)(auh/ax;)I1L1 and IICf(auh/ax,AIL2 as earlier (see

§3.5. LINEAR HYPERBOLIC EQUATIONS

191

(21.33)), we get that OB,h 8uh

ex, Ox; 8

< C6 A(t)[K2 (tJu h )1 /2 + K (tlu h )1/2] ,

(21 . 41)

< C6 A(t)[K2 (tIu h )1 /2 + K (tlu h )1/2 J .

(21 . 42)

1

L2

h

1

Ch 8x

L,

Finally, to estimate 11(8Ch/ex,)uhI1L, we again use the Holder inequality

and the imbedding theorems. If n > 4, then v1 = (n + e)/2 > 2 and by the Holder inequality and the imbedding theorem in 2 of §8 in Chapter 1, we have "Ch II

8 x,

uh11L2

dx1... dx,,

< Cf' 8xt n U) IeChl

I/q

)2/(fl+)

(n+t)/2

n{

luh1° dx 1... dxn/

< B3A(t)Iluh11",

< B3A(t)[Ko(tluh)1

/2

+ K1(tluh)1/2 + K2(tl uh)1/2]

where, this time, q _ ( - n+t)-1 <

, and B3 is the constant in the

imbedding theorem.

If n = I , 2, or 3, then v1 = 2 and, by the imbedding theorem in 1 of §8 in Chapter 1, 1

:5

2

/'

dxl ... dxn)

18x, I < B4A(t)Il uhJJWi:,

1/2

Iluhllc

.

Thus, for I = 1, ... (21.43)

[N,(t)]1 /2 < C7[Ko(tluh)1/2 + KI(tluh)1/2 + K2(tluh)1 /2J.

An estimate for [No(t)]1/2 is obtained similarly. Taking into account all the foregoing, we get from (21.40) that K2(tluh) < C1 f K,(Olu,) + f [KO(zIIuh)'2 + J1C1(tlluh)1/2

+K2(tluh)1/2+. (t1)JK2(t11uh)1/2dt,}

.

(21.44)

If

Y2(t) = C7 {K2(oIuh) + 2 fo 1y2(t1)1/2[ o(tl)1/2

+y1(11)1/2+y2(11)1/2+F(t,))d11}

,

(21.45)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

192

then, as above, (21.46)

xz(tluh)
For y2 we have the equation y2 = 2C7y

1/2

1/2

IYo

1/2

+ y,

1/2

+ y2

+ F] .

(21.47)

Taking y1/2 = z2, we find

z2=C7(zo+z,+z2+F).

(21.48)

The lemma is proved (with M = max{ 1, C2, C3, C,)). (5') 5. Solution of the generalized Cauchy problem. Let us proceed to the proof of the basic theorem. (58) PROOF. In view of our estimates, the family uh has a bounded integral for the sum of the squares of all the second-order derivatives over (Y . In other words, uh E LZ) , and IIuhUL,:) < M, where M is a constant independent of h. We conclude from the boundedness of II uh II and Iluh IPL2 that IIuhII22) <_ MI .

(21.49)

By the imbedding theorem, the uh and 8uh/8t form compact sets in WZI) and L2 , respectively, on W(t), and this pair of functions, which depends on 1, represents a trajectory equicontinuous with respect to h in the pair of spaces W ( 1 ) , L2 .

Consequently, from this family of trajectories we can select a subsequence

converging uniformly with respect to t to a trajectory u, 8u/2t. Passing to the limit in (21.25) and using the theorem on the existence of generalized derivatives (see 2 in §5 of Chapter 1), we get

K,(tlu) < y,(t),

K0(tlu) < yo(t),

K2(tIu) < y2(t).

(21.50)

The family uh obviously satisfies the integral identity

fuhLhvdz=fvFhdc1 ,

(21.51)

where v i s an arbitrary function of the variables x, , ... , x , t that is continuous with its derivatives up to second order and vanishes outside some domain V, such that Vu c fl (denote by Lh the operator adjoint to L,). Passing to the limit as h - . 0, we obtain

fuJJvdczd:=fvFdczdz.

(21.52)

This equality, by the definition, says that the function is a generalized solution of equation (21.5). It is easy to see that this function satisfies the initial conditions.

The continuous dependence of the solutions upon the initial data is also proved without difficulty.

§3.5. LINEAR HYPERBOLIC EQUATIONS

193

If F(t) is sufficiently small and K0(OIu) and K1(01u) are small in turn then the functions yp(t) and y1(t) will be small. By the inequality (21.50), there follows the smallness of K0(t1ju) and K1(tju). The theorem is proved. (57)

The Cauchy problem is solved in the first formulation. We pass to the second formulation. 6. Formulation of the classical Cauchy problem. We shall seek a solution of equation (21.5) with conditions (21.7) having continuous derivatives of second order.

On the coefficients A,j, B; , C, and F, as well as upon the initial conditions, we must now impose some different conditions, which we shall call the conditions n). The conditions n) are the following:

1) The coefficients A., and their first derivatives are continuous and satisfy the inequalities r aA;i

A00 >m>0,IA1jI<-A,

aAv


A, axk

at

(21.53)

where m and A are constants; 2) the generalized derivatives of A,1 satisfy for some e > 0 the inequalities a I Ail

at°aax;, ax:"


dx,,

dx1

(1=2,...,k1-1),(21.54) where 2;=(n+e)/(1-1) for n>21-2,and A,=2 for I 0 the inequalities I!/'

JAff>

I

a'B; at°uax 1 .. ax°, .

v,

1 /k,

dx dx 1

I


n

k

1

-t),

(21.56)

where p,=(n+e)/1 for n>21,and p,=2 for I 0 the inequalities (Z I

UPI

ax;a1c

at°o

,

axa I)


dx1 ... J

(1 = 0,... ,k,-1), (21.57)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

194

where v1 = (n + e)/(1 + 1) for n > 21 + 2 ,and v,=2 for 1
Inc

,.. a xn. (E'_O'F at °oa


(21.58)

where F is a constant; 7) in Q(0) uo E W(k'z);

(21.59)

u E W2(k'-1)

(21.60)

8) in Q(0) 1

If k1 = 2, then the conditions n) coincide with the conditions o). We can now formulate the main theorem. THEOREM. The Cauchy problem (21.5), (21.7) has a solution that is continuous with its derivatives up to order m > 2 in some neighborhood of the set f2(0) if the conditions n) hold for kl = m + 1 + (1) .

The proof of this theorem is based on the use of the imbedding theorems and the same inequalities that enabled us to prove the theorem in item 2. For m = 2 we get a solution of the Cauchy problem in the second formulation. Before proceeding to the proof, we make some remarks. As in the proof of the theorem in item 2, we construct averaged functions for A;j , B. , C, F,

uo, and u1 and form the new equation

Lhuh=Fh,

(21.61)

whose solution is sought under the conditions 8 u8th

uh1r=o=uoh' I

r.o

-ulh

(21.62)

and, as above, we let

8u

'I

Llhuh =Fh -EBIhau - Chute. ras

i

Simultaneously with the equation (21.61) we consider all possible equations for the functions v°o...°.h =

atUk

at°oax°' ... axn 1

(21.63 )

§3.5. LINEAR HYPERBOLIC EQUATIONS

195

Differentiating, we have

.8x

ai at°oa

n

1

_

E

n

P

n

a#Aijh fiuaxI at ...8xn

co-p-EE

°...°

o
i=1 ,=I at-,8+2uh

x

at

-floax;1-al

.. axn fl-Ox;axj a"Bih

PO--B, It at-Q+1

u.

x at°a-0oax; , -fl,

...ax°n-6ax;

aftch atfioaxrt I

at-huh

axfi^ at

Qoax°'-'I ... ax n -fi. 1

(21.64)

where C00

are the binomial coefficients.

Let 2

P

[atyoaxy, w

Ko(tI w) = IWO) Eyi=

1

axy.

dx1... dxn.

(21.65)

n

7. A lemma on estimating derivatives. LEMMA. Under the conditions n) the functions uh satisfy the inequalities

Kp(tluh)
(21.66)

where I j2

dYO

dt

= M[yo/2 + y.112 +... +YP'2 + F(t)],

(21.67)

if yoll=o = MKP(OIuh), where M is a positive constant independent of h, and F(t) is the function on the right-hand side of the sixth condition.

PROOF. For p = 0, 1, 2 this estimate was established earlier. Let us proceed to the proof of the lemma for the remaining p. First of all we estimate "L1

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

196

If n > 2l +2 , then v. = (n+e)/(fl+ I)> 2, and by the Holder inequality,

aPCh

JC =

aXn

Gv8')ax$-

a"u

1/2

2 h

dX, ... dXn

al°°-f0ax'-d, ... I

<

(P+I)/(n+t)

(n+t)/(f+1)

afiCh

dx, ... dxn

al"0axpl ... axn

I /q

q.

a'- uh

dx1 . dxn

In-M a1-0U h

r (21.68)

where q' = (1 - n+6)-1 . Since q' < q = 2n/(n - 2fl - 2), it follows from the imbedding theorem in 2 of §8 in Chapter 1 that JC < A1(KO(t1U,)1/2 +KI(IIUh)1/2 +...+Kl+l(tIuh)1/2].

(21.69)

If 1
J

I

C-

I

azPoaxr' ... ax-:0. II

1/2

2

a6 Ch

dx

1

dxn

al-0U h

01fi-ax°'-fl' .. ax n 1

-0^

11C

l-Buh


11

II

C

According to the imbedding theorem in 1 of § I of Chapter 1, we again get (21.69).

§3.5. LINEAR HYPERBOLIC EQUATIONS

197

Similarly, a0A,jh

l'

a&axip, ... axn. ar-'*2u °

X

at°.-,%ax

1/2

2

h°

dx1 ... dx,,

.

1+1

< A,

(21.70)

EKj(tluh)"',

=o 2

JB

{L.(t)

atB"ax".... axn.

)

+lu

'<<

2

X C at°0-p0axo'-fi'.

axe.-P.ax;

1+1

< A3 E K,(tluh)Ir2 .

(21.71)

f=0

Combining these estimates, we get _

r+1 II

oo...

II L2

< A4

KI(1luh)112 + F(t)

(21.72)

i=O

(AI , ... , A4 are constants independent of h) . We now return to (21.22) , set w = v° ..,°R in it, and sum over all derivatives of order 1. We get

c2 E

KI(tIv°0...0.) < C2 E KI(Oly°o...°.)

2 =r

£ =r

+CI JI(K1(t,ly

°^)+K1(1 Iv

°q)1/2110(!;...°,IIL,}dt,

c2, C,,C2=coast>0. Using the estimate for C3Kt+1(t, u):5

I1L2 and the fact that

KI(tly°o...°.) < C4Kr+1(zIu), C3, C4 = const > 0,

(21.73)

3. HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

198

we have finally that KI,I(tluh) 5 M I Kt+l(Oluh)

+IlK,+1(tlluh)l 2

dtiJ

(21.74)

J

=1

(1=1,... , kI-1), where M is a corresponding constant. Setting

yt+l(t) = M ly t +1(0)+2 J

(Eil+tl)} dill

rYt+l(tl)I/2 {Y(()1/2 r

l

1

we can use this to prove the inequalities Kt+l(tluh) 5Y1+1(1).

(21.75)

For the functions yo = 1, ... , kl , d d t2p = L'OJ2 + y112 +... + y /2 + F(t)l ,

(21.76)

Yolr=o = MKP(Oluh).

The lemma is proved. (60)

8. Solution of the classical Cauchy problem. We can now prove the theorem formulated in item 6. (59) Consider the sequence of solutions uh of the averaged equations with the averaged initial conditions . (61)

These functions have bounded integrals Ko (tl uti) , p = 0, 1, ... , m + I + (1) and hence also bounded norms in W,("). The theorem on imbedding in the space C on f (t) and the estimates of the form (11.15) for the functions uti and their derivatives up to order m give us that these derivatives form compact sets in the space C on cf . Choosing from the set {uti} a sequence uniformly convergent with their

derivatives up to order m, and passing to the limit in the basic equation (21.61) with respect to this sequence, we get that the limit function u satisfies

the equation Lu = F and has continuous derivatives up to order m. It is useful to note that the inequalities (21.75), which we established for uh , remain valid also for the solution u. This follows from the theorem on the existence of generalized derivatives (see 2 in §5). (62)

Comments 1. The theorem is valid also for unbounded domains fl. Here it suffices to take into account for any e > 0 there exists a ball C such that fn_c 19p(P )I° dv < e and to use the assertion proved for bounded domains. P 2. In a number of problems it becomes necessary to use averaging operators

of a more complicated structure. Different types of such operators were introduced and studied in [53], [154], [40], [44], [159], [246], and [265]. 3. The theorem is also valid for unbounded domains. In this case a dense set is formed by the functions of the form P,Xk , where P, is a polynomial with rational coefficients, and Xk is the characteristic function of the ball of radius k about the origin (k = 1, 2, ...) . 4. It was proved in [80) that any pair of bounded domains is summable. Therefore, this theorem is valid for any pair of bounded domains, and in view of the local character of the generalized derivative it is true also for a pair of arbitrary domains (including unbounded domains). We note that there exist pairs of unbounded domains that are not summable. 5. The theorem is valid for A < 0 when p = I. The arguments simplify: the inequalities (6.3) and (6.2) can be established directly without using the Holder inequality. 6. Under the assumptions of Theorem 2 the function U(Q) is not necessarily bounded for all Q . It is clear from the proof carried out that U(Q(s))

can be defined as the limit in L.. on E, of the integral ft<, A, then U(Q(J)) is summable (over any bounded domain E$ in an s-dimensional hyperplane) to the power q' < q = s/A. The proof is carried out with simplifications according to the same scheme.

7. For the case s = n this theorem was proved by Hardy and Littlewood (see [109]) in a stronger formulation (for the limiting exponent q' = q) for

n = 1 , and by Sobolev for n > 1 [285]. For s < n Sobolev proved the theorem with p = 2 and q' = 2, Kondrashov [139) proved it for q' < q,

199

200

COMMENTS

and 1l'in [117] proved it for q = q . Detailed proofs of these assertions can be found in the book [29]. Adams [2], [3] established that if 1 < p < q < oo and A > n/p', then

(fu()jd)

I/q

<- C,IIIIILI

(1)

in the whole space for some constant C, independent of j if and only if the measure u satisfies the condition sup sup p"lp -A(P(B(P , P)))I /q < oc p>0 T

(2)

where B(- F, p) is an open ball of radius p about the point P . 8. Reshetnyak [252] and Burenkov [42] showed that the equality (7.12) can be written in the form

(P) =

(-1)°

X (Q)dvQ +n

1 8°((x1 -y1)°I ... (x" -Y")°^v(Q))

1

(Q. P)ay°,9y°^ dvQ (3)

r"-

Following [29], we derive this formula directly from the Taylor formula

G(P)-

a , ! ay

o

'a'

(xl

-Y,)°I...(x"-Y")°^

eye ....ayn

x(x1-y,)°'...(x"

dt,

where Q'(y , ... , y;,) E C. Multiply both sides of this equality by v(Q') and integrate with respect to Q' over the whole space, taking all the functions

COMMENTS

a,,

ar^

201

to be defined by zero outside Q. Then 1

SA(P)=

1

aI I ... and K

ay1

-yn)°^v(Q')dvQ,

x (x1 _ y'1)°i ... (xn

+l

'°t ...aye°^ n

fC

1

1(1al (4)

The first term on the right-hand side of (4) becomes the first term in the formula (3) after integration by parts. Denote by J°,... the integrals after the second summation sign in (4). Changing the order of integration and replacing Q' + t(P - Q') by Q (here xi - y,' is replaced by

(I - t)-I(x1 - y,), and dvQ, by (1 - t)-"dvQ), we get, again changing the order of integration, that

r '.(Q)

= J ay,,

(x1 - y1)°i

... (xn - yn)°^

ayn

x

P+Q-P

v

Carrying out the change of variables t = r, , where r = I

that (setting ! =

dt t)n+l

0

V,

QI , we find

as as before) (XI-y,)°i...(xn-yn°^

8'q(Q) _=1rayj,...ayn^

rn 00

x

r v(P +r,! )r1-I drl

"Jr

dvQ.

Thus, we arrive at the formula (3) with (XI - y1)°'...(xn - yn) I

n'

r OQ

xK f v(P+r,1)rn,-Idr,

(5)

(it is not hard to see that the function W0.0 in (7.12) has precisely this form).

202

COMMENTS

Different derivations of the integral representation (7.12) can be found in the books [271) and [ 129) (second edition).

9. We note that in the equalities (7.12) and (3) it can be assumed that v is an arbitrary function having continuous derivatives up to order 1 on Z7, vanishing together with its derivatives up to order 1- 1 on the boundary of

the ball C, and such that K = fc v(Q) dvQ 00. In (4) we can take v to be any function summable on C and such that

K#0. For functions rp having on Q continuous derivatives up to order / and vanishing outside bounded domains a with V,, C Q, (4) implies the simpler integral representation

1f

SD(P)=an

1

1

r nX

(x1-yI )°i...(xn-yn) rI

a'p(sl)

8yall...ay.. AVQ

(6) 1

where a,, is the area of the unit (n - 1)-dimensional sphere.

To get (6) we assume that, for a particular point P E S2, C in (4) is a ball 17 - 'U1 < R containing V, , and we replace v in (4) by a function Vk such that Vk (Q) = 1 for R- k < I P - Q I
Kk f vk(P + r1 -Or"_ dri -. an

For I = 1

,

as k - oo.

n = 1 , and S2 = (a, b) the equality (6) reduces to the

obvious equality p(x) _ j; sgn(x - y)q'(y) dy. A direct derivation of the representation (6) can be found in the books [208), [3031, and [ 193). In [ 193] another representation is obtained, valid for an arbitrary bounded

domain f and the broader function class W(l) n W (k) with 2k > 1, where W Ik) denotes the closure, in the norm of the space of the set Co of all infinitely differentiable functions V on A that vanish in a neighborhood of the boundary, and in a neighborhood of infinity if Q is unbounded. From this integral representation it follows that the imbedding theorems in §8 remain valid for any bounded domain Q if the functions in WP') belong

in addition to the space W( Pk) , where 2k > 1. This fact no longer holds for 2k < 1. 10. Reshetnyak (see (98)) and Besov [261 obtained a representation of the form (7.12) for a broader class of domains involving the so-called "flexible cone" condition.

COMMENTS

203

11. There are also constructions of integral representations of the functions q in terms of nonmixed derivatives of fo of different orders with respect to different variables. We present one of these, due to II'in [121] and illustrating a way of deriving such representations based on the use of averagings. Suppose that p is continuous on Cl together with its derivatives 81' cp/exi'

(li are positive integers, i = I , ... , n), K(P), P = (x1 , ... ,

is an infinitely differentiable function satisfying the following conditions:

a) K = 0 outside n = [a, , b,) x ... x [a,,,

where aibi > 0, i =

b) fK(P)dvP=1;

c) f0x'K(P)dxi=0, (s=1,...,1;-I)forall P(x1,...,

.

x,_1,x;+1,...,xp) and for each i = 1,..., n. Then the function N1(P) =

(x1K(P )) satisfies the relations

5=0, 1,. for all P

1i-1

and the function

-+

1

x

Li(P)=(1.-1)1 f Ni(X1,...,xi-1,

dt

is infinitely differentiable and satisfies the conditions L. = 0 outside n, and 8i'L1/8x;, = N1.

