Incomplete Systems of Partial Differential Equations

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666

P'ROC. N. A. S.

6 Cf. Whyburn, G. T., "Concerning the Complementary Domains of Continua," forthcoming in the Annals of Mathematics, theorem 15. For the theorem for the case where the sets RI, R2 and Rs are open subsets of M, see Whyburn, G. T., these PROCEEDINGS, 13, 650-657 (1927), theorem 3. 6 Moore, R. L., Trans. Amer. Math. Soc., 21, 333-347 (1920). 7 Moore, R. L., and Kline, J. R., Annals of Mathematics, 20, 182-223 (1919). 8 Cf. Whyburn, G. T., these PROCEEDINGS, 13, 31-38 (1927) and Bull. Amer. Math. Soc., 33, 305-308 (1927). 9 This follows from the fact that K is the M-boundary of each of the sets R, and Rb. Cf. the abstract of my paper "On Irreducible Cuttings of Continua," presented to the American Mathematical Society, December 28, 1927. 10 Wilder, R. L., Fund. Math., 7, 340-377 (1925), theorem 1. 11 Moore, R. L., Trans. Amer. Math. Soc., 27,416-428 (1925). 12 Lubben, R. G., Bull. Amer. Math. Soc., 32, 114 (1926) (abstract). 13 Moore, R. L., Bull. Amer. Math. Soc., 29, 291-297 (1923). 14 Moore, R. L., Math. Zeit., 15, 254-260 (1922). 15 Cleveland, C. M., these PRocEzRDiNis, 13,275-276 (1927). 16 Moore, R. L., Ibid., 13, 711-716 (1927), theorem 2. 17 Ayres, W. L., Bull. Amer. Math. Soc., 33, 565-571 (1927), theorems 4 and 5. For an additional hypothesis needed in these theorems, see a note in the Bulktin of the American Mathematical Society, 34, 107-108 by W. L. Ayres. It is of interest to note that while, as shown by theorem 8 of the present paper, Ayres' condition (4) in his note makes the point set K such that it is a subset of a simple closed curve, his condition (4') does not do this. His condition (4) implies (4') but (4') does not imply (4).

INCOMPLETE SYSTEMS OF PARTIAL DIFFERENTIAL

EQUATIONS' BY JOSZPH MILLIR THOMAS DXPARTMNTr OF MATHMATrICS, UNIVERSITY OF NEUNNSYIVANIA

Communicated July 5, 1928

The theory of a system of total differential equations

_a aJxi

=

aa

a =

1,2,..., r; i = 1, 2,... ,n

(1)

which are completely integrable, that is, for which the conditions

baai+ rbaa aXj X = 1 aU+

=

+

j a2) Jaaj

au, api

='1 f(2)

are satisfied identically in u and x, is known to correspond exactly to the theory of a jacobian system of simultaneous linear homogeneous equations,2 but the usual treatment of the incompletely integrable case for the two

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667

types of system does not preserve this correspondence. It is the purpose of the present paper to give a treatment displaying the equivalence of the two sorts of systems in the incompletely integrable case. Consider a simultaneous system of n independent linear homogeneous equations of the first order in a single unknown function f. Without loss of generality we can assume the system in the form

Xtf _-+ axj

EI a,,iauCr-

O.

a=

l1 2, . .., n.

i

(3)

Any solution of (3) will also satisfy

(Xi, Xj)f

E bij

(4)

= °,

where the parentheses denote the Poisson operator. Equations (4) contain none of the derivatives if/axi and, therefore, cannot be algebraic consequences of (3). Either they are identically satisfied, or the system is incomplete. In the former case, the coefficients of af/lu in (4) vanish, that is, the relations

iJ,,i bai+ Er

?bXj

5=1 I

ba* ba,j 6UD

r ?a* + E baas

bJXi

a

10=i

a

(5

,

U86

are satisfied identically in u and x. System (3) is then jacobian and has r integrals distinct with respect to the variables ui, u2, . ..., u,. In the latter case, equations (5) constitute additional conditions on f, and (3) has fewer than r distinct integrals of the type ordinarily considered, namely, functions of u and x each of which substituted in (3) for f reduces the left members to zero identically in u and x. We may seek, however, an integral of (3) in an extended sense, that is, a function f and a set of relations among the variables u and x having the following properties: the relations are distinct with respect to the 14's; and whenf is substituted in (3), equations (3) are satisfied, not necessarily identically, but as a consequence of the relations among u and x. Any such integral must satisfy (4) in the same manner. If we require that (3) have r integrals of this type distinct with respect to the u's, then regarding i and j as temporarily fixed in the linear homogeneous equations

c we

r

h -is a

o I

conclude from the non-vanishing of

X*

I(flbu| thlat bpfi;

= 0 =

O,

or that

(5)

PROC. N. A. S.

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668

must be satisfied. We denote this system by F1 = 0, and build up a sequence of relations

F1 = 0, F2

=

0, ..., FN = 0,

(6)

in which the set Fa comprises all the equations XS(Fa- 1) = 0. Since = 0, our integrals must satisfy (6). There must exist an integer N for which the corresponding systems (6) are equivalent to a system

X,(0)

fe(u, x)

=

=k +1, ...

0,

(7)

r

which can be solved for r-k of the u's, say Uk + 1, . . .* ur in terms of the remaining u's and the x's and whose solution substituted in the equations of FN + 1 = O converts them into identities in x and ul, u2, . .., Uk. To prove this a necessary condition for the existence of the integrals of (3), let us choose any value for N, and let the functional matrix of the corresponding system (6) with respect to the u's have the rank 1. We can solve certain of equations (6) for I of the u's, say ur - + 1 , Ur. The results, on being substituted in any of equations (6), give identities in u..., u, - I, and hence must also give identities in x because no relation among the x's is implied by the set of relations among u and x admitted. The expressions for ur I + 1 ..., ur when substituted in FN + 1 = 0 must give either identities in ui, ..., u, -_ and x or relations involving ul, .... ut _ I since relations among