For h > 0, 2 = (Al, ... , 2n), 2i = 1/l;,

and P/hx =

IA)

(x1 /ha- , ... , x,, /h'') we consider the following anisotropic average of the function 9: (ph,(P) =

hlz

f K ( Plhz

Pl

(o(P1)dvP

.

The equality

9,1(P) = 4'h'(P) - f hatp,,(P)dt

(7)

is valid for any a and h such that 0 < e < h. Since qy(P)

fo(P) as e - +0, we get an integral representation for to at the point P by passing to the limit in (7) as a -» +0. On the right-hand side of (7) use is made of the values of ip only at the points I + Q where t" "d E P for 0 < t < h, i.e., at the points of some "horn" V_

P+Q:QE U r, ( 0<1
t

t 1'

COMMENTS

204

with vertex at P . The parameters a. , b; , and h of the horn should be chosen so that VP c it for any point P E fl. This imposes a definite condition of a geometric character on fl. Let Q = (y, , ... , y,,) . Considering that 8

at

t-121K (Q)) _

--

tlxl AQ to

we arrive after integration by parts in (7) at the integral representation

(A.z_fL.

rQ)dv. ay,

dt

(8)

for arbitrary P E Q. 12. Other integral representations of functions are obtained and used in [9], [27], [272], [252], [322], and (240). A detailed exposition of the topic of integral representations of functions can be found in the books [29] and [98].

13. The theorem is valid for l > n when p = 1. This follows from the corresponding estimate for integrals of potential type (see (5) ). 14. For n > Ip the theorem is true also for the limiting exponent q = q . To prove it for p > 1 , it suffices to invoke the corresponding estimates for integrals of potential type (see (7) ). A proof was given for the case q' = q by Gagliardo when p = I and I < n [78]. He proposed another approach to

the proof of the imbedding theorems (for p > 1 and q' < q) that did not use integral representations nor properties of integrals of potential type.

15. For n = Ip, p > 1, the theorem ensures the imbedding of Wit) in L4. for any q' < oc. This result was improved in [339], [245), and [317], where it was proved that in this case exp

2

(IcsI)") dv


(9)

fn

for any functions rp E W(1) , where C2 is a positive constant independent of

V (the exponent p' cannot be replaced by a larger exponent). The case of unbounded domains was considered in [55]. The sharp value of the constant

C2 in (9) was obtained in [206] for I = I and p = n > 2 for functions 9 E Wol) on an arbitrary bounded domain 0.

16. If s < n , then the value of Ib on the section of n by an s-dimensional hyperplane is called the trace of qi on this section. For other definitions of the trace see the books [217], [303], and [29]. 17. The theorem on imbedding W, in LQ. on sections by s-dimensional

hyperplanes was proved under the assumption that s > n - 1p. For s <

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n - Ip the trace in the sense indicated in (16) does not exist in general for an individual hyperplane. However, it is possible to characterize the behavior of 9 on such sections if we use the space with mixed norm defined by II 9'IIL(p,.....p.) -

((fT00...

7.

dxl)

p2/p,

L:

(x1, x2, ... lip.

p,lp2

dx2)

...

) P - /P -1

dx0 I

I
Under the same assumptions about f as in the theorem in 2 of §8 we have for I < s < n - lp and q' < p(n - s)/(n - s - !p) the inequality 1191IL(p,...,p.q',...,q') :5 C411911RW,l,

(10)

which characterizes the behavior of the norm

... CJ

l

Ila(xl , ... ,

dx1... dx,)

I/P

as a function of xs+1, ... , x, . The inequality (10) was stated as a conjecture in a report by Sobolev and S. M. Nikol'skii [296); it is a special case of the inequality (11)

where 1
18. A domain Q is said to satisfy the cone condition if each point in it can be touched by the vertex of a cone in 0 with fixed height and angle of opening (this condition was apparently first considered in [279]). It was shown in [89] that the class of bounded domains satisfying the cone condition coincides with the class of domains representable as a sum of finitely many domains, each starshaped with respect to its own ball. 19.

It follows from results of Reshetnyak and Besov (see (10)) that the

imbedding theorems in §8 are valid for the broader class of domains satisfying

the "flexible cone" condition. We remark that the cone condition and the weaker "flexible" cone condition on i2 are not necessary for the validity of the imbedding theorems (corresponding examples can be found in the book [193)).

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20. Maz'ya obtained [ 188]-[191], [193] necessary and sufficient conditions for the validity of the imbedding theorems in terms of isoperimetric or capacity characteristics of the domain i2, along with sufficient conditions of a geometric character. Following [ 1931, we present one result of this kind. Let

0 be an open set, and u a measure on Q. Then for all q' E Co on a

('dy)

u9

<
(12)

if C6 = sup(k(g)'lg/a(8g)) < 0,

(13)

{g}

where q > 1 , and {g} is the collection of bounded open sets g such that g c 12 and the boundary 8g is an infinitely differentiable manifold, and a(8g) is the area of the surface 8g. The constant C6 cannot be replaced by a smaller constant. 21. In [188] and [189] the concept of capacity and isoperimetric inequalities are used to obtain imbedding theorems for domains satisfying the degenerate cone condition, which is defined, as in (Is) , by replacing the cone by a solid body of the form {x2 + . . + xn < a2xu , 0 < x1 < a), where d > 1 . Similar theorems were obtained independently in [88] by the method of inte-

gral representations. Here the relation between 1, p, and n in the theorem on imbedding in C and the exponent q in the theorem on imbedding in Lq depend on A. 22. Denote by C the space of uniformly continuous bounded functions on fl, with the same norm as in the space C. Under the assumptions of the imbedding theorem in 1 of §8 a stronger assertion is valid: if Q is a bounded domain starshaped with respect to a ball, then for any 9 E W(l) the function {b is in C . We remark that if Cl is a union of finitely many domains, each starshaped with respect to a ball, then this assertion is not true. 23. The norm IIVII(o),, is used most often to norm the spaces W(l) . Also widely used is the norm 11

to W, 11

-

,

II8x.... exa' LD

which is equivalent to the norm II9II(0 l, for the domains under consideration

in the present book (a proof that these norms are equivalent for a broader class of domains-bounded domains with continuous boundary-is given in the books [209] and [158]; the norms are not equivalent for arbitrary domains-corresponding examples can be found in the book [193]). In a number of cases use is made of the norm

+E r=i

Ilatw

ax, L,

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For domains that are unions of finitely many bounded domains starshaped with respect to a ball, this norm is equivalent to the norm Ilgl1,,,1) if 1 < 0 p < oo (equivalence fails even for a cube when p = 1 and when p = oo) . A proof of this assertion and corresponding examples can be found in the books [29] and [193]. 24.

If for an unbounded domain i2 we define the space Wol) to be

the space of functions ry for which the norm IIq?IIw, in 3 of §10 is finite, then for such spaces the imbedding theorems in §8 are valid under the same assumptions about the parameters (see, for example, [217] and [29]) also for unbounded domains satisfying the cone condition (in the theorem in 2 of §8 it is necessary to assume in addition that q' > p). 25. Sharp (smallest possible) constants in inequalities of the form (8.1) and (8.2) have been found in a number of cases. In the case of the whole space the inequality <

(p- 1 \(v')l'

I

r( + 1) r(n)

n-p

r ;)r(n-n+

i/n II

IIL.

holds for functions (0 E Co (see [188] and [69] for p = 1 , and [11] and [308) for p > 1). For convex domains S2 of sufficiently small diameter d (d < do, where d0 is determined by the parameters 1, n, and p) it has been proved [47] that for 1 > n/p ll(vllc 5 (measQ)-'/PII,,IIw;1>,

while for 1 < n/p and I < q' < q = np/(n - lp), (measf2)'/4*-'/PIIpIIw;

Il9'IlL.- <-

for any lD E W(l) (the norm 119'1Iis defined in (20) ). 26. In the case of the whole space (see(24 ) ) (14)

IIwIIL);) <_

v

v

v

which is analogous to (10.7), where the constant M is independent of io E W (') . Using it for the functions rp (e' /P P) , where e > 0, we get an inequality with an arbitrary parameter 11011L(-)

M(e-(I")

<-

II9IIL, +e"U9'IIto)),

f

(15)

where v = }(1-m-v+Q) . Minimizing the right-hand side of this inequality over e > 0, we get the multiplicative inequality 11011L(T) 5

v°(1

II9II%)".

(16)

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Since ( Y ( 1 . ) ' ' < x + y, (14) follows from (16). The inequality (15) is also valid in the case of bounded domains £1 that are unions of finitely many domains starshaped with respect to a ball, though

not for arbitrary e > 0, but for 0 < e < ea , where co depends on n, p, 1, and Q. The inequality (16) for functions in W(l) on S2 does not hold, but the following modification of it is valid: II0IIL(T, <_ i11 I9IILo(II9IILo + IIwIILo

(17)

a

Inequalities of the type (17) were first obtained by Il'in [116] and Ehrling [64].

Different inequalities of this form were proved by Gagliardo [79],

Nirenberg [222], Solonnikov [299], and others.

27. Suppose that n > lp and n - 1 > s > n - 1p. Then by the imbedding theorem in 2 of §8, for the case of the whole space (see (24)) every function 91 E WP(l) has a trace (see (16)) in LP on an s-dimensional hyperplane. The problem of constructing functions on the whole space from a given trace was posed by S. M. Nikol'skii, who solved it in the scale of spaces (introduced by him) H') (1 < p < oo, l an arbitrary positive number) consisting of functions belonging to Lo together with all derivatives of orders less than 1, with the highest derivatives satisfying in LP a Holder condition with exponent 1- [11 for I not an integer or the Sigmund condition for 1 an integer (see [217]). Necessary and sufficient conditions on the trace of a function in H1' consist in the requirement that the trace be in the space H(1- n-21P) on

an s-dimensional hyperplane (it is assumed that l > (n - s)/p). For a positive integer I and a number e > 0 we have that HDl+e) c W<<1 e Ho(l) . This and the results of S. M. Nikol'skii yield necessary conditions on

the trace of a function f E W«) , as well as sufficient conditions close to necessary conditions on a function to on a hyperplane for it to be extended to the whole space as a function in W(l) An exact characterization of the trace of a function in Wpl) for I = 1 and s = n - I was obtained for p = 2 independently by Aronszajn [8], Babich and Slobodetskii [ 15], and Freud and Kralik [74], and then for p > 1 by Gagliardo [77). These results were extended by Slobodetskii [269], (270) to

thecases s(n-s)/2 ands=n-l,p> 1, 1>(n-l)/p.

In the case p > 1 , p 34 2, 1 > (n - s)/2 the corresponding final result is due to Besov [22], who obtained an exact characterization of traces of functions in W(l) in terms of the spaces (introduced by him) B,11 (1 < p < oo, 1 an arbitrary positive number): in order that for I < p < oo, I < s < n, and

I > (n - s)/p a function to on an s-dimensional hyperplane be the trace of some function in W(l) it is necessary and sufficient that q' E BD 1-"-S/v1 on this hyperplane. This result was obtained independently somewhat later by Solonnikov [298].

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We present a definition of the spaces BP(l) (for 1 > 0 not an integer it coincides with the definitions Aronszajn, Gagliardo, and Slobodetskii gave

for the spaces W(/)): a function 9 defined in the whole space belongs to Bpl) if for nonintegers I the norm

{JJ E .a)n

alt1S)(P)

aX7' ... axa. a('1(#(Q)

axial ... ax:.

dvPdvQ

°

IF -

1,1P

Q I n+0-111),

is finite, and if for integers I the norm

(IIIeX`I...ax ((

ll

8x0 ...84^

Vv

v

dvPdvQ

el_I,(Q) 8x" ... 8x.*

I

17 -

Q In+v

is finite. The properties of the spaces H<<) and B!') were thoroughly studied in the books [217] and [29].

Analogous results are valid also for a bounded domain Q that is a sum of finitely many domains starshaped with respect to a ball and with boundaries simple in the sense of 2 in § 10. If the boundary of 0 consists of s-dimensional manifolds S. (s = 0, ... , n - 1), then for a function 9 defined on the boundary to be the trace of some function in W(!) on f2 it is necessary and sufficient that to E Bol-"-Sl0) on S. (s = 0, ... , n - 1). See

[29] for the definition of the spaces Bo on manifolds and a proof of this assertion.

28. In the limiting case I = (n -s)/p, p = I , Gagliardo gave a description -s of the traces of functions in W, on an s-dimensional hyperplane [79]. Questions involving the nonexistence in this case of a linear operator of extension from the hyperplane are investigated in [46], [239], and (105]. 29. The imbedding operator is not completely continuous in the theorem in 5 of § 11 for the limiting exponent q' = sp/(n -1 p) . Corresponding examples were constructed in [14). The book [4] gives sufficient conditions ensuring that the corresponding imbedding operator is completely continuous in the case of unbounded domains, and for bounded domains 0 satisfying the cone condition the book [193] gives conditions on the measure µ necessary and sufficient for the operator imbedding Wol) in the space LQ , (IIcIIL4 M = ( fn I9IQ dµ)'11) to be completely continuous.

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30. A theory of function spaces consisting of functions having "smoothness of noninteger order 1 " in a definite sense has been constructed. The most thoroughly studied of such spaces are the Nikol'skii-Besov spaces BD1)e (Bp1)00 = H (1)I Bo )c = BD1) (see (27)) and the Lizorkin-Triebel spaces FoB (Fp1 z = WD 1 ) when I is a positive integer). A detailed exposition of results relating to these spaces can be found in the books [217], [29], [303), [315], and [316].

For p = 0 = 2 these spaces coincide, and an equivalent norm in them is the norm II(1 +

g12)1120(4)IIL2

(18)

'

where 0 is the Fourier transform of the function (P. Spaces with the norm (18) were first considered by Slobodetskil [269]. Spaces with noninteger I are closely connected with the general theory of interpolation of spaces (see 1150], [169], [151], [315), [19], and elsewhere). Further generalizations of the spaces W(1) are connected with the study of spaces in which the differentiability properties are characterized by a certain smoothness function (see [61], [114], [335), [318), [94], [127), [128), [137], [95], [210], the survey [ 174], and others).

31. In connection with various problems in the theory of partial differential equations and mathematical analysis, the need arose to construct an imbedding theory for different generalizations of the spaces considered in the present book. There have been detailed studies of anisotropic spaces of type W(1) with norm defined by the equality P0.P1 ....Fn

L" LXJLP,

(the spaces L(, of LP, ; see

(17) ).

v with mixed norm have also been considered in place

Imbedding theorems, trace theorems, and extension theorems for anisotropic spaces of differentiable functions were first obtained by S. M. Nikol'skit [213] (for the spaces H' 1n) . The study of the anisotropic spaces W11 .....1,

(p0 = pl = = p,, = p) was begun by Slobodetskii [269] and developed by various authors (see [118], [172], [27], [299], [134], [66), [323], [26), [152), and others). In formulating the corresponding results on the domain 0 it is necessary to impose the so-called "horn condition" (see(")), which replaces the cone condition (see(")) in the case of anisotropic spaces. The parameters of the horn are chosen in dependence on the parameters 11, ... , 1n, p1, ... , pn . A detailed exposition of the results can be found in the books [217), [29], and [323). 32. S. M. Nikol'skii [218] introduced the spaces SLW with dominant mixed derivative, in which the norm for functions defined in the domain f2

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is given by

190f

II! soW-a_a1 =

axa

.

af a ..ax.

L,

where 6, takes the two values 0 or 1. The theory of such spaces and generalizations of them was developed in [ 175], [6], [60], and elsewhere. Further generalizations are connected with the study of the spaces W("M with norm a of

J (0

.

axn

(9xe' )EK

Lvo,..

where K is some set of indices (al ) ... , a,,) (the a, are nonnegative integers); see [119], [ 177), [131), [178), and others. 33. Imbedding theorems for spaces of functions of type W(1) with values in a Banach space are proved in [288]-[290]. Further results in this direction have been obtained in [87] and [149].

34. For applications to the theory of differential operators it is useful to consider the spaces W(1) , where 1 is a negative integer, which is defined ° (-1)

as the dual space of W11. (1 < p < oo). For p = 2 such spaces were introduced and used in work of Leray (see the book [165]), Lax (163], and Schwartz [262], and for p $ 2 in work of Lions and Magenes (see the book [169]). It is possible to give a unified definition of the spaces W(t) suitable both for positive and for negative 1. This question was treated in detail in the book [316] in terms of the spaces v(1e (W(l) F(l2) . The spaces Bn1)e (see (27)) have also been studied for arbitrary real 1, in particular, the spaces

(# Lo for p 0 2 or 9 # 2) ; see [220), [217]). 35. A considerable number of papers have been devoted to the study of imbedding theory for spaces of type W(1) with the Lo-norm in their norms replaced by the norm in an Orlicz space (or the norm in a rearrangementinvariant space). A detailed exposition of results can be found in the book [ 157]. Spaces of this type arise in the study of nonlinear partial differential equations. 36. In many problems the necessity arises of studying weighted spaces of type WD1) with the Lo-norm in them replaced by the norm BP(0e)

Un

(P)I°P(P)dvP) I

where p(P) is a nonnegative weight function, or by the norm 1 /p

(fnl9(P)°d#)

212

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where p is some measure. A systematic study of weighted spaces was begun by Kudryavtsev [ 154]. Surveys of results in the theory of weighted spaces and their applications to the theory of partial differential equations can be found in the papers [28], [108], [13], [12], [232), and [135] and in the books [217] (2nd ed.), [315], [157], [323], and [193].

37. The theory of spaces of the type W«) has been generalized also in other directions. In particular, a systematic study of spaces of infinite order has been made in connection with applications to the theory of partial differential equations of infinite order (see, for example, [57), [58]). Spaces of type W(<) consisting of functions of infinitely many variables have been investigated (see, for example, [76]). Spaces of type W(1) of variable (depending on the point) order have also been studied [ 17], [319]. More general considerations in this direction involve the theory of pseudodifferential operators (see, for example, [313]). Spaces W(1) with 0 < p < 1 were studied in [237) and (316). Analogues of imbedding theorems for differential forms were obtained in [97). 38. Different geometric characteristics of imbedding operators have been

studied in a number of publications (see, for example, [297], [126], [133], [31], [314], [122], [130), and Chapter III of the survey [312]). 39. It was established in [53] and [196] that for any domain f the space W(l) (1 < p < oc) coincides with the completion of the set of functions P 1P E WD1) infinitely differentiable on 0 in the norm of this space. It was proved in [40] and [44] that for any domain Q any function 50 E WI) can be approximated arbitrarily closely in the norm of this space by infinitely differentiable functions on f taking the same boundary values as p in a certain sense. It was established (78) that for domains Q with locally continuous boundary the set of functions infinitely differentiable in the whole space is dense in WP(1) (generalizations of this result can be found in the book [29], §19). For a certain class of domains ( for which this assertion fails, [34) gives necessary and sufficient conditions on a function p E W(1) under which it can be approximated arbitrarily closely by infinitely differentiable functions on the whole space. 40. The book [295] (p. 283) presents necessary and sufficient conditions

on a function 9 E W(1) on i2 (1 < p < oo) under which it can be approximated arbitrarily closely in the W(l)-norm by functions in Co on a bounded domain 0 that is a sum of finitely many domains starlike with respect to a ball when the domain has a simple boundary in the sense of 2 in § 10. Corresponding necessary and sufficient conditions in the case of an arbitrary domain f) were found in [41] for p > n. In the same place there are conditions on C for p > n necessary and sufficient for the possibility of approximating functions to E WP(l) arbitrarily well by functions in Co to be

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equivalent to the membership in WD l) on the whole space of the extensions

of functions f E WP(l) on S2 by zero outside 0. For p < n a number of results in this direction were obtained in [ 110] and [ 111 ]. The problem of "spectral synthesis" for the spaces W(l) , which is closely connected with this question, was solved in [I I I]. 41. In [291]-[2931 theorems were proved on approximation of functions

in Lt l) on R" (I < p < oo) by functions in Co . Analogous questions for unbounded domains are considered in [24]. Questions on approximation of potential and solenoidal vector fields with applications to hydrodynamics are treated in [ 112], [18 1 ), and [ 182].

42. Important in the study of the spaces Wu> is the question of extension

of functions in W«) on a domain Q to the whole space with preservation of class, i.e., the question of constructing for any 97 E W(<) on S2 a function

4) E W(l) on R" coinciding with ip on 0. For domains with a sufficiently smooth boundary the extension theorem was proved for I < p < oo in (219], for domains with boundaries of class Lip I and for I < p < oo this result was obtained in [48], and for 1 < p < oc in [31]. Further results were obtained in [43], [38], [332], [125], (148], and [329]. For certain values of the parameters 1, p, and n necessary and sufficient conditions on i1 have been obtained for the validity of the extension theorem (see [332) and [ 148)).

For domains with boundary of class Lip y (0 < y < 1) a theorem on extension with preservation of class does not hold in general. For such domains a theorem was proved in [43) on extension with minimal worsening in class, namely, deterioration to the space Wilr] . Theorems on extension with preservation of smoothness exponent and with minimal worsening of summability exponent have been proved for various types of domains with boundary of class Lip y (see [194], [68]). Extension theorems for the anisotropic spaces Wpo:v1 . v. (see (31)) were proved in [45], [67], and [268]. 43. Let to : R" R" be a homeomorphism and q7' the operator defined

by the equality (to' f)(p) = f(()). The question of necessary and sufficient conditions on fi under which qi' implements an isomorphism of the spaces Wol) was studied in [330], [333), and [255). 44. Numerous articles have been devoted to the solution of boundary value problems by variational methods (see [204) and the bibliography given there, and also [198], [199], [154], and [72], and others). 45. See (27) about necessary and sufficient conditions on ci under which this function is permissible.

46. This assertion was proved in [282], published in 1937. In the same place an analogous proposition, given in 4 of §14, was proved for polyharmonic equations. For the Laplace equation this assertion was also obtained

214

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in 1940 by Weyl [336); it is often called Weyl's lemma in the literature. (A simple proof can be found in [225], p. 71.) 47. An analogous example was constructed in 1871 by Prym [250], but his paper remained unnoticed for a long time. The example constructed by Hadamard in 1906 [106] influenced investigations in the area of the calculus of variations and the theory of functions, and led to the formulation of the problem of finding necessary and sufficient conditions for a function to be extendible from the boundary of a domain into the domain with the requirement that the function belong to the space W(l) on the domain (see (27) ).

48. Many monographs and papers have been devoted to the study of generalized solutions of boundary value problems for second-order elliptic equations, and the basic questions have been included in textbooks (see [521, [2051, [161] and its bibliography, (2001, [197], [225], etc.). A survey on second-order equations with nonnegative characteristic form can be found in [231). 49. See [29) about necessary and sufficient conditions on the system

of functions under which it is permissible. 50. A further study of boundary value problems for elliptic equations with boundary conditions on manifolds of various dimensions was carried out in (143], [229], [147), [304), and [305]. The study of the biharmonic equation with two independent variables, which arises in the theory of plates, is of great interest for applications in elasticity theory. The behavior in a neighborhood of boundary points of a generalized solution of the fundamental boundary value problem considered in §§ 14-15 for the biharmonic equation was investigated in detail in [143]-[147] and [226], [229], along with the question of smoothness of the generalized solution in a closed two-dimensional domain. These questions are closely connected with the Saint Venant principle important in mechanics (see (228)). Analogous questions were considered in [310] for a polyharmonic equation. A survey was given in [144] of publications on the investigation of boundary value problems in nonsmooth domains. Expositions of the contemporary theory of elliptic boundary value problems can be found in the books [114], [21], [169], [311], and (313). 51. Variational methods have found wide application in the spectral theory of operators, and have been further developed (see, for example, [51], [242], [ 199], (1), [170], and (103) and its bibliography). 52. At present there are several monographs [81), (166], (107), [1651; (see also [123), [202], [114], [115], [183]) in which various methods are used to investigate the Cauchy problem for hyperbolic equations and systems. The work of Petrovskii (241) in which the concept of a hyperbolic system was first introduced and the Cauchy problem was first investigated for it, is contained in Russian translation in the book [243]. In the same place there are comments by Volevich and Ivrii on this work [334), and a survey is given of

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publications on the Cauchy problem and the mixed problem for hyperbolic equations. 53. An analogous investigation of the Cauchy problem for the wave equation with initial data in the spaces B (l) was carried out in [25]. 54. The integral equation (20.22) obtained for the solution of the Cauchy problem (19.1), (19.40) was used by Levitan [1671, [168] in the study of the asymptotic behavior of the spectral function in the eigenvalue problem for a selfadjoint second-order elliptic equation with Dirichlet boundary condition. 55. In the present section the solution of the Cauchy problem is reduced to the Volterra integral equation. All considerations are carried out "in the small," i.e., in a sufficiently small neighborhood of a particular point, which ensures the absence of singular (focal) points on the conoid of characteristics. In [184] and [185] Maslov proposed a method that enables one to reduce the Cauchy problem globally to a Volterra integral equation, i.e., in any domain

of existence of a solution of the problem. By this method he proved the existence and uniqueness of a solution of the Cauchy problem "in the large," investigated the structure of a fundamental solution, and studied the question of propagation of singularities of the initial functions uo and u1 . 56. In the past few decades much work has been devoted to the investigation of the Cauchy problem for second-order equations of the form (21.5) when the condition (21.6) is replaced by the condition E" _1 AJjplpj > 0. Such equations are called degenerate hyperbolic equations or hyperbolic equations with multiple characteristics. The Cauchy problem and boundary value problems for such equations and for higher-order hyperbolic equations with multiple characteristics have been studied in [32] and [224] and elsewhere (see the survey [334]).

57. With regard to the theorem on imbedding in L.. for the limiting exponent (see (7)) it can be assumed that e = 0 in the condition o) in 3) and 4) for n > 2, and in 5) for n > 4. To prove the lemma in item 4 it suffices to use the imbedding theorem for the limiting exponent. 58. We show that the functions u, have a common domain A' of defini-

tion such that Q.(0) c tY c S2 for all h, 0 < h < ho, ho = const > 0. This is essential for the proof of the theorem.

According to the constructions in §20, the Cauchy problem for equation (19.1) with smooth coefficients and smooth initial functions uo and u1 defined on S2(0) has a solution at those points of the neighborhood of S2(0) that are vertices of characteristic conoids belonging to f for 0 < x0 < x0 o, intersecting the hyperplane x0 = 0 at points of Q(0), and reducing by a nonsingular change of the variables x0 , ... , x,k;1 of the form (19.26) to the circular cone (19.28). As shown in I of § 19, the determinant D(x0 ..... . y2k+i) is nonzero at the vertex of the conoid, and it is possible to determine a ball of radius z about the vertex of the conoid where this determinant is nonzero; moreover, r does not depend on x2k.1)/D(yo..

216

COMMENTS

the location of the vertex. This is easily seen from the formulas (19.25) if the smoothness of the coefficients of the leading derivatives in equation (19.1) is taken into account.

We prove that for each particular h with 0 < h < ho the solution uh can be defined in a domain Q' consisting of a union of circular cones K c Q such that the angle at the vertex of K formed by a generator of K and the t-axis is equal to a, and the intersection of the cone with the hyperplane t = 0 belongs to Q(0) ; here a is a certain number with 0 < a < X/2. It is easy to see that a can be chosen to be independent of h and the location of the vertex of K in such a way that the characteristic conoid with vertex coinciding with the vertex of the cone K lies interior to K for t > 0. By what was proved in 8 and 9 of §20, the solution uh of the Cauchy problem for the averaged equation with the averaged initial conditions for t =

0 exists in each cone K for 0:5 t < y, where y = T cos a (r may depend on h). Further, solving the Cauchy problem with the initial conditions obtained for uh when t = y, we extend the solution uh on the cone K for y < t < 2y, then for 2y < t < 3y, and so on. In finitely many steps we construct uh in the whole cone K. Note that the domain fY, which is determined by the number a and in which the uh are defined, depends only on f2 and the maximum moduli of the coefficients A;, of equation (21.5). 59. By taking into account the theorems on imbedding in L. for the limiting exponent (see (7)) it can be assumed that e = 0 in conditions n) in 2) for n > 21- 2, in 4) for n > 21, and in 5) for n > 21 + 2. To prove the lemma in item 7 it is necessary to use the imbedding theorem for the limiting exponent. 60. Leray's separating operator method [165] is an analogue, for the case of higher-order hyperbolic equations, of the method given here for obtaining a priori estimates for the solution of the Cauchy problem for a second-order hyperbolic equation. 61. See (58) about the domain of existence of solutions. 62. We remark that by using estimates analogous to those obtained in 4 and 7 it is possible to construct solutions of the generalized and classical Cauchy problem (21.5), (21.7) by the Hopf-Galerkin method [113]. Here an approximate solution is sought in the form of a linear combination of finitely many

functions depending on x and taken from some complete system of functions, with coefficients depending on t. The coefficients are determined as the solutions of the corresponding system of ordinary differential equations. The imbedding theorems imply convergence of these approximate solutions to a solution of the Cauchy problem (see, for example, [325)).

APPENDIX

Methode nouvelle h rcisoudre le probleme de Cauchy pour les equations lineaires hyperboliques normales t Introduction

Le probibme dont nous nous occupons dans ce memoire a ete trait6 par differents auteurs.

Pour Ie cas le plus general ce probleme etait resolu en premier lieu par M. J. Hadamard dans ses memoires bien connus et dans son livre elegant "Legons sur le probleme de Cauchy pour les equations hyperboliques" [ 107]. La seconde solution appartient h M. Mathisson [ 187].

Ici nous donnons une autre solution de ce probleme. L'idee principale de noire methode, differente de la methode des auteurs cites, presente un developpement des idees de Kirchhoff appliqu6es par lui A une equation d'onde aux coefficients constants dans l'espace A trois dimensions (quatre variables independantes). M. Gogoladzb et l'auteur ont deja applique les idees enoncees dans cet article $ quelques cas sp6ciaux [273], [274], [9l], [92], [93]. Trois petites notes sur ce sujet ont et6 deji publiees dans les "C. R. de I'Ac. des. Sc. de 1'U.R.S.S." [275], [276], [277].

Dans ce memoire nous allons exposer la methode en detail. Le premier chapitre contient la deduction de noire formule fondamentale qui sert 'd construire le premier algorithme de calcul. Cette formule est une identite int6grale qui lie la fonction quelconque u avec les valeurs initiales u1r.o = U (0) et

8u l = uu) 8t 1.o de la fonction u et de sa derivee premitre sur la surface initiale et avec la fonction

p = Lu

(0.1)

This appendix is the article by S. L. Sobolev [278] published in 1936. A short exposition of results of this paper is contained in [277], published in 1935. See the "Comments on the Appendix", written by V. P. Palamodov, on pages 253-267 of this volume. 217

APPENDIX

218

qui est le r6sultat d'une operation diff6rentielle lineaire hyperbolique sur u. Puis nous donnons quelques calculs 6lementaires des approximations successives qui permettent de deduire la second identite principale. Cette dernibre identite exprime la valeur de la fonction u h 1'aide des fonctions p, U(O) et u(') citees ci-dessus. Les autres chapitres contiennent quelques generalites concernant 1'existence de Ia solution cherch6e. En nous servant de quelques conceptions de l'analyse fonctionnelle nous

representons notre problbme dans une forme nouvelle et demontrons que dans cette forme le probltme admet toujours une solution unique. Si la solution cherchee existe dans le seas classique, alors notre solution se confond avec celle-ci.

Chapitre I. L'identit fondamentale 1. Probl6me de Cauchy. Dans ce chapitre nous nous occupons de la generalisation d'une formule due h M. Kirchhoff qui aura une grande importance pour tout ce qui suit. L'equation la plus generale du type lineaire hyperbolique normal A un nombre paire des variables independantes, comme it est connu, peut etre reduite a la forme 2u

2k++12k+1

Lu = i-1 j=1

AijBx i

2k+I

8x +

8u

BiBx +Cu i-1

2

t2

ou les coefficients Aij = Aji , B1, C sont des fonctions des variables independantes x1 , x2 , ... , x2k+1 , t et la fonction F est une fonction donnee de ces variables. Le probl8me de Cauchy consiste dans la recherche d'une solution de cette equation qui satisfait aux conditions uir-0 = u

(XI , X2, ... , x2k+1),

(1.2)

8l

= u(')(x1 , x2 , ... > x2k+1) . 11=0

Dans ce qui suit nous nous bornons seulement au cas of les coefficients Aij , Bi , C sont des fonctions analytiques des variables independantes, mais cette supposition ne joue aucun role dans 1'enonce suivant et peut We remplacee par la seule condition que ces coefficients aient un certain nombre de derivees continues.

Pour que l'equation (1.1) soit du type hyperbolique normal iI faut et it suffit que la forme quadratique 2k+1 2k+I

Aijpipj Li-1 Lr j=I soit positive definie. Supposons que cette condition est toujours remplie.

APPENDIX

219

2. Remarques preliminaires. Rappelons maintenant quelques traits principaux qui se rattachent a la construction d'un conoide caracteristique avec le

sommet dans un point donne M° aux coordonnees x° , x2 , ... , XU+1 , 1 L'equation de la surface caracteristique pour ]'equation (1.1) sera, comme on sait bien: 2k+ 12k+ I

A= E E =AjPPj-q2=0, .=1

(1.3)

j=1

ou par pi, q sont designees des quantites proportionnelles aux cosinus directeurs de la normale a la surface caracteristique. Soit ]'equation de cette surface

G=O;

(1.4)

alors pi et q peuvent We choisies de telle facon que aG

p'

- ax. '

OG

q - ar

(1 5) .

Les equations differentielles des lignes "bicaracteristiques" c'est-a-dire, les "caracteristiques des caracteristiques" seront ds =

dx. _ -dp. _ dt _ -dq -q - T81-01 15/8p, 1a/axe

(1.6)

Les bicaracteristiques de (1.1) sont celles des solutions du systeme (1.6) pour lesquelles A=O. (1.7) La surface integrale de ]'equation (1.3) sera engendree par une multiplicite Mk+1 de points M avec les coordonnees x1. x2, ... , x2k+1, I appartenant a une telle famille de bicaracteristiques, satisfaisant it la condition (1.7) et dependant des 2k parambtres v1 , v2 , ... , v2k de telle fawn que

+qdt = 0.

p1dx1

(1.8)

De la theorie des equations du premier ordre on sail que Ia condition (1.7) sera satisfaite le long de chaque courbe integrate de (1.6) si title est remplie

dans un point quelconque s = 0 de cette courbe. D'une facon analogue la condition (1.8) sera aussi satisfait automatiquement sur toute la multiplicite M2k+l si title est vraie sur quelque sous-multiplicite M2k qui correspond a une valeur constante de s. Construisons maintenant la famille de bicaracteristiques passant par It point M°. 11 est clair qu'on peut definir ces lignes comme les courbes integrales de (1.6) satisfaisant aux conditions initiales 0

0

X. I, o =x. tls=o =1

P, IS=° =P; 0

0

,

qi3 0 =q

(1.9)

220

APPENDIX

Ici les p° et q° sont des parametres arbitraires qui, comme nous avons deja vu, doivent satisfaire a la relation 2k+I 2k+I

F,E Avp, ° q ° - q ,=I i=I

z

= 0.

(1.10)

(La signification de A° est evidence.) Nous obtenons de cette fawn une famille dependant de (2k + 1) parametres arbitraires, mais it est aise de verifier point de vue des valeurs de x, et t un de ces parametres est illusoire parce qu'il entre comme un facteur de p,

et q. En effet, en remplacant dans (1.6) p, et q par ap" et aq' et s par s'/a nous voyons que les equations restent invariantes, c'est-a-dire, si le systeme des fonctions x,(s), t(s), p,(s), et q(s) satisfaisait (1.6), alors le systeme des fonctions

x'(a)' t la/ , ap' la/ ,

et

oq (a)

satisfait aussi ace systeme (1.6). Nous voyons ainsi que pour conserver dans l'espace R2k+2 tous les points qui appartiennent aux bicaracteristiques passant par M° on peut se borner settlement de la consideration de telles courbes pour lesquelles, par exemple, q0 = 1

.

Dans cette hypothese notre famille satisfera a toutes les deux conditions

(1.7) et (1.8) pour s = 0, c'est-a-dire au point M°, et, par consequent, la multiplicite M2,+I , des points de notre famille, presenters une surface caracteristique dans 1'espace R2k+2. On appelle cette surface le conolde caracteristique. Pour preciser quelques proprietes de ce conoide nous ferons une analyse plus detaillee des integrales de ('equation (1.6).

Ayant suppose qu'au voisinage du point M° les Aj -, B, , et C peuvent etre developpees en series des puissances de (x, - x,°) et (I - t°) nous voyons que la solution generale de (1.6) aux conditions (1.9) peut We aussi developpee en series des puissances de la variable indbpendante s dont les coefficients sont des fonctions de xo , to , po , q0 11 est facile de montrer que les coefficients de s" dans le developpements de x; et t seront des polynbmes homogenes du n-iCme degre des quantites p,° et q0, tandis que les coefficients de s" dans les developpements de p, et q seront des polynbmes homogenes du degre (n+ 1) de ces mimes quantites. En effet, le procede de calcul de ces coefficients consiste dans la determination de ]a valeur de la n-iCme derivCe de la fonction inconnue par rapport

a s au point initial. Si nous etablissons que la n-ibme derivee de x, et

APPENDIX

221

(ou respectivement pi et q) s'exprime dans chaque point comme un polynome homogene du degre n (respectivement n + 1) par rapport aux p, et q, alors notre proposition sera demontree. On le demontre par induction complete, parce que la derivee par rapport A s d'un tel polynome nn_, (p, , q) s'exprime de ]a maniere suivante: t

dnn_, = ds

ann_1 dxi i-1

Ox

ds

ann- dpi +ann-1 d9+ann-1 dt api ds aq ds at ds

LL.

(1 11

)

En remplacant ici les , d., , , et , par leurs expressions tirees de (1.6) nous voyons que cette derivee est un polynome du degre n ce qu'il fallait demontrer. Un calcul immediat nous montre que les developpements des x,, t, Pi, et q seront les suivants: 2k+ I

0

00

0 o n (1) 0 0 xi=xi +SEA,jpj+ES Xn (pj,q ),

n-2

i=1

x

t°-sq °+ES Tn(pj,9 n-2

(1.12)

00 eEsnnn(i)(pj

Pi =P1 +

,

90)

n=2 r00

0 0 q=q +cEsPn(p,q) 0

n

1

n=2

11 faut bien noter que les series obtenues seront uniformement convergentes

dans un certain voisinage de s = 0 non seulement pour p° et q0 reels, mais aussi pour toutes les valeurs complexes de ces parametres, bornes par un certain nombre M. Introduisons maintenant au lieu de xi et t les variables normales de Lipschitz, c'est-&-dire les variables

Q=s9

P; =spo,

(1.13)

Comme nous voyons maintenant les variables x, - x° et t - t° peuvent titre representees par les series uniformement convergentes de P, et de Q 2k+1

00

j=1

n-2

0 i (xi-xi)= EA0 ijPj+ EX, (P;,Q),

(t-t°)= -Q+1: TT(Pi,Q), n=2

(1.14)

APPENDIX

222

tandis que p, et q seront representees par les series 1

P;+nW(P;, Q)

P; = s

n-2

q

(1.15)

IQ +Pn(P1, Q)1 n=2

La relation entre P; et Q et x; - x° et t - to est reciproque, c'est-adire, on peut former les developpements des fonctions P; et Q suivant les puissances de x; - x° et I - t° . Ces developpements seront 2k+1

00 (xj-x0)+

H0

,j

P;=

n=2

j=1

(1.16)

00

Q= (t°-t)+EH,,, n=2

oiu les 8(') et H. sont des polynomes homogenes de x; - x° et t - to et la matrice H° est. reciproque a la matrice A° , c'est-a-dire

r,

2k+1

A;jHj,=6;,={111 0, i=

i

(1.17)

Dans les coordonnees P, et Q 1'equation du conoide caracteristique sera reduite a la forme 2k+1 2k+1

A°P;P1-Q2=0.

G

(1.18)

;_ i=1 Cela suit immediatement de la condition (1.10) si nous la multiplions par s 2 . Pour tout ce qui suit il sera necessaire de resoudre cette equation par

rapport a 1. Soit le resultat de cette solution, c'est-a-dire l'equation du conoide, resolue

par rapport a t :

t-z(x1,X21...,x2k+1)=0.

(1.19)

Nous passons maintenant au calcul de quelques derivees partielles de t par rapport a x1 , x2 ° , x2k+I .

APPENDIX

223

II est evident que aT

P; + _°n°__211h'1

P; q

ax!

--

1

Q + En=2 Pn

I H (x1 - x,) + E°°2 Kn

(1.20)

t - t + n=2 Ln P;-)-Q(-)+...

a2T

8X; axe

Q2 +...

-(t - t) H; 0

2

0

2k+1

2k+1

0

0

0

H;.H)J(x; - x;)(X - Xi) + .. .

_ -H°Q2+P.P. + Q3+...

(1.21)

ou dans le numerateur sont omis tous les termes du degre superieur ou egal h trois et dans le denominateur sont omis les termes du degre superieur $ trois. Le calcul des derivees d'ordre superieur est evident. Les numerateurs et les denominateurs des fonctions ainsi obtenues seront

evidemment convergents dans une region qui est la meme pour toutes les derivees. Le degre du premier terme du denominateur sera egal au degre du terme correspondant du numerateur plus (m - 1), oil m est l'ordre de derivation. 3. Quelques relations entre les derivees successives d'une fonction donnee. Introduisons maintenant au lieu de la variable t une nouvelle variable independante

Convenons d'indiquer le resultat de changement des variables independantes dans une fonction quelconque p par le symbole , c'est-h-dire,

V(xl,x2,...,x2k+1, 11) = q(X1 , X2 , ... , X2k+1 , 11 + T(x1, X2 , ... , x2k+l )) .

En designant par

(1.22)

. la derivee partielle par rapport a x, , dans l'hypothCse

que les variables independantes sont x, , x2, ... , x2k+1, It et par D la derivee par rapport I la meme variable, dans l'hypothCse que les variables

APPENDIX

224

independantes sont xI , x2 ,

, x2k+1 , t1 , nous aurons evidemment

au _ Du 8t Dtl '

a2u

D2u

Dt '

auDuDuaT ax;

Dtl ax; '

Dxi

D2u 0T

D2u Or

Dxi Dt axj

Dxj Dt ax,

2

a u _ D2u ax; axj Dx; Dxj

D2u Or Or

(1.23)

Du a2T Dtl ax;axj

D12 Ox, axj

Aprbs le changement des variables independantes notre operation Lu sera de la forme D2u

2k+1 2k+1

2k+1

Du

Lu = i-1E Fj-1Arj Dx Dx.! + L=l

B,

i

-2

2k+1 (2k+',,,,,9-r

D2u

ax.l

j=1

Dx;Dt

F, r, Aa2Z r-1 j-1

E

+

AaT--Or F, F j-1

;-1

'ax;

i=1

2k+12k+I

.

.-

2k+1$0 t

ijOx,Ox,

2k+1 2k+I

+ Ci

,

D2u 1

axi axj

Dt'

Du Dtl (1.24)

.

Introduisons maintenant quelques notations. Considerons 1'espace R2k+I coordonnees xl , x2, ... , x2k+l , et certaines operations lineaires dans cet espace. Soit 2k{

M( )v =

2k+ arA; j l

-, ;-1 atl

it =0

Or Or -art ] v I

ax; axj

2k+1 {2k+I a'A.

,=1-1

2arA'I 8t'

i =1 j_1

I

r =0

an 8xj

Dv Dx;

a2 Z ri=0

axi 8xj

D2v

2kk++12kk++1

D(')v = Z.., Z.., i-1 j=1 atI'

at;

2k+1

+

at'B; I

aT

ate

Ox.

i=1

2k+I ar

a'A+.j

t,=0

Dx; Dx,

=1

at;

r,=0

r,-0

Dv Dx;

v,

aCI +-p

APPENDIX

225

Convenons encore d'indiquer par us la valeur de acteristique

sur le conoide car-

U.'(XI,X2,...IX2k+l)_

sU(XI,X2,...,X2k+I,t(XI,X2,...,X2k+1))'

(1.26)

En differentiant notre identite (1.24) s fois par rapport A r et en posant I, = 0, nous obtenons evidemment s

I

r! (sS- r)!

f{M(r)us-r+2 + L(r)Us-r+1 + DrUJ_r) _ FLU I _

(M 2/ = 0). o

(1.27)

En reuniant les operations faites sur la meme fonction et les designant par D(S)v = D(s)v 3

,

D(s) v = sD(s-')v + L(s)v, ....................................................

D(s)v

=

s!

D(r)v +

L(r+1)v

s!

(r+1)!(s-r+1)!

r!(s-r)!

(r+2)

S!

+(r+2)!(s-r+2)!M

(1.28)

v

I(s)v = L(0)v +SM")v, nous obtenons finalement (1.29) s=°

(r=0, 1,...,k-1).

Le systeme d'identites (1.29) joue un role essentiel dans ce qui suit. 4. La construction des fonctions ar . Les operations Mk_, sont k operations lineaires sur k + 1 fonctions u°, UI , ... , Uk U. Introduisons maintenant un systeme de k + 1 operations lineaires N' , sur les k fonctions a1, a2 , ... , ak qu'on peut nommer adjointes aux operations M,-,. On ob-

tient ces operations adjointes par ]'integration de ]'expression k

ff

c,Ms d R2k+I s=1

par parties, l'integrale ctant prise sur une certaine portion de 1'espace R2k+I , de telle fagon que l'integrale restante ne contient plus de derivees des u. Les coefficients de ur dans le terme restant seront des operations Nr N.

APPENDIX

226

Un calcul assez simple nous donne I(k-1)N'a k 1= a2 ,..., ar) = I

Nk-r+1 (al

1

r-2

(k-r)' ar +

L

(k-r+s+1)'

Ds

ar-s+l

s=0

(r=2,3,...,k),

(1.30)

k-1 s=0

of par le signe ' est designee l'operation adjointe au sens classique. Nous avons par definition k

k

k

j.°

s=l

ajMjujN, _ L(ak-t+11

j=I

us - us1

ak-s+I)

k k-j+1

+ FE (ajD(k ')sus - u,D( i)saj)

(1.31)

j-1 s-0 2k+I

aU

rsI axr

Dans la suite nous allons preciser les fonctions U, qui sont facilement calculables.

Construisons maintenant un systbme de fonctions ar , solutions particulieres du systbme

Ni =0 (1=1,2,...,k). (1.32) Choisissons ce systbme de fonctions de telle fagon qu'il possbde les proprietes suivantes: a) Chaque fonction a, est une fonction analytique uniforme des variables p2 , ... , p2k+II. q° , et s, dependant seulement des produits , pis pis , . , p2k+l s , q0s. II faut remarquer que ces variables ne sons pas independantes, car sur noire multiplicite caracteristique on peut toujours prendre q0 = I et, d'autre part, 2k+I 2k+I

EApip ,

-q°z

=0. i-l j=1 b) Chaque fonction a, comme fonction de la variable s peut We presentee comme un polynbme de degre (r - 1) de log(q°s) : 0,)]r-1. (1.33) ar = RO(s) + RIr)(s) log(q°s) + ... + R(r) I (s)[log(q Les coefficients de log(q°s) seront des series de puissances de s contenant un certain nombre de termes de degre negatif et convergeant dans une region determinee.

APPENDIX

227

Lt premier terme dans la serie R(')(s) sera de la forme

CS

- (k+r-1-1)

et dans la premiere serie, c'est-A-dire R( )(s) , ce terme sera precisement egal A

(-2)k-'r(k + r - 1)I-(k)

(90S)-k-r+1

/I'(k - r+ 1)I-(2k - 1)F(r)

- ak)(Q0S)-k-r+1 - a(r)Q-k-r+l (1.34)

oil par AO est designee la valeur du discriminant de la forme A au point 0

X1

0

0

2 , ... , X2k1 . 1

0

En particulier, le premier terme dans la serie

sera

2k+1

.

1

R(k) (S)

Q-

(1.35)

Do

11 est evident que ces premiers termes dans Ro) seront prealables dans toutes les expressions de or,.. Il est aise de voir que le systeme (1.32) est un systbme de recurrence et que les fonctions a, peuvent We determinees successivement. Pour chaque fonction a, nous aurons de cette facon une equation dilferentielle lineaire du pre-

mier ordre, dont le second membre depend de toutes of (I = 1, 2, ... , r 1). Donc, toutes les a, peuvent titre obtenues par de simples quadratures. Pour demontrer 1'existence de la solution ci-dessus nous procedons par ('induction complete. Supposons que les fonctions of , 02 , ... , o,_ 1 sont determinees de la facon

indiquee. L'equation pour Ia fonction o, sera de la forme

o = - r-2

I:D(k-r+$+1)0

(1.36)

:-o

Integrons cette equation par la methode de variation des constantes arbitraires. L'equation homogene correspondante sera

I(k-')'u, = 0.

(1.37)

APPENDIX

228

L'operation 1(k-')-Q, est de la forme suivante: l(k-r)' Q = L(o)' o` + (k - r)MI )- a r r r 2k+1

2k+1

D

at

2 =11 DX' E;=1Aii

j

2k+1 2k+1

19X;

r,=0

+2k+1

a2r

I

A 11

or

B

aX,aX;

,=019X,

'I

=0

2k+1 2k+1 af1,;

- lEE t ,-O

Q - o,

19X;19X;

c'est-A-dire

at

19Q,

19X;

19X;

1(k-r)

°r = 2

A,, j=1

i=1

r,=0

alt i=1

19X; 19X,

r

j=1

(22 +2

=0

u 19X; j=l

i=1

2k+1 2k+1

+(k - r) 1=I

aA,;

E j=l

at

11

'J

19X`

r ,-O

at at

BA,)

ar

_B

I

ax.r ax1 o,. (1.38) r,=0

En tenant compte de

`

2k+1

i=1

at 2k+1 1 ax ;j p A'jIr =019X. - - E I; qI = --qds i= ff

J

[voir (1.6)] et en designant par S2, le coefficient de a, daps l'equation (1.38) nous aurons 9

as' - '2,o, = 0

(1.39)

De ('equation (1.39) on obtient

Q,=- ,(P/,P2,....P2k0 +1'q°)eJ '

dr=c,ef

'

.

(1.40)

Precisons d'abord Ie second facteur daps la formule (1.40). La fonction gsi2, est evidemment la valeur sur le conoide caracteristique dune fraction de deux series reguliZres par rapport aux P, , P2, ... , P2k+1 Q Q. Ceci sera as , a sons des evident si nous ramarquons que les coefficients A,; , B1, series regulitres en PI , P2, ... , Q et que les derivees de r sons des fractions comme nous aeons vu daps le no. 2.

APPENDIX

229

,

q0

Or, gsf2, comme une fonction de s dependant des parametres po peut titre developpee en une sbrie de puissances de cette variable convergente dans un certain cercle IsI < p (1.41) dependant seulement du premier zero du denominateur de toutes les derivees de T, c'est-a-dire, du premier zero de la fonction 0

0

0

et des rayons de convergence d'autres series qui entrent dans gsCl,. (En tenant compte de ce que la valeur initiale de q est l'unite nous voyons que ce rayon est different de zero.) En calculant le premier terme dans gsf2, nous voyons qu'il s'obtient de ]'expression

a2T

2k+1 2k+1 i=1

;_1

et qu'il est egal A Q

-2k+I

i-l

- k'1A?H,°Q2+.zki1FJ2_k'IA°PfP;+... Q3+...

Q-(2k+1)Q2+Q2+...

Q3+...

=-2k.

(1.42)

En nous servant de ce resultat nous pouvons preciser la formule (1.40) en posant 6 -_ r

1

(q°S)k

efu +k

(1.43)

L'espression (1.42) est une fonction analytique des variables P11 P2 , ... ,

P2k+ I ' Q et une fonction de s qui peut titre dCveloppee en une serie de puissances de s dont le rayon de convergence est non nul. Il en suit qu'en posant I

°1-

(-2)"1I'(k)

r(2k-1)

Q,,

(1.44)

a1 sera de la forme indiquee. Passons au calcul de la fonction a, meme. En designant le second membre de ]'equation (1.36) par X, nous pouvons obtenir la solution generale de cette equation dans la forme 0

$ qsX, ds 2ff, s

(1.45)

Considerons maintenant de plus pres !'expression X,. Les derivations par rapport aux x1, x2, ... , x2k+ 1 qui entrent dans les operations (1.30) peuvent titre remplacbes par des derivations par rapport aux variables normales

PI, P2, ... , P2k+1, Q. D'autre part, la derivation par rapport a P; ou Q

APPENDIX

230

d'une fonction qui depend settlement de ces arguments, c'est-A-dire des pro-

duits des p° et q° par s peut etre reduite is la derivation par rapport A p° et q0 au moyen de la formule simple

8F(PI,P2,...,P2k+l,Q)8F _r ap°

d,ou

aPi

8F1aF

s apo

8P;

(1.46)

11 suit de cette remarque et de la structure des fonctions of , 02, ... , or-1

que X, sera aussi un polynome en log(q°s) du degre (r - 2) : Xr = TO )(s) + TT')(s) log(q°S) + ... + Tr(')2(s)[log(q°s)]'-2 .

(1.47)

Les coefficients de ce polynome T(') seront evidemment des series de puissances en s convergentes dans le meme cercle (1.41). II est facile de verifier que Ie terme principal dans 1'expression T(') sera de la forme CIS-(k+r-1+1),

(1.48)

tandis que le terme principal dans T(r)(s) sera equivalent 3 2k+1 2k-1 =1

J=1

N

(-2)* -r+1

D2

Q_k_r+2 r(k + r - 2)r(k) r(k - r + 2)r(2k - 1)r(r - 1) - 2k+12k+1 Ao8 2r (-2)k -r r(k + r - 2)r(k)(-k - r + 2) Q_k-.+2 'l T' -A x ri r(k - r + 2)r(2k - I - 1) =1 j=1 A

0

' Dx, Dx,

AT

E qo ar 8r (-2)k +E of ',8x'8"

r(k + r - 2)r(k)(-k - r + 2)(-k - r+ 1) Q-k-r

r(k-r+2)r(2k-I)r(r-1)

I

(-2)k-'+1

f-U

-

r(k + r - 2)r(k) k-, r(k-r+2)r(2k- I)r(r- I)I-2k(k+r-2)+(k+r)(k+r-2)]Q-

r(k + r - I)r(k)

_ vrAT

Q-k-r

r(k - r + I)r(2k - I)r(r - 1)

et sera donc egal 6 (-2)k-r+1

r(k + r - 1)r'(k)

T r(k - r+ l)r(2k- )r(r- 1)

o

s)

-k-.

(1.49)

L'expression qsX'

(1.50)

20,

sera aussi un polynome de log(q°s) de la forme qsX, = (r) (r) 0 2or L o (s) + L I (s) log ( q s) +

(r)

0

+ L -2 (s)[log( q s)]

r-2

(1 . 51)

Le terme principal dans Ll')(s) sera de l'ordre s-'-1+1 , tandis que Ie terme principal dans Lo') (s) sera

(-2)k Vo

r(k + r - 1)F(k) F(k - r + 1)F(2k - 1)r(r - 1) (q

o s)

-r+1

(1.52)

APPENDIX

231

On peut demontrer notre proposition en integrant terme ii terme (1.45). II est utile de noter que 1'expression No coincide avec -x,+1 et que ]'on peut aussi appliquer A cette expression la formule (1.47). Il faut remarquer encore que le terme principal dans ok+1) s'annule, car la fonction r(k-r+1) qui se trouve dans son denominateur devient infinie. 5. La premiere identite fondamentale. Maintenant it West pas difficile de deduire notre premiere identite fondamentale. Dans ce but considerons de

nouveau notre conoide caracteristique avec le sommet au point M°. Ce conoide s'entrecroisera avec le plan t = 0 le long de la multiplicite qui est un contour de la region D°:

0
(1.53)

dans notre espace R2k+1 Enlevons Ie point x° , x2 , ... , X2k+1 de la region Do au moyen d'un petit ellipsoide 2k+1 2k+1

H° (xi - x,°)(xj - x°)e2

H° i=1

(1.54)

j=1

et designons Ie domaine restant par D.. Dans Do les fonctions a out les derivees continues et satisfont au systeme (1.32). Or, dans Do nous aurons, en comparant (1.29) et (1.31), 1' identite k

8k-fLu

E aj 8t k-j j=1

2k+1

8U

- -,No =-E 8xr .

(1.55)

.=1

Pour deduire maintenant notre premiere formule fondamentale it faut integrer l'identite (1.55) dans la region Do et ensuite transformer le premier membre en une integrale prise sur le contour de Do . Nous obtiendrons

ffD

k

E of

o j:1

8k-'Lu 8k 8tk-j

dR2k+1 -

J, A. uoNo dR2k+1

k

__ - I E U, cos nx, dS2k+1 -

f

k

°=

_° :l

U, cos nx, dS2k+1

1

(1.56)

r=1

oil les cos nx, sont les cosinus directeurs de la normale interieure dans le sens euclidien et par dS2k+l est designe l'element de la surface dans R2k+1 aussi dans le sens euclidien.

Il reste maintenant de passer a la limite pour a 0. Dans ce but faisons une transformation des variables independantes x1, x2 , ... , X2k+I introduisant les coordonees y1 , y2, ... , y2k+1 de telle facon que ]a forme positive definie H° soit reduite A la somme des carres 2k+I

H°

_ Fy j=1

.

(1.57)

APPENDIX

232

Les coefficients aij satisfont evidemment k l'equation

F= ! a!k

2k+1 2kk++1

i=1

10 k,

°

Hijalkaj!

j=1

{

0' 1=k. 1

(1.58)

Introduisons encore les coefficients de la transformation inverse 2k+1

yj =

Yji(xi - x,°

(1.59)

i=1

qui sont lies avec les aij par la formule 2kk++1

L aijfljs = bis

(1.60)

J.1

La transformation inverse transposee 2k+1

Pi = E pjisi

(1.61)

!=1

reduit la forme A°(P,) a la Somme des (2k + 1) carres. Pour demontrer cela it est utile de servir des definitions de la theorie des matrices. Soit a la matrice Ilaij II et a la matrice transposee, c'est-a-dire

aij=aji Alors, si d est la matrice unite, la condition (1.58) peut titre ecrite

UH°a = 6, d'ou, en prenant l'inverse,

d'oi enfin 2k+I 2k+1 i=1

E Ai fislfikj = dsk ,

(1.62)

j=1

ce qui demontre noire proposition. Il faut remarquer que Ie determinant de la substitution lineaire aij s'exprime ainsi

laijl = NO. Cela suit de la formule (1.58). 11 est utile pour ce qui d'evaluer la valeur de q°s , c'est-e-dire la valeur de Q dans les coordonnees yi . Dans ce but it suffit de remarquer que sur noire conoide caracteristique 2

2k+I2k+1

AIJP,PJ, i=1

j=1

APPENDIX

233

or, it se transforme en Q2

=

2k+I

2 S".

i=1

D'autre part, les variables S1 soot liees avec y, par des formules 00

Y;=Sl+F, Y.(SJ); n-2

it s'ensuit que 2

2k+1

Q=>

00

y;2

(1.63)

+ E R' (Y;) n=3

r=1

En designant 2k+ I

p =

Y2 t-I

nous voyons que la valeur de (q°s) est du meme ordre de grandeur que p. De notre evaluation de No et des a, it suit que les integrales dans le second membre de (1.56) sont uniformement convergentes dans le domain D° tout entier, et qu'il existe une limite determine de ce membre si a tend vers zero.

Cela est evident par exemple, si nous introduisons les "coordonnees polaires" par les formules yI = Pcos291,

y2= psin t9lcosi92, ........................................

Y2k = p sin 191 Sin a2

(1.64)

Sin 62k_ I COS 9 ,

Y2k+I = p sin $1 sin 192 ... sin 192k_ I sin 91 .

Nous aurons dR2k+I

- D(yl

Y2 , ... , Y2k+1) D(p, $1 , 292, ... , 192k-I , V)

xdpdO1dt92 dq1 = VAOP'k

sin2k-1

29 1 sin

2k-2

192

sin 292k-I

x dp d191 dt92 . . d192k-I do .

(1.65)

En tenant compte de ce que la valeur absolue de No (a) et de tous les a, Ap-2k ne surpasse pas logp nous voyons qu'ils sons integrables sur R2k+1 ce qu'il fallait demontrer. Remplacons maintenant les variables x1, ... , X2k+1 dans le premier membre de (1.56) par les y1 , y2 , ,Y2k+I

APPENDIX

234

Remarquons que 1'expression

fk

U, cos nX,d.S2k+1

r-1

se transforme en k

1: V, cos ny$ddS2k+I S=1

ou les V, sont lives avec les U, par les formules 2k+I

F, flj,U,.

(1.66)

r=1

dx dx ... dx

Pour la demonstration it faut remplacer cos nx,d,,S2k+I par d; et faire un changement des variables independantes. Dans les coordonnees y; 1'ellipsoide (1.54) deviendra une sphere, les cos ny, seront egaux i3 v et dYS2k+I sera egal A p2k

sin2k-119 1

sin2k-2192

sint92k_I d$1 d82... dd2k-I dq,.

On peut demontrer que le terme du degre e- 2k clans chaque expression U. provient de l'integration par parties de akD( )uo - uoDJ)-°k Plus precisement ce terme provient de 2k+1

E t[0f1s 8xk j-1

I

Si nous tenons compte seulement du terme

of

.

Bans ak on peut

remplacer cette expression par u° 2k+1 8Q-(2k-1) o P,

aQ

V U° J=1

A'j 4

ou par 2k+1

Qo (2k -

1)uo(gs)-2k+1

E Ao Pj 1=I

et finalement U,

(2k -

1)u°s-zk+1 (x,

- xo) + o(s-2k+1)

(1.67)

.

A 1'aide d'un calcul simple nous pouvons trouver que 2k+I

E U, cos nxdXS2k+1

lim

e0 40=49

2

S=1

(2k+1)/2

=2(2k-1)lt

t°).

(1.68)

APPENDIX

Designant la constante 2(2k - 0'

r(u. rux

235

par K nous obtenons de la formule

)

(1.56) l'identite fondamentale 2k+1

u(M°, t°)

K L(M ;M°,t°)=0

U( M; M°, to)cosnx,d,,S2k+I

t-l

k-'_

u(M, t) NN(M; M°, t )dR2k+1

+ KI ff

k Ea (M; Mo, to)8

K fJO
j=I

1

Lu(M1t) atk-J

dR2k+1

(1.69)

oiu M est le point $ coordonnees xl , x 2 , ... , x2k+1 et M° le point h coordonnees x°, x2 , ... , x2k+I . Cette identite integrale est la base de toute notre methode. 6. La methode des approximations successives et la seconde formule fondamentale. Transformons la formule (1.69). Portons dans le second membre

de (1.69) la valeur de u tiree du premier membre. Nous obtenons 2k+l

I

u(M°, IO)

0 E U,(M ; M, 0t) cos nxsd,,S2k+I

K T(M;M°,(°)=0 $-I

-K +

k

a,(M;

KIfO
Mo, to)e k-j Lk(M, t)

ff0 M°
dR2k+1

at ,

to)

2k+I

LM;M,O=O

U, (M('); M, t)cosnxs')dS;, S=I rrk

K

fJ0
(M(I)' x

M, t)

ak-iLu(M(,)

1(W))

dR2k+I (')

8t()"

+1 (J

No(M(u;M,t)

K f 0
x u(M('), t('))dR?I k

dR2k+1 .

(1.70)

La formule (1.70) est la formule fondamentale it6r6e. En r6petant ce procede, c'est-a-dire en placant dans la dernibre integrals de (1.70) la valeur

de u(M('), t(')) tiree de la formule (1.69) nous obtenons la formule deux fois iteree etc.

APPENDIX

236

Demontrons maintenant que les seconds membres des formules iterees convergent vers une limite si le nombre d'iterations croft infiniment. Nous obtenons ainsi notre seconde identite fondamentale. Dans ce but it suffit de demontrer qu'un seul terme, contenant la fonction u elle-meme dans le second membre dans la formule n fois iteree:

f fO
U"(M0, 10) = In ;71 x

ff

N0"(M(I ); M, t)

O
x 0
x

M(2), M (') .

t(1) )..

M("-I) t("-I)) ffo:s,(M(-), M(. u

i

'

ne t"

x u(M("), t(")) dR(") 2k+1 x dR2k+i) ... dR2k+l dR2k+I ,

(1.71)

tend vers zero. Remarquons en premier lieu que les U"(M°, t°) sont lies par la formule de recurrence suivante:

(I"(M0, 0) _

' ff

No(M; M°, t0)U"-1(M, t)dR2k+t (1.72)

La fonction UO est evidemment borne:

IU01<M,. Si nous introduisons les coordonndes polaires

X, = pcosfl1 , X2 = psint91 cos12, (1.73)

X2k = p sin 0, sin 02

sin 02k- I cos q ,

X2k+I = p sin t9l sin 62 .

sin 02k_ I sin 9,

alors l'element dR2k+I peut titre ecrit dR2k+I =

p2k

sin2k-2

sin2k-1

dI

t92

sin t92k_ l d p d t91

d t92k_ 1 d o .

De la formule (1.47) it suit que No (M ; M°, 10) ne surpasse pas

(1.74) MQ-2k-u

ou M est une constante et a est une constante positive arbitraire. D'autre part de la formule (1.18) it suit que Q:5 Ap. La formule (1.72)

APPENDIX

237

nousdonne IU"(M°,10)1 IC

I O

R

rx

...J

J0

0

x

p(61,fl=,...,1

2x

MApGIU-ll dp}

J xd

ip{sin2k-1

d1 d61

(1.75)

} dt92k-1 .

Remarquons maintenant que pour d1, 62, ... , go fixes la fonction p1 = t°-r est une fonction de p satisfaisant a l'inegalite p1 5 L11p1 et

dp1

5L2.

dp En introduisant au lieu de p la variable p1 nous aurons Kfx

f ...fi x

x

_lU"(M

2x

{j'°MAL1L2Iu_1IP°aP1}...

dt 91.

(1.76)

Suppos ons maintenant que U"-1 satisfait 3 I'inegalite

lU"-'(M 't )1< r((n-1)(1-a)+1)' et demontrons que U,, satisfait 6 I'inegalite

lU"(M ' t

r(n(1 - a) + 1)

)1

Cette condition pour Uo etant remplie nous allons voir que les fonctions U^ la remplissent aussi et, par consequent, tendent uniformement vers zero. Dans ce but it suffit de placer dans la formule (1.75) l'evaluation supposee vraie pour U^_1 . Nous aurons U"(M0, 10) <

F"-IC r((n-1)(1-a)+1) to(l -

p(l))(n-I)(1-o)pi

a dp1

O.u-e1 "-1C

- I)(1 - a)+ 1) en-Ic1W1-.)r((n

B((n - 1)(1 - a)+ 1, (1 - a))

- 1)(1 - a) + I)r(1 - a)

- r((n - 1)(1 -a)+ 1)r((n- 1)(1 -a)+2-a)' et en supposant que Cr(1 - a) = 8' nous aurons l'in6galit6 cherchee. Passant maintenant 6 la limite dans le identites iteratives nous arrivons a notre seconde formule fondamentale qui s'exprime comme une serie de la forme

00

u(M°, 1°)

=E n=0

(1.77)

APPENDIX

238

oil

2k+I

1

J°= --K r(M;MO.1O)=0 ( E US(M;M Moto) cosnx$dS2k+I 0

$=I

J.

K

k

Ea ffO
;

at

1°)NO(M; M°, 1°)J.JJO_I(M, t)dR +I.

f
(M Mo, 1O ) 8 k- Lu(M, t)

: M°,

(1.78) dR2k+I

(1.79)

=t

7. Le premier algorithme de solution. Ce que nous avons expose ci-dessus nous permet de construire un algorithme pour la solution du probleme de Cauchy pour notre equation (1.1) en supposant qu'une telle solution existe et admet des derivees bornees jusqu'h l'ordre (k + 1) par rapport aux variables independantes. En effet, dans la serie (1.77) tous les termes sont maintenant connus, car

les UJ dependent settlement des valeurs des derivees de u sur la surface t = 0. Mais ces derivees, comme it est bien connu, peuvent titre calculees des donnees de Cauchy (1.2). La valeur de Lu et de ses derivees peut aussi titre obtenue de l'equation meme. Remarquons encore que la convergence de la serie (1.77) peut titre constatEe immediatement, par des evaluations tout i fait analogues it celles que nous avons faites plus haul. Nous aurons 1J"1 ` r(n(1 - a) +

Maintenant it nous reste de demontrer 1'existence de la solution. Cette question sera l'objet des chapitres suivants. Chapitre II. L'espace fonctionnel (' )' 1. L'espace fonctionel fondamental. Pour ce qui suit la consideration d'un espace fonctionnel que nous designons par c aura une grande importance. Les elements de cet espace seront des fonctions 9 de 2k + 2 variables independantes: x, , x2 , ... , X2k+t , t qui sont continues avec quelques leurs derivees. Ces fonctions doivent titre telles qu'e chaque fonction to corresponde un certain domaine borne V it 1'exterieur duquel la fonction lD s'annule. L'ensemble des fonctions de notre espace qui ont Ies derivees continues jusqu'a l'ordre s constitue un sous-espace 41, . Determinons la conception de la convergence pour les fonctions de notre espace. Nous dirons que la suite rp,, de fonctions du degre s converge dans (') These numbers refer to the "Comments on the Appendix" by V. P. Palamodov (on pages 253-267).

APPENDIX

239

0, vers Is fonction p si 1) ii existe un domaine borne v qui contient tous les V41 dans son interieur, 2) les fonctions gyp,, avec toutes leurs derivees jusqu'A I'ordre s convergent uniformement vers rp et ces derivees correspondantes. Nous designons cette convergence par

& $ q?.

(2.1)

Il est evident que Ia formule (2.1) entraine (pR

to pour chaque r < s.

Determinons Is conception d'une fonctionnelle lineaire dans 0. Soit 47) le nombre qu'on obtient par l'application de Is fonctionnelle p h Is fonction to. La fonctionnelle p est dite lineaire de Is classe s si elle est determine pour chaque fonction de (b, et satisfait aux conditions (p

(p ac'1 + bq'2) = a(p 91) + b(p c'2),

(2.2)

et

(p 9)

(p

Si

'P"

4 p.

(2.3)

L'ensemble des fonctionnelles de Is classe s sera designe par Z, . Evidemment on peut definir Is somme de deux fonctionnelles et le produit d'une fonctionnelle par une constante. Introduisons Is conception de Is limite d'une suite de fonctionnelles (convergence faible). Nous disons que Is suite p converge vers Ia fonctionnelle

p dans Z, si (p. 9) - (p

gyp) pour chaque tC de 4>, . Cette convergence

sera designee par

p f' p. Considerons maintenant les operations lineaires dans nos espaces. Soit Lip Is fonction qu'on obtient par l'application de l'operation L A Is fonction {o . Une operation Lp dans 4) est nommbe lineaire si elle est 1) additive et 2) continue: L(aip, + bcp2) = aLgo1 + bLSo2,

Lc 4 Lrp

si

c'

V,

(2.4) (2.5)

c'est-a-dire si cette operation fait correspondre a 1'ensemble 40, quelconque un sous-ensemble de 4>, avec Is conservation de convergence. La conti-

nuite d'une operation est done determine par Is fonction s, (s2) qui fait correspondre au nombre s2 un autre nombre s, . Cette fonction que nous nommons Is fonction de continuite doit exister au moins pour une valeur de S2 .

APPENDIX

240

D'une fawn analogue on peut construire les operations lineaires dans 1'espace des fonctionnelles. L'operation lineaire est une operation qui fait correspondre h chaque fonc-

tionnelle de ZS une fonctionnelle de la classe ZS et qui est additive et continue:

L(ap, +bp2) = aLp, +bLp2, Lp,, s'+ Lp si

p,, - p.

(2.6) (2.7)

Le caractere de la continuite de ces operations est defini aussi par la correspondance entre les deux nombres s, et s2 . 2. Quelques exemples importants des fonctionnelles lineaires. Theoreme d'approximation des fonctionnelles. Parmi les fonctionnelles lineaires se trouvent quelques-unes qui ont la forme bien simple. Elles peuvent titre presentees sous la forme (P (P) =1

J!

p(M) c(M) d R2k+2 ,

(2.8)

ob p(M) est une fonction du point M, sommable avec ses derivees jusqu'h I'ordre s dans chaque partie bornee de 1'espace. Nous nommons ces fonctionnelles des fonctionnelles du degre s. Les proprietes de ces fonctionnelles ont une grande importance pour ce qui suit, car on peut toujours construire une suite des ces fonctionnelles de n'importe quel degre qui converge vers chaque fonctionnelle p dans Z,. Construisons, en effet, cette suite. Soit &)(M') (M') une fonction qui est determine par les formules w,n

(M')=0

si

r(Mi M2) =

2k+ I

r=i

(X

it _X+2))2 + (tug

ke 1

i,r

'

- t(2))2 > (2.9)

si

r(M',M2)<17, la constante K est determine comme K

=

III

w,, d Rik+z

(2.10)

Etant donnee une fonctionnelle p construisons la fonction p,I(M) comme (2.11)

APPENDIX

241

Demontrons que les fonctionnelles p,r pour n infiniment petit convergent vers la fonctionnelle p dans Z,. En effect, construisons

(P= JfJ91(M)P(M)dR2k+2.

(2.12)

Comme it estfbiien connu, cette integrale peut titre remplacee par la somme

f J fo(M)P,;(M)dR2k+2 =

+e (0', )AV + e i=1

oit a est une quantite infiniment petite pour les AV assez petits. Nous aurons de cette fagon

(

fff

1

AV I +e.

=P I

Passant iA la limite et remarquant que pour Ail tendant vers zero la somme copl')AV.

i-l

tend uniformement vers la foncttiioonr limite 4'(

P,I(M) = JJJ avec ses derivees jusqu'fi l'ordre s , nous obtenons

(PI*9)=(P'91)-

(2.13)

Passant i la limite encore une fois et en tenant compte que c,, tend vers p uniformement avec ses derivees jusqu'A 1'ordre s , nous voyons que

c'est-n-dire

P,1 S-'

P.

Les fonctionnelles p. sont evidemment du degrb infini. Donc, notre theoreme est demontre. 3. Quelques exemples importants des operations Iin6aires. Operations adjointes. Donnons quelques exemples des operations lineaires dans l'espace des fonctions. La plus simple entre elles est ]'operation de multiplication par une fonction donnee co(M) Lcp = tog.

(2.14)

APPENDIX

242

Il est aise de voir, quelle est la fonction de continuite. Si la fonction Co a des derivees continues jusqu'A I'ordre s2 alors Sl = min(s2 , sZ) .

Considerons encore l'operation de derivation. Nous avons

Llo = ap ,

(2.15)

x

la fonction de continuite dans ce cas sera

SI=s2-1. Par addition et multiplication de ces operations on obtient des operations differentielles ordinaires qui presentent pour nous un interet tout A fait special. On peut construire des exemples d'operations lineaires dans 1'espace des fonctionnelles, en se servant de la conception de l'operation adjointe. Considerons Ie produit (2.16)

Il est facile de voir que ce produit depend de 9 lineairement, c'est-a-dire

(p L(a'p1 + b(P2)) = a(p Lips) + b(p L92), et

(p . Lc)

(p . Lq,)

si

p E Z., et i° Donc, ce produit peut etre ecrit comme

9P

(2.17)

ou X est une fonctionnelle.

La fonctionnelle X depend de p aussi lineairement. L'additivite est evidente,car a(p, Lln) + b(p2 - Lq) = (ap, + bp2 Lip) .

(2.18)

La continuitC suit du fait que

si

IP E (b

$2

et p,,

s

(2.19)

donc la fonction de continuite est l'inverse de la fonction s1 = p(s2) (plus precisement A chaque valeur de sI correspond la plus grande valeur possible de s2). Donc,

X=L*p. L'operation L* p est nommee l'operation adjoint pour l'operation Lip.

APPENDIX

243

Pour les fonctionnelles de la forme speciale que nous avons considerees dans le no. 2 on pent naturellement construire des operations de multiplication par une fonction quelconque co et de differentiation. Si

(P . P) =

J fJ P9' dR2k+2

(2.20)

alors nous posons (.co) axP=fff

(2.21)

et

(wP 9) = J ff wpb dR2k+2 .

(2.22)

On peut verifier la formule

axP c') =

(pxP)

(2.23)

qui suit de ('integration par parties de (2.20). D'autre part, (2.24) (wp 9)) _ (P (01p). De la remarque sur les operations adjointes it suit que les operations 57

et w peuvent titre etendues aux fonctionnelles de type plus general. Il faut evidemment considerer le second membre de (2.23) et (2.24) comme la definition du premier membre. s'applique a chaque fonctionnelle et sa fonction de contiL'operation

nuite sera s2 = si + 1 . L'operation de multiplication par to, quand co est une fonction qui a des derivees continues jusqu'a 1'ordre s', sera determines pour les fonctionnelles jusqu'a la classe s', sa fonction de continuite sera s2=sI S.

L'extension faire est unique, car l'ensemble sur lequel la definition etait valable est partout dense. En se servant des remarques deja faites, on pent aussi determiner routes les operations differentielles daps l'espace des fonctionnelles. Ce sera d'un interet tout special de trouver la solution de 1'equation

Lu = p

(2.25)

pour certaines conditions initiales, en supposant que p et It ne sont pas des fonctions, mais des fonctionnelles. Le but principal de tout ce qui suit est d'Etablir 1'existence et l'unicite d'une telle solution. 4. Problems de Cauchy pour 1'espace des fonctionnelles. Revenons main-

tenant au problems de Cauchy qui etait pose daps le chapitre 1. Soil u la

APPENDIX

244

fonction inconnue qui satisfait A l'equation 2k+I2k+1 2k+I a2u aU

a2u Lu= E EA;;axaX.+EB'8x +Cu-ate =F

(2.26)

[1'equation (1.1)) et aux conditions initiales ult.0 = u(0)(XI , X2, ... , X2k+1),

aul Ft

=u

(I)

(2.27)

(X1 , X2, ... , X.,k+1).

r=0

Considerons la fonction discontinue u qui coincide avec la fonction u pour t > 0 et s'annule pour t < 0. Construisons la fonctionnelle correspondante urp dR22. (u rp) = Jffio dR2k+2 (2.28)

= 1J I>o

Tachons maintenant de trouver I'operation Lu pour cette fonctionnelle. Par definition

(Lu 9;) = (u L*tp) = f/f u Llo dR2k+2 >0

En se servant de l'integration par arties, nous obtenons

(Lu p) =

IlL0 Luq dR22 + li=o {udR2I . r=o

aurons

En tenant comppttede (2.26) et

(Lu - r,) = f/f

FSo dR2k+2

{u(°)

jt

-

dR2k+1

(2.29)

+ t=0 L'operation dans le second membre de (2.29) etant connue, nous voyons que la valeur de Lu est egale a une fonctionnelle donnee p. Nous avons donc Lu = p (2.30) >o

oil (P - 92) °

fJfFc,dR2k+2 + >o

ff (u(0! -dR2k+1

.

(2.31)

0

La propriete essentielle de notre fonctionnelle consiste en ce fait que le produit (u rp) ne depend pas des valeurs de la fonction 9 pour t < 0. Si nous convenons de dire qu'une fonctionnelle p s'annule dans un domaine D si (p r) ne depend pas des valeurs de to dans D, nous pouvons formuler le "probltme de Cauchy generalise": Trouver la solution de 1 equation

Lu = p,

Plt
(2.32)

aux conditions initiales ulr
(2.33)

APPENDIX

245

Remarquons que si une telle solution existe et si elle s'exprime par la formule (2.28) avec la fonction u admettant les derivees sommables du deuxieme ordre, alors la fonction u satisfait a l'equation (2.26) et aux conditions initiales (2.27). En effet, nous aurons alors pour n'importe quelle rp : dR2k+2 + J Jr=o t u

"

8t } d R2k+I

- JJf 0 Fq dR2k+2 +ff=o

U(O) 8

_u(1) )

dR2k+I

Mais cela peut avoir lieu seulement, si (2.26) et (2.27) sont satisfaites ce qu'il fallait demontrer.

Chapitre III. Probleme de Cauchy generalise 1. Examen de l'operation inverse fondamentale. La formule (1.77) nous donne une representation de la fonction It arbitraire, admettant les k + 1 derivees continues, etant donnee la valeur de Lu pour t > 0 et les valeurs initiales de la fonction u et de la derivee e pour t = 0. Nous allons etablir quelques proprietes concernant ('operation dans le second membre de cette formule. Nous demontrons la proposition suivante. THEOREME. Si nous mettons daps le second membre de 1'equation (1.77) au

lieu de Lu unefonction arbitraire que admet des derivees partielles continues

jusqu h l ordre s + k - 1, au lieu de u(0) une fonction arbitraire admettant les derivees continues jusqu d l ordre s + k + I et au lieu de u(1) unefonction arbitraire admettant les s + k derivees continues, alors la somme

E J. M-0

devient unefonction, ayant les derivees d'ordre s continues. DEMONSTRATION. Pour la demonstration nous derivons la serie (1.77) terme 3 terme en tenant compte des formules (1.78). Pour simplifier ce procede it est bien commode de reduire les integrates (1.78) aux integrates etendues A des domains fixes. Dans ce but on peut, par exemple, introduire de nouvelles variables t 11

to,

o,t91,$2,...,82k_I'9r

(3.2)

oil 191, d2 9 ... , 82k-I , 9 sont determines par les formules analogues aux

APPENDIX

246

formules (1.64) S1 = pcost91 ,

S, = psinfll cos d2, sin 0, _ I cos 0, ,

S, = p sin 191 sin 02

(3.3)

........................................

S2k = psin191

sin

t92k_ I sin rp .

Les variables S, sont des variables normales de Lipschitz pour les coordonnees y introduites par les formules (1.62). On voit de ces formules que les variables 191 , 192 , ... , 192k_I

rp

restent constantes sur chaque bicar-

acteristique.

Le determinant fonctionnel qui entre dans les integrales (1.78) apres ce changement des variables independantes, sera une fonction analytique. Les variables x1 , x2, ... , x2k+1 , t seront remplacees par des fonctions analytiques des variables 0

0

xl ' x2'

0

, x2k,I , t

0

, a, 192, ..

, 192k_I 1 91 tI

J0 sera une integrale qui depend des parametres x°, ... , t0 seulement par l'intermediaire des fonctions o, et vall" ul0j , et u11I .

La differentiation sous le signe d'integrale est evidemment possible. Puisque ak_, se presente comme une serie, contenant des termes de s en puissances negatives et logs qui se transforment apres notre changement de variables en termes aux puissances negatives de (I - t1) et log(1 - t1) qui ne dependent point des coordonnees x°, x2 , ... , x2k+1 t° la differentiation ne change pas l'ordre de ('infinite pour les fonctions ak_, . Alors la possibilite de derivation de la serie (1.77) terme A terme peut titre demontree par l'induction complete. La convergence uniforme des series des derivees resulte de pareilles evaluations. Notre theoreme est demontre. Tous nos resultats auront encore lieu, si nous considerons au lieu de t = 0 une nouvelle surface, portant les donnees de Cauchy, choisies de telle facon qu'elle ait l'orientation d'espace, c'est-A-dire qu'elle decoupe de chaque conolde caracteristique une region finie, contenant le sommet du conoide dans son interieur. Considerons le probleme de determination de la fonction u qui s'annule avec ses derivees sur une telle surface. Le second membre de l'identite (1.77) depend alors seulement de la fonction Lu. Ce membre nest autre chose qu'une operation additive sur Lu. Designons cette operation par G. Notre equation (1.77) sera ecrite comme

u = GLu.

(3.4)

APPENDIX

247

L'operation G fait correspondre A chaque fonction yi de (Ds+k+I de noire espace fondamental une autre fonction to = Gw,

(3.5)

qui admet des derivees continues jusqu'A l'ordre s et qui est definie et differente de zero dans le domaine infini que nous precisons plus loin. 2. Espace hyperbolique. Pour simplifier l'expose de noire methode it sera bien utile d'introduire un nouvel espace fonctionnel que nous nommons 1'espace hyperbolique et que nous designons par IF. Cet espace sera construit de la manitre suivante. Nommons domaine hyperbolique direct un domaine, dont la partie commune avec l'intsrieur de n'importe quel conoide caracteristique direct (oil la valeur de t au sommet est maximum) est finie.

Le domaine hyperbolique inverse sera le domaine pour lequel la partie commune avec chaque conoide caracteristique inverse est finie. L'espace hyperbolique direct est constitue des fonctions tiv qui sont differentes de zero chacune Bans son domaine hyperbolique direct correspondant

V. et qui admettent les s derivees continues (s = 0, 1, ...). L'ensemble des fonctions de telle sorte admettant les derivees continues jusqu'b l'ordre s constitue un sous-espace IF,. La conception de la convergence dans `F3 est analogue b celle qu'on a consideree dans le chapitre II. Nous disons que la suite yr converge vers la fonction limite V dans IF, si les derivees de yr jusqu'b I'ordre s convergent uniformement vers les derivees correspondantes de yr dans chaque partie bornee de 1'espace et s'il existe un domaine hyperbolique direct PW contenant tous les Vr,^ dans son interieur. II est evident que noire espace fondamental considers dans le chapitre 11 entre dans 1'espace hyperbolique. L'espace des fonctionnelles dans 1'espace hyperbolique sera designs par W. La signification du symbole W3 est evidente. II est clair que chaque fonctionnelle dans 1'espace hyperbolique sera aussi une fonctionnelle dans noire espace fondamental. II faut remarquer que la convergence faible d'une suite de fonctionnelles

p,, appartenant en mime temps aux W et Z n'est pas identique dans ces deux espaces. Une suite convergente dans Ws est evidemment convergente

dans Z., tandis que pour qu'une suite p convergent dans Zs soit convergence aussi dans WS it faut et it suffit qu'il exists un conoide caracteristique

b I'exterieur duquel bus les p s'annulent. En effet, dans I'hypothCse contraire on pourrait construire une fonction w qui appartient b `3' c'est-a-dire est differente de zero dans un domaine hyperbolique direct, pour lequel la suite (p,, yr) n'a aucune limite. Dans nos espaces hyperboliques on peut aussi introduire les operations

248

APPENDIX

lineaires qui seront tout A fait analogues aux operations lineaires dans les espaces fondamentaux. Nous pouvons maintenant caractcriser plus precisement la nature de l'oper-

ation G. L'operation G est une operation lineaire dans 1'espace hyperbolique direct qui transforme dune maniere continue chaque ensemble `I':+k_I en un sousensemble de 1 espace 'YS

.

La demonstration est evidente. En revenant pour le moment iA la formule

GLu = u,

(3.6)

nous voyons qu'elle est valable, si (2) u E ` $+k-I

(s> 2).

(3.7)

3. Operation adjointe is l'operation G. Puisque 1'espace des fonctionnelles dans 1'espace hyperbolique est plus etroit que 1'espace des fonctionnelles dans 1'espace fondamental, l'operation G' adjointe h l'operation G est determinee non pas Bans tout 1'espace de fonctionnelles fondamentales, mais seulement pour de telles fonctionnelles qui sont encore des fonctionnelles Bans 1'espace hyperbolique direct.

Nous allons voir qu'on peut etendre cette operation sur un ensemble de fonctionnelles un peu plus generales. Pour qu'une fonctionnelle p de 1'espace fondamental soil aussi une fonctionnelle dans 1'espace hyperbolique direct, it faut et it suffit que cette fonctionnelle soit differente de zero seulement a l'interieur d'un conoide caracteristique direct fixe. Dans le cas contraire on pourrait construire un domain hyperbolique di-

rect K dont la partie commune avec le domain oil p est different de zero est infinie.

En deduisant de cette partie une suite de domaines D,, qui s'eloignent infiniment et en construisant Bans chaque D la fonctoin 9,, de telle facon que (p 9,,) = 1 on obtient la fonction 00

41=E4',,,

(3.8)

(p - 91)

(3.9)

-I differente de zero seulement Bans K, pour laquelle le produit n'a pas de sens.

Nous demontrons maintenant que l'operation G' peut We etendue A toutes les fonctionnelles qui sont differentes de zero Bans un domaine hyperbolique inverse. (3)

Dans ce but it suffit d'etablir que si les deux fonctionnelles pI et p2 different seulement a 1'exterieur d'un conoide inverse fixe (c'est-a-dire coinci-

APPENDIX

249

dent pour chaque fonction que diflhre de zero A I'intCrieur de ce conoide), alors G' p, et G' p2 peuvent titre differents seulement a 1'exterieur de conoide eux aussi.

Cela pose la demonstration suit de la definition de G'. En effet, si la fonction yr est differente de zero seulement dans un conoide inverse fixe C alors (1.77) ('operation Gyi sera de meme differente de zero seulement dans ce conoide. Nous avons donc

(p, Gw) = (P2 Gw),

(3.10)

(G'p, w) = (G*p2' w)

(3.11)

c'est-a-dire,

ce qu'il fallait demontrer. Introduisons maintenant un nouvel espace abstrait Q dont les elements sont des fonctions w s'annulant en dehors d'un conoide caracteristique inverse correspondant 1 V. Soit 1'ensemble cz un sous-espace de Q. Nous disons que la suite w', converge vers to dans ffs , si dans chaque partie finie de 1'espace euclidien (XI , x2 , ... , z) to, converge vers to avec toutes ses derivees jusquh l'ordre s et si d'ailleurs it existe un conoide caracteristique Vw, contenant tous les V,^ h son interieur. D'une manibre tout i fait analogue A la prbcedente on peut construire des fonctionnelles dans its que nous nommons Y,. Les Y, sont des ensembles lineaires dans Z,. La conception de la convergence dans Ys (convergence faible) sera un peu plus etroite que dans Z,. La

relation entre Ys et W. est analogue $ la relation entre Zs et Y,. 11 est facile maintenant d'etablir que l'operation G* est une operation lineaire dans Ys qui fait correspondre h chaque Ys un sous-ensemble de Y:+k-I

On peut obtenir la formule fondamentale analogue A (3.6) qui done la propri6t6 principale de l'op6ration G' :

L'G'p=p.

(3.12)

Cette formule est 6videmment valable pour n'importe quelle fonctionnelle. 4. Le problbme inverse.

Notre problbme qui est l'analyse de 1'equation

Lu = p, (3.13) 11 est bien utile de considerer en nous a conduit aux operations G et G* . meme temps 1'6quation

L'v = X,

(3.14)

et de construire d'une manibre analogue les operations G" et G' en remplacant seulement partout le conoide direct par le conoide inverse et vice versa. L'opdration G" est dCtermin6e dans 1'espace hyperbolique inverse des

fonctions w' .

APPENDIX

250

Sa loi de continuite et telle qu'elle fait correspondre a ]'ensemble `' +k_ I un sous-ensemble de `P en conservant la continuite. D'autre part, elle transforme chaque sous-ensemble L:+k-I de `P:+k-I de fonctions w' , differentes de zero seulement a 1'exterieur d'un conoide inverse correspondant, en un autre sous-ensemble de its en conservant la continuite

dans le sens de 0. L'operation G' est determinee sur les fonctionnelles de 1'espace fondamental qui sont differentes de zero dans le domaine hyperbolique direct. Elie transforme chaque ensemble des fonctionnelles de Ys+k-I possedant cette propriete en un sous-ensemble de Ys qui la possede aussi et conserve

la continuite dans le sens de Y. Pour les operations G' et G" ont lieu les formules fondamentales suivantes:

G''L'w = w

(3.15)

LG'p = p.

(3.16)

et

5. Les inversions droites et gauches de ]'operation L dans l'espace fonctionnel. Dans les paragraphes precedents nous avons vu que pour ]'operation L dans 1'espace des fonctionnelles differentes de zero dans un domaine hyperbolique direct, it existe toujours une operation inverse droite G' satisfaisant a ]'equation (3.16) et que sur ]'ensemble de quelques fonctionnelles speciales it existe une operation gauche inverse G satisfaisant a (3.6). Nous demontrons le theorCme suivant.

THEORtmE. LopEration e effectuee sur lesfonctionnelles qui peuvent 2tre

representees dans la forme (2.8) avec la densite p appartenant a I espace hyperbolique direct et presentant unefonction d'ordre s + k -1, coincide avec /'operation G du numero precedent, c est-h-dire G'p donne la fonctionnelle representable dans la forme (2.8), dons la densite est Gp(M) . DEMONSTRATION. Considerons la quantite

GLG'p

(3.17)

pour la fonction p de 1Y2k (a) et en nous Traitant le resultat G'p comme une fonction de `Yk+I servant de la formule (3.6), nous voyons que la quantite (3.17) est egale a G'p. D'autre part, en considerant p comme une fonctionnelle et en nous servant de (3.16) nous voyons qu'elle est egale a Gp.

Donc, Gp = G'p

ce qu'il fallait demontrer. De meme on peut demontrer la coincidence des operations

G' et

(3.18)

APPENDIX

251

sur routes les fonctionnelles qui peuvent titre presentees sous la forme (2.8) avec la densite p qui appar;ient h 'Yk de l'espace hyperbolique inverse. Dbmontrons maintenant que l'operation G' est aussi une operation droite inverse de L pout routes nos fonctionnelles. La demonstration est baste sur la remarque que nous avons faire dans le chapitre H. Nous avons demontre 1$ que ]'ensemble des fonctionnelles representable dans la forme (2.8) avec la densite derivable n'importe quel nombre de fois, est partout dense. 11 West pas difficile d'etendre cette remarque aussi pour les espaces W, , Y, ,

Ws ,et Y, . Cela fait, considerons la quantite G'Lp. De la continuite de G' et L it suit que G'Lp = lim G'Lp,,

(3.19)

(3.20) oiu p,, sons des fonctionnelles representables au moyen des fonctions de `Y2k

D'autre part,

lim G'L p = limp,, = p,

(3.21)

G'Lp = p

(3.22)

d'oii finalement

ce qu'il fallait demontrer.

Nous pouvons maintenant supprimer le sign ' pour les operations G et G' et formuler le resultat final. Il existe une operation lineaire G qui est en meme temps l'opEration droite et ]'operation gauche inverse de ]'operation L determine dans l'espace Y et 1'espace W. De meme it existe ]'operation G' qui est en meme temps ]'operation droite et l'operation gauche inverse de V. 6. L'existence et I'unicitf du probleme generalise de Cauchy. Maintenant it ne presente aucune difliculte d'etablir l'existence et l'unicite de la solution du probleme de Cauchy pour 1'espace des fonctionnelles.

Ce probltme qui etait pose dans le no. 4 du chapitre II consiste en la recherche de la solution de ]'equation

Lu = p (3.23) quand la fonctionnelle donnee p et la fonctionnelle cherchee doivent s'annuler en dehors d'un domaine donne hyperbolique direct. L'existence d'une solution suit de l'existence de ]'operation droite inverse. En effet, par definition meme la fonctionnelle u = Gp (3.24) satisfait h ]'equation (3.23). D'autre part, en effectuant la multiplication gauche de ]'equation (3.23) par G nous voyons que ]'equation (3.24) suit immediatement de (3.23). L'unicite est ainsi demontree.

252

APPENDIX

7. L'existence de Is solution dens le sens classique. II serait interessant de repondre maintenant A la question de 1'existence d'une solution du probl8me de Cauchy classique. La reponse suit du theorbme precedent. Cette solution existe dans le cas seulement ou noire fonctionnelle

u = Gp est representable daps la forme (2.8) avec la densite qui admet des secondes derivees continues. Les resultats de ce chapitre nous donnent aussi une condition suffisante qui peut We utile pour certains cas speciaux. Au moyen du theoreme du no. I de ce chapitre nous voyons qu'une telle solution existe si la fonction F = Lu admet des derivees continues jusqu'h l'ordre k + 1, la fonction u(0) = ul,=o admet des derivees continues jusqu'h

l'ordre k + 3, et la fonction u, _ Sk, Ir=o admet des derivees continues jusqu'a l'ordre k + 2. (5)

Comments on the Appendix 1. This chapter contains the first detailed study of Sobolev's theory of generalized functions (there is a brief sketch of the theory in his preceding note (277]). Its basic ideas and constructions have entered into the contemporary theory practically without changes. Let us go through the most important of them.

(1) The definition of a generalized function as a functional on the space of smooth functions with compact support.

(2) The order of singularity of a generalized function-the class of the functional in the Sobolev terminology. (3) Regularization of generalized functions by means of convolution, and the possibility of approximating infinitely differentiable functions. (4) Linear differential operators in the space of generalized functions as adjoints of operators on the space of test functions.

(5) The flexible use of other spaces of test and generalized functions needed for the given problem. These spaces are distinguished or described by conditions on the supports of ordinary or generalized functions. Although the author does not introduce this concept explicitly, he really makes constant use of it. (6) Reduction of the Cauchy problem to the solution of an equation with a right-hand side but without initial conditions by turning the initial data into sources delta-shaped in time. Such a device was later widely used in diverse variants.

The emergence of the theory of generalized functions was prepared for by developments in mathematical analysis and theoretical physics. The wellknown ideas of Heaviside and Dirac, and especially the work of Kirchhoff

and Hadamard, stimulated its appearance. However, the work of the predecessors did not contain concepts and constructions similar to the rigorous constructions of Sobolev. The theory of generalized functions took on a modern form in the book [262] of Schwartz, where a Fourier analysis was constructed for distributions (generalized functions) of slow growth. Schwartz included in the theory im-

portant ideas of Hadamard and M. Riesz about the "finite parts" of diver253

254

COMMENTS ON THE APPENDIX

gent integrals, as well as a number of new ideas, in particular, locally convex topologies and partial regularity (see, for example, [176]).

The new possibilities that had arisen in the Sobolev-Schwartz theory as a result of the synthesis of diverse ideas led to a rapid development of its applications. We mention some of the main directions of these applications. Malgrange and Ehrenpreis used the theory of entire functions to further the development of the method of Fourier transformation of generalized functions and applied this method for proving the existence of fundamental solutions of partial differential equations. The methods of the theory of entire functions were employed by Gel'fand and Shilov ([84]-[86]) to solve another well-known problem-the determination of the broadest classes of uniqueness and well-posedness of the Cauchy problem for systems of equations of evolution type with constant coefficients. These authors extended the concept of a generalized function, including in consideration a whole scale of spaces of test functions, both smooth and analytic. Under the name of the theory of ultradistributions this direction was later developed by Roumieu and Beurling. The next step in the evolution of this direction is the theory of hyperfunctions constructed by Sato (see, for example, [266]). The division problem (Schwartz [262]) acquired its natural form in the framework of the theory of generalized functions-the possibility of dividing a generalized function by a polynomial or analytic function. This problem is also connected with the solution of differential and convolution equations; in its original form it was considered as far back as Bochner. The solution of this problem was obtained by Hormander, Lojasiewicz, and Malgrange. The deep theory arising here, which goes far beyond the boundaries of the original problem, is reflected in the book [ 179]. By combining the theory of generalized functions with the new analytic

and algebraic methods, a Fourier analysis was constructed in the space of solutions of an arbitrary system of differential equations with constant coeffi-

cients. The result of this analysis-exponential representation of solutionscontains a general method for investigating different problems in the theory of such systems of equations, in particular, local properties of solutions, uniqueness of the Cauchy problem, and solvability of the system with righthand sides. An exposition of this theory and its generalizations is contained in the books of Palamodov [233] and Ehrenpreis [63] (see also item 5). In the 1930's Sobolev developed the theory of generalized functions in a new direction. On the basis of the concept of a generalized derivative of an ordinary function he introduced and studied the scale of spaces WDI) now called Sobolev spaces. He applied this theory to the investigation of the Dirichlet problem for certain higher-order elliptic equations ([280], (281], [282], [286]). These ideas of Sobolev were also broadly developed and applied, and the basic facts of the theory of Sobolev spaces and their generaliz-

COMMENTS ON THE APPENDIX

255

ations-imbedding theorems and trace theorems-became some of the most important tools of modem mathematical analysis. For Sobolev, generalized functions were first and foremost a language important for applications. He later used this language in a quite new area-the theory of cubature formulas [295]. The language and methods of the theory of generalized functions have by now become an essential part of the theory of linear partial differential equations; they are broadly'used also in other areas of analysis and mathematical physics, and they have become more and more common in applied investigations. One can make judgements about the role and applications of the theory of generalized functions from the monographs [83], [263], [327], and [328], the survey [235], and the fundamental work [115]. 2. Actually, only the following result local with respect to t is proved here: if the function u is nonzero in the half-space t > t°(x) , then the

relation (3.6) holds for t < t1(x), where 1,(x) is a continuous function of the point x = (xi , ... , x2k+I) satisfying the condition that for every x° the null bicharacteristics emanating from the point (x°, t°(x00)) do not have focal points in the strip t°(x) < t < tI (x) . This implies that all the forward characteristic conoids with vertices in this strip do not have singular points in it. Essential use is made of this property of conoids in the arguments. Simple additional considerations enable us to pass to a global result under certain assumptions about the behavior of the bicharacteristics. Consider a continuous function 11-t(X) < tl(x), close to tI (x) and an infinitely differentiable function h(x , t) equal to I for

t < tI_t(x) and to 0 for t > tI(x) For every function 00

WoE`.=n''s the function hGW0 is defined for t < tI , is infinitely differentiable and equal

to 0 with all its derivatives in a neighborhood of the hypersurface t = ti . Denote by GI Wo its extension by zero in the half-space t < 1I . It follows from the theorem proved in this article that the function WI =W° - GILWo

is equal to 0 in the half-space t < ti_t(x) . Next, it is possible to find a continuous function t2(x) > 11(x) such that on the null bicharacteristics emanating from the points (x°, t,_t(x°)) there are no focal points if t < t2. By using similar arguments, it is possible to construct an operator G2 such that the function W2=WI-G2LWI belongs to `' and is equal to 0 for t < t2_t , and so on.

COMMENTS ON THE APPENDIX

256

Continuing this construction, we form a sequence of functions to < t1 < that tends to infinity on each compact set. This is possible if none of the bicharacteristics goes to infinity in finite time. Assuming that this condition is satisfied, we get a sequence of operators Gk acting in IF.. The desired operator G is defined on the space of functions equal to 0 for I < 10(x) by the formula

G=GI+G2(1 -LGI)+G3(l -LG2)(1 -LGI)+ +Gk+I(I

LGI)+

This series clearly converges in 'P.. and satisfies the relation (3.6). By constructing a suitable sequence of functions < t_2 < t_I < to and corresponding operators... , G_21 G-1 , Go it is possible to extend G to a continuous operator on the whole space 'l' satisfying (3.6). Thus, this theorem and all the subsequent ones based on it are true also in a global variant. A different, clearer method of constructing the operator G is given by the construction of a parametrix with the help of the canonical operator of Maslov (see [183]-[186]), and also the construction in (59], which uses Fourier integrals. These methods enable one in principle to investigate in detail the structure of the parametrix in a neighborhood of focal points. However, for such an investigation it is necessary to use the theory of generalized functions, whose foundations were laid in this paper of Sobolev (see [82], [234]).

3. It is actually proved below that the operator G' can be extended to the space of functionals defined on an arbitrary backward characteristic conoid K, i.e., to the quotient space of W by the subspace consisting of the generalized functions equal to zero on K. 4. This assumption is not necessary. The desired relation (3.18) can be established with the help of the following lemma. LEMMA. For every I > 0 the image of the operator L : w!+2 -

in'P1.

T,

is dense

PttooF. We choose an arbitrary function w e 'P, and make an analytic change of variables y = y (x , t), s = s(x, 1) such that 1) w is nonzero only on the half-space s > 1 ; 2) each hypersurface s = o is noncharacteristic for equation (1.1). The function w can be approximated in the sense of convergence in 'P, by a sequence of functions

aj(y,s),

j=1,2,...,

each belonging to 'P,.2, nonzero only in the half-space s >- 0, and analytic for each particular s with respect to the variables

Thus, it suffices to show that for every function a having the properties described there exists a function b E Y'7+2 such that Lb = a.

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We construct such a function with the help of the Duhamel principle. Consider the auxiliary Cauchy problem

Lb, = 0,

b0(y, a) = 0,

ab(as a) = a(y, a),

which depends on the parameter a > 0. Since a(y, a) is an analytic func tion of y, while the coefficients of the operator L are by assumption analytic functions of all their variables, this problem has by the Cauchy-Kovalevskaya theorem a solution bQ defined at least in a neighborhood of the hypersurface s = e r. Using this theorem repeatedly, we can extend this solution to an analytic function bQ on the whole space R2k+2 when none of the bicharacteristics goes to infinity in a finite time. By investigating the dependence of this solution on or it is not hard to establish that it has continuous derivatives

up to order I + 2 with respect to the variables y, s, a jointly. Setting b(y, s) = fosb,, (y,s)da, 0

we get the desired solution of the equation Lb = a. PROOF. Let us return to a proof of the theorem. We show that G is a right inverse of the operator L on the space T. For every 1 > 0 and every function w E Pf+2 let (p = Li. We have that

LG9 =LGLw=Lw=4p. Thus, the operator LG coincides with the identity operator on the image of By the lemma, this image is dense; hence the new operator L : Y'1+2 LG is the identity on the whole of 'Y in view of continuity, and this is what

was required. The operator G' is a left inverse of L (since G" is a right inverse of L' by what was proved). In the preceding notation

(G-G')(p =(G-G')Lw= w-w=0, hence

G-G'=O on a dense subspace of T,; whence G = G. 5.* The reader following Sobolev's arguments can appreciate the simplicity and conciseness of the style of the last chapters, their pragmatic character of presentation. It is to be emphasized that here are presented the foundations of the Sobolev-Schwartz theory that has become one of the fundamental events in analysis in our time. It is perhaps meaningful to give some thought to questions arising in this connection: what is the pre-history of this theory, and what is its significance and what is the reason for the role it plays?

An attempt will be made to answer these questions, without any pretense to completeness. A whole book has been written about the pre-history of the theory of generalized functions: Liitzen 1176). The author of this book 'Added in print.

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considers the pre-history and analyzes many publications that attempted to extend the framework of the traditional notion of a function and a solution of a differential equation, in particular, the well-known discussion among Euler, d'Alembert, and Lagrange in the eighteenth century about the concept of a solution of the equation of motion of a string. It should be mentioned that Euler interpreted the concept of a solution most broadly, while Lagrange had the embryo of the idea of using test functions in connection with the study of solutions. In the nineteenth and early twentieth centuries inducements toward the creation of a theory of generalized functions arose in various areas of analysis, in particular, in potential theory and other branches of mathematical physics. Somewhat by themselves are the papers of de Rham on currents on smooth manifolds-a generalization simultaneously of the concept of a singular chain and the concept of a differential form. The concept of a current found adequate expression only in terms of the theory of the generalized functions (distributions) of Sobolev and Schwartz. Why did the foundations of the theory of generalized functions not arise earlier or later, why was it in this paper of Sobolev where it appeared? Of course, here is evidence of the scientific boldness of the author and his tendencies toward general functiontheoretic constructions. These qualities were later reflected in the creation of the very important theory of the function spaces now called Sobolev spaces. Sobolev valued this theory, and returned later to it repeatedly. In addition to these subjective reasons for the origin of generalized functions at this moment and in this paper, there are in my opinion important objective reasons. Briefly, the fact of the matter is that in this paper the theory of generalized functions appears for the sake of solving one of the central problems in the theory of hyperbolic differential equations with several independent variables. We remark that many of the "pre-historical" papers that, objectively speaking, led up to the theory of generalized functions but did not give it impetus were connected with elliptic equations in one way or another. Relevant here are investigations in potential theory, as well as work at the beginning of the century associated with the fundamental theorem of integral calculus. Also relevant here is the theory of de Rham, which to a significant degree is the theory of the Laplace operator on a manifold. Although the modern theory of elliptic (pseudo)differential equations makes broad use of the concept of generalized functions, there were apparently not enough internal stimuli in this theory for these concepts to originate. This paper of Sobolev relates to one of the most important directions in analysis, going back to the work of Huygens in the seventeenth century and including the names of Riemann, Poisson, Kirchhoff, Volterra, and Hadamard.

It was the work in this direction that contained the impulses and ideas that led Sobolev to conceive the foundations of the theory. As far back as Euler it was proposed to go beyond the bounds of a narrow conception of the equation of a string and to regard as a solution any function of the form

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u(x, t) = f (x - t) + g(x + t), where f and g are arbitrary functions as understood at the time. However, this generalized understanding of a solution could not become a common idea, since it depended on the concrete form of the equation. For the closely related telegraph equation a clear concept of a generalized solution was given and investigated only at the beginning of the twentieth century by Wiener. He defined such a solution as an arbitrary function satisfying the equation in the weak sense, i.e., as a functional on compactly supported infinitely differentiable functions. Closely related ideas were expressed also by Gunther and Kochin. Hadamard's work at the beginning of our century [107] on the Cauchy problem for hyperbolic differential equations is in the strict sense of the word a precursor of the Sobolev theory of generalized functions. To a large degree Hadamard's work served as the origin of the idea of generalized functions, although, as should be emphasized, it did not contain any general definitions from the future theory. At the same time, important concrete examples of generalized functions appeared there under the name "improper integrals of a new form". In the simplest form, Hadamard wrote such an integral as e

1

a(- x) dx

(b

(1)

where p is an arbitrary integer, and 0 < p < 1 . He defined this integral by passing to the limit in the proper integral with upper limit b - C, adding the sum of p suitable terms with a power singularity as a 0. It later became clear that these integrals form an important family, elementary generalized functions of a distinct kind; however, they did not appear in Sobolev's paper, because this would have gone outside the framework of the problem solved there. The special role of hyperbolic differential equations can be explained by the fact that they yield a large diversity of singular solutions. In particular, the fundamental (elementary for Hadamard) solution describing the propagation of a wave from an instantaneous point source is such a solution. In contrast

to the case of an elliptic equation, for which the fundamental solution has a point singularity, the elementary function of a second-order hyperbolic equation has a discontinuity on a hypersurface serving as the trajectory of the wave front due to the source. The fundamental solution goes to infinity near this hypersurface (though it remains locally integrable) in the case of a wave solution on the plane. If the motion takes place in three-dimensional space, then the fundamental solution has a delta-shaped singularity just as a delta-function appeared in the work of Kirchhoff at the end of the last century. The "order of singularity" increases like n/2 as the dimension n of the physical space increases. If the equation has variable coefficients, then the wave front itself (a conoid of rays) can acquire singularities at which there is a (partial or complete) focussing of rays, and the singularity of the fundamental solution becomes more complicated. This paper of Sobolev

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does not touch upon the question of possible singularities of the wave front nor on the corresponding singularities of the fundamental solution, but the complexity of the analytic picture is implicitly present in any method of construction. In the final analysis, this complexity led Sobolev to think of finding a simple and effective axiomatic scheme that would enable him to accommodate this complicated picture. Taking into account that from the very beginning the axiomatics of generalized functions was adapted to express arbitrary singularities of a fundamental solution (as well as of any solution of a hyperbolic equation with generalized Cauchy data), we can now understand why this axiomatic scheme turned out to be the extension of classical analysis that was able to accommodate, in particular, the concept of a weak solution of an elliptic equation. However, this conclusion, drawn half a century after the pioneering work of Sobolev, was by no means obvious at the time, nor even twenty years later, and the success of the theory was startling. This success was ensured also by the enormous contribution of Schwartz [261], who introduced a number of important ideas into the theory, combined diverse approaches, and posed many problems that stimulated the subsequent development of the theory. The achievement of Schwartz is first and foremost

the combination of the theory with the Fourier transformation. The prehistory of this idea includes the work of Fourier himself (1822) in which he wrote a formula for the recovery of a function from its Fourier series as a convolution with the symbol 6, where

8(x) _

2+

cos ix

is what we now call the delta function. Inversion of the Fourier transformation leads to a similar formula. These formulas, which do not make sense from the point of view of classical analysis, were subjected to criticism by Darboux. However, the criticism of mathematicians did not stop Heaviside, who already used the modern form to write these and similar formulas for inverting the Bessel transformation, and even "proved" them. Bochner [33), who defined the Fourier transform of a function of one variable having no more than power growth to be a sum of formal derivatives of suitable continuous functions, i.e., in modern language, to be a generalized function of finite order, came close to the construction of the generalized Fourier transformation. It was not clear, however, how this definition could be extended to functions of several variables. The construction proposed by Schwartz for the Fourier transformation on the space of generalized functions of slow growth included the approach of Bochner and gave a precise meaning to the arguments of Fourier and the computations (at least some of them) of Heaviside. From the Sobolev-Schwartz theory it became clear that the delta-functions of Fourier, Kirchhoff, Heaviside, and Dirac and analogous symbols of other authors were one and the same if they were given the meaning of generalized functions. The theory of Schwartz simplified considerably the use of

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the Fourier transformation in the theory of differential equations, in the first place in the theory of equations with constant coefficients. The possibility of unbounded differentiation in the space S'(R") turns this space into a module over the ring A of polynomials in n variables, with the basis elements of the ring acting as partial derivatives with respect to the independent variables.

The Fourier transformation, which preserves the space S'(R"), turns this structure into a different module structure over the same ring, in which the 27ri#,, . With the help of basis elements act as multiplication by these structures Schwartz formulated the well-known division problem: can any generalized function u E S' be divided by a nonzero element a E A, i.e., can the equation av = u always be solved in S'? Thus, this equation can be understood both as an algebraic and as a differential equation. The positive solution to the problem (Lojasiewicz, Hormander, Malgrange) led to the development of a new area of analysis (see [179)) which later came to play an important role in the theory of singularities of smooth mappings (the Malgrange preparation theorem). Another important idea introduced in the theory of Schwartz is the broad use of duality methods for locally convex spaces. The essence of these methods is that certain important properties of a linear operator acting on generalized functions (for example, a differential operator) can be translated in terms of the adjoint operator acting in spaces of test functions. For example, if the base or dual space is metrizable and complete (a Frechet space), then the operator is epimorphic if and only if the adjoint operator is an isomorphism of the dual space onto its image (the Dieudonne-Schwartz theorem). The possibilities opened by duality theory methods caused (especially at first) enthusiasm and led to a number of general theorems on solvability and local properties of solutions of differential and convolution equations with several

independent variables. A more critical understanding of the achievements based on duality theory later emerged. In the final analysis every duality theorem is based on the Hahn-Banach theorem, which does not contain a constructive method for finding the desired linear functional (provided one is not concerned with a Hilbert space). In its full generality the Hahn-Banach theorem is based on transfinite induction, which is equivalent to the axiom of choice. The latter, as is well known, is independent of the other axioms of set theory and can be assumed or rejected, depending on the taste of the researcher. As a result there arises a contradiction between purpose and method: the original problem-a certain concrete problem relating, for example, to a differential equation-is solved by means completely extraneous to this problem. However, the contradiction does not affect the validity of the results obtained by these methods. The basic results, at least, can be re-proved without resorting to duality theory, but by somewhat more complicated arguments. The duality theory continues to be used even now, but there has undoubtedly been a shift in the direction of more concrete effective

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methods as well as formulations of problems in the last ten to fifteen years. Subsequent progress in the theory of generalized functions has occurred

in several major directions. One of them is the extension of the theory of the Fourier transformation to functions (ordinary or generalized) of arbitrary growth. This problem was considered in the 1950s by Ehrenpreis and Malgrange in connection with the division problem, and by Gel'fand and Shilov ([85], [86]) in connection with the study of the characteristic Cauchy prob-

lem. The Fourier transforms of such functions are defined as continuous linear functionals on suitable spaces of entire functions. Such functionals are not generalized functions of Sobolev, but more complicated objects. In particular, the concepts of support and localization are not defined for them, because there are no functions with compact support in the test space except for the function identically zero. The space of such functionals as well as the space of generalized functions are included in the naturally constructed family of Gel'fand-Shilov spaces (s.-fl)', in which the parameter a characterizes the growth of elements at infinity, and fi is the degree of singularity of these

functionals. Here there appears an effect, absent in the Sobolev-Schwartz theory, of dependence between the quantities a and fi : if a + ft < I , then the test space S1 turns out to be trivial, i.e., it reduces to the single function identically equal to zero. Another major direction-the theory of pseudodifferential operators-is an evolution of the idea that a differential operator with smooth coefficients can be defined in terms of the Fourier transformation. The next step in this direction is the theory of general Fourier operators (Maslov and Hormander), which is a synthesis of the theory of pseudodifferential operators, the usual Fourier transformation, and operators of solution of the Cauchy problem for hyperbolic equations (see, for example, [115]). This direction has enriched the theory of generalized functions, with development, in particular, of the concept of a wave front and of a generalized function associated with a conical Lagrangian manifold (Hdrmander). Let us look at one more idea, introduced by M. Riesz and Schwartz in the theory of generalized functions. In his major paper [254) Riesz employed the method of analytic continuation for giving a meaning to divergent integrals. In simplest form this method relates to the Hadamard integral 1107). Riesz regarded it as a convolution operator acting on a function a whose support belongs to the right half-line. The difficulty encountered by Hadamard in the case when the exponent p +,u becomes an integer was avoided by Riesz and by dividing the integral by the gamma function, having poles at the points µ = 0, -1 , -2 , ... , which were excluded from consideration by Hadamard. As a result he obtained a family of convolution operators X

I°a(x) =

f a(t)(x - t)°-' dt.

r(a) These integrals converge in the usual sense only for Re a > 0, but if the

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function a is infinitely differentiable, then they have an entire analytic continuation with respect to the parameter a, and thus admit continuation to

negative integer values of a. In particular, Io turns out to be the identity operator. These operators have the remarkable property I. * 10 = I°,a for all values of a and fl, which means that the correspondence a ,-. 1. is a group homomorphism. For a negative integer a the operator 1° is a differentiation, and for a positive integer it is a multiple integration operator (the Dirichlet formula). We remark that while remaining in the framework of classical analysis we can observe only a part of this picture, namely, we can define 1° only for Re a > 0 and separately for a = 0, -1, ... , without seeing that both cases are included (in a unique way) in a common family of operators that is an entire function of a ! This example is a simple and convincing testimony to the fact that the theory of generalized functions is the most natural (and economical) extension of classical analysis. The main result in Riesz's paper is the construction of an analogous group

homomorphism connected with a quadratic form q(x) of signature (1, n 1). He starts from the family of functionals O q°{o dx ,

fq >0

(2)

where 0 is a function equal to I on one convex component Q of the cone q > 0, and equal to 0 on the other. For Re a > 0 the integrand is continuous, and these functionals are defined on the space 9 of infinitely differentiable functions p whose supports belong to a set of the form K + Q, where K is compact (depending on ip). Riesz observed that this family has a meromorphic extension to the complex plane with poles located on the two

arithmetic progressions a = - 1, -2, ... and a = - '" , - i - 1, .... The first of them is connected with the nonsingular part of the boundary of Q ; in a neighborhood of a nonzero point x E 8Q the integral (2) reduces to the Hadamard integral (1), whose poles are -1 , --2, .... The second series of poles is connected with the singular point x = 0 of the set of roots of the polynomial q. To compensate for these poles Riesz divides this family by the product of two gamma functions and arrives at the family of functionals

_ Z°

2nq°_q

7r"/24°r(a)I-(a

+ I -5)

This family has an entire analytic continuation with respect to the parameter as a family of continuous functionals

where q(x) = x,2, - xI -

on the space 9Q . The convolution operators defined in . with the help of these functionals have the group property

Z.*ZB=Zo+p,

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with Zo = a giving the identity operator, and Z_k = 0 k8 , where a2 ax,,

a2

axI

a2 axn_

1

is the wave operator in n-dimensional space. The last property, in combination with the group law, implies the relation

kZk =,5. This leads to a conclusion that is a main goal of Riesz's paper. the generalized

function Zk is the fundamental solution for the kth power of the wave operator. This elegant construction completes the cycle of investigations, begun as far back as the nineteenth century, devoted to constructing fundamental solutions for wave equations with constant coefficients, and simultaneously explains the reason the Huygens principle in the wide sense is valid only in a space of an even number of variables. Namely, if n > 2 is even, then q+_ q and at the same time a pole the value a = 1 is a pole of the family

of the function I'(a + 1 -

U. The value of the family Z at this point is

proportional to the ratio of the residues of the numerator and the denominator, but the residue of the family q+-I at the point a = I is a functional whose support belongs to the boundary of Q, since this generalized function coincides in the interior of Q with the ordinary function q°-f . Thus, Z. is a generalized function with support belonging to (actually coinciding with) 8Q, which implies the satisfaction of the Huygens principle in the wide sense (the presence of a backward wave front) for the wave equation. Here again the theory of generalized functions served as a natural language, in which it became possible to expound the construction of Riesz. This construction was developed in the framework of a more general direction that can be defined as a combination of the theory with real algebraic geometry. The generalized theorems of Riesz on the possibility of meromorphic ex-

tension of the family f+ for an arbitrary polynomial f (the problem of Gel'fand) was solved by means of powerful tools in algebraic geometry, in particular, the theorem of Hironaka on resolution of singularities of analytic hypersurfaces. The poles of this extension are located on finitely many arithmetic progressions with difference equal to 1. The coefficients c.,, in the Laurent expansions at these poles are certain generalized functions whose supports determine an algebraic stratification of the set of roots of the polynomial f. This stratification, like the arrangement of the poles, undoubtedly plays an important role in the algebraic geometry of this set. However, these deep questions have not yet received enough investigation. The progress in the theory of partial differential equations achieved on the basis of methods in the Sobolev-Schwartz theory has led to the statements of more general problems going beyond the bounds of classical questions. This development resulted in the formulation of a general problem that in

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the language of differential equations is the problem of exponential representation of solutions of equations with constant coefficients, while in the dual language (in the sense of duality of vector spaces and simultaneously in the sense of Fourier duality) it is called the "fundamental principle", following Ehrenpreis. The exponential representation is a distant analogue of Euler's result that every solution of an ordinary differential equation with constant coefficients can be written as a sum of exponentials satisfying the same equation. In the case when the characteristic equation has multiple roots this representation includes corresponding exponential polynomials. The problem is to determine whether every solution of a partial differential equation on R" with constant coefficients or of a system of such equations can be represented as an integral with some measure over the manifold of exponentials of the form exp( x), E (R')', or of exponential polynomials satisfying

the same system. Such a formulation does not use (not explicitly, at least) the theory of generalized functions, and, formally speaking, could have been posed by Euler himself two hundred years ago. However, this problem was actually formulated only at the end of the 1950s as a result of progress in the theory. At the same time, physicists already were making broad use of the possibility of exponential representation for diverse equations of wave type in the theory of quantum fields, with results from the theory of generalized functions as justification. The difficulty in the general statement of the problem consists, in particular, in the problem of how to understand the integral itself. Indeed, an exponential representation in the trivial case when the differential operator is equal to zero is the decomposition of an arbitrary function on R" or on a domain in this space into an integral with respect to exponentials with linear phase functions, i.e., into a Fourier integral. As we noted, the decomposition of functions of arbitrary growth at infinity into a Fourier integral was obtained only as a result of the development of the theory of generalized functions. The problem of exponential representation and a circle of connected problems were solved at the beginning of the 1960s (Palamodov (233], Ehrenpreis

[63]); in particular, it was established that any solution of an arbitrary system of differential equations with constant coefficients on an arbitrary convex

domain U c R" can be decomposed. The conclusion of this result, briefly speaking, is based on a combination of two methods: 1) the development of the theory of the Fourier transformation for functions of arbitrary growth (or functions defined only on the domain); 2) the development of algebraic geometry. As we noted, the Fourier transform of a rapidly increasing generalized function is no longer a generalized function, but is a functional defined on the space Z of entire functions satisfying specific growth restrictions at infinity. Such functionals cannot be localized, because there are no partitions of unity in Z . If we want to describe the functionals annihilated by the operation of multiplication by a polynomial (a special case of the problem of

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exponential representation), then we must employ different methods. One of these methods uses a construction similar to that used in sheaf theory: the space Z is imbedded in the right resolution formed by the holomorphic tech cochains that are obtained by the same restrictions of growth at infinity. A variant of this method employs the flabby resolution formed by the smooth

8-differential forms on C". On this path there arose a combination of the language of the theory of generalized functions with methods in sheaf theory and homological algebra. One more new idea was included in the theory of generalized functions

at the beginning of the 1960s, though in essence it goes back to the work of Cauchy and Sokhotskii. The well-known formula of Sokhotskii can be interpreted as the equality of the generalized functions 1

2-.i I

-

1

x-i0 x+i0

'

(3)

where I/(x

i0) are the limit values of an analytic function on the real axis from the direction of the lower and upper half-planes, respectively. In general, if f is a function analytic on C \ R, then it is possible to consider its limit values f(x i0) E 2' on the real axis from two directions (under the condition that such values exist). It turns out that every generalized func-

tion can be written in the form u = f(x - i0) - f(x + i0) with a suitable analytic function f (an analytic representation); in particular, the equality (3) is an analytic representation of the delta function. For analytic representation of the generalized functions of several variables one uses various tubular coverings of the space C" \ R", and every function defined on R" or on a domain in this space has at least one analytic representation. Starting from such representations, Martineau [180] constructs the left resolution of the space of generalized functions on R" formed by the tech cochains on the indicated tubular covering whose coefficients are holomorphic functions having at most power growth with respect to the distance to R" (the condition for the existence of limit values on R"). In this language the theorem on analytic representation is an isomorphism of the space 2' and the (n -1)-dimensional cohomology group of the cochain complex described (the equality to zero of the (n - 2)-dimensional cohomology is a variant of the well-known edge-of-the-wedge theorem of Bogolyubov [326]). Closely connected with this realization of generalized functions is the idea of hyperfunctions proposed earlier by Sato [260]. By definition, a hyperfunction of Sato is an element of the (n - 1)-dimensional cohomology group of the space C" \ R" with coefficients in the sheaf of holomorphic functions. By comparing this definition with analytic representations of generalized functions it is possible to express the idea of the hyperfunction as follows: this is a combination of limit values of R" of holomorphic functions that do not in general have such limit values (since no restrictions are imposed on their growth near R"). The Sato hyperfunctions are the most distant gener-

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alization of the concept of a Sobolev generalized function, in some respects too distant, because the Sato theory not only does not have the concept of a support, but also does not have any reasonable topology or at least convergence. In the space of hyperfunctions there is not the usual analysis, but on the other hand there are unusual algebraic properties and sheaf properties; in particular, the sheaf of hyperfunctions is flabby, i.e., every section over

an arbitrary open subset U C R' has an extension to a hyperfunction defined on the whole of R" . The powerful impetus given to analysis by the theory of generalized functions led to the emergence of new directions containing concrete achievements and methods using unusual combinations of analytic, topological, and algebraic ideas. The theory of generalized functions itself, opening in these directions, has become to a large extent a language that is universal for the whole of analysis. This language has become as necessary as the language of Banach spaces. The role of the theory of generalized functions as a universal language and a meeting point of diverse ideas has few parallels in mathematics. In algebraic topology a similar role is played, perhaps on a lesser scale, by homological algebra as a unified language and a collection of general methods. In algebraic geometry a similar place is occupied by the theory of Grothendieck schemes. Although the theory of generalized functions now has more than a half century of history, I submit that the possibilities for its development and influence on analysis are not yet exhausted. V. P. Palamodov

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332. S. K. Vodop'yanov, V. M. Gol'dshtein, and T. G. Latfullin, A criterion for extension of functions of class Lz from unbounded planar domains, Sibirsk. Mat. Zh. 20 (1979), 416-419; English transl. in Siberian Math. J. 20 (1979). 333. S. K. Vodop'yanov, V. M. Gol'dshteln, and Yu. G. Reshetnyak, On geometric properties offunctions with first generalized derivatives, Uspekhi Mat. Nauk 34 (1979), no. 1 (205), 17-65; English transl. in Russian Math. Surveys 34 (1979). 334. L. R. Volevich and V. Ya. Ivrit, Hyperbolic equations, in: I. G. Petrovskil, Selected works. Systems of partial differential equations. Algebraic geometry, "Nauka", Moscow, 1986, 395-418. 335. L. R. Volevich and B. P. Paneyakh, Certain spaces of generalized functions and imbedding theorems, Uspekhi Mat. Nauk 20 (1965), no. 1 (121), 3-74; English transl. in Russian Math. Surveys 20 (1965). 336. H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 1940, no. 7, 411-444. 337. G. N. Yakovlev, Boundary properties of functions of the class W(2) on domains with angular points, Dokl. Akad. Nauk SSSR 140 (1961), 73-76; English transl. in Soviet Math. Dokl.

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338. A. Yoshikawa, On abstract formulation of Sobolev type imbedding theorems and its ap plications to elliptic boundary value problems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17 (1970), 543-558. 339. V. 1. Yudovich, On certain estimates connected with integral operators and solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961), 805-808; English transl. in Soviet Math. Dokl. 2 (1961).

Subject Index weak, 27 Completeness, 10 54, 64 of

Anisotropic spaces, 210, 213 Arzela's theorem, 27, 30 Averaged equation, 187, 216 Averaged functions, 12, 141 derivatives of. 33

weak, 27

Cone condition, 205, 210 flexible, 202, 205

Bicharacteristics, 146, 147, 152. 219, 255 Bolzano-Wcierstrass principle, 27

Continuity in the large, II

Convergence, 24

strong, 10, 26, 239 Cauchy problem, 129, 218 classical, 193, 198. 252 existence of solution of, 182, 183 for an even number of variables, 177 for an odd number of variables, 179 for degenerate hyperbolic equations,

weak, 24, 26, 239, 247, 249 Cubature formulas, 255

Dieudonne-Schwartz theorem, 261 Dirichlet problem, 81, 254 solution of, 85, 87 Division problem, 254 Duhamel principle, 257

215 for higher-order hyperbolic equations,

215

for hyperbolic equations, 214, 217 for hyperbolic differential equations.

c-net, 29

Equivalent functions, 2 Euler equation, 96 Extension theorem, 213

259

for linear equations of hyperbolic type, 146, 154

for linear equations with smooth coeffi-

Frbchet space, 261 Functional, 16

cients, 162

for the space of functionals, 243 for the wave equation. 129. 215, 217

bounded, 16 linear, 16, 21, 240

generalized, 137

norm of. 16

generalized. 144, 181, 192, 244, 245, Gauss-Ostrogradsky formula, 184 Gel'fand-Shilov spaces, 262 Generalized derivatives, 32

251

Cauchy-Bunjakovsky inequality, 132 Cauchy-Kovalevskaya theorem, 257 Characteristic conoid, 147, 153, 169, 215, 219, 220, 255 Characteristic surface, 146 Clarkson's inequalities, 17 first, 17

existence of, 33 Generalized functions, 253 of slow growth, 253

Hadamard integral, 262, 263 Hadamard's example, 90. 214 Hahn-Banach theorem, 261 Holder inequality, 3, 5 generalized, 4

second, 19 Compactness, 27 of integrals, 71, 73

strong, 27. 29

285

SUBJECT INDEX

286

reverse, 7 Hopf-Galerkin method, 216 Horn condition, 203, 210 Hyperbolic space, 247 inverse, 249

Hyperfunctions, 254, 266

Imbedding theorems, 54, 57, 64, 215, 255 Inner integral, I Integral representations of functions, 51.

203, 204 Integrals of potential type, 39, 204

Kondrashov's theorem, 70 Lebesgue integral, 2

Lebesgue measure, 2 Leray's method, 216 Lizorkin-Triebel spaces, 210

imbedding, 55, 70 complete continuity of, 77, 80, 126, 209

geometric characteristics of, 212 in polar coordinates, 156

integral, 167 adjoint, 172, 175 inverse, 169, 245 projection, 45, 57, 59 spherical, 61, 62 pseudodifferential, 262 wave, 129, 264 generalized, 138 Orlicz spaces, 211

Permissible boundary values, 98 Permissible function, 82 Poincare's inequality, 64 Polyharmonic equation, 95, 96 basic boundary value problem for, 103

Malgrange preparation theorem, 261 Method of successive approximations, 169,

Riesz-Fischer theorem, 10

235

Minimizing sequence, 82, 117 Minkowski inequality, 3, 6 for numerical series. 6 reverse, 7

Saint-Venant principle. 214 Separability, 16 Sobolev spaces, 254, 258 Sobolev-Schwartz theory, 254, 257, 260, 264

Neumann problem, 92, 94 solution of, 94, 95 Nikol'skii-Besov spaces, 210

Norms, 8 equivalent, 9, 57,

Space-like, 151

Summable functions, 1, 2 Summable pair of domains, 38

in Lo ', 43

Time-like, 151 Trace theorems, 255 Triangle inequality, 9, 43

in S1, 46 in 47, 69

Ultradistributions, 254

59

in Lp, 10, 16

in a function space, 10

mixed, 205 natural, 63 permissible, 8

Operator, adjoint, 162, 241 averaging, 199 Fourier, 262

Variational problem, 81, 213 solution of, 82, 93, 98 Volterra integral equation, 215 Weighted spaces, 21 I, 212

Weyl's lemma, 214

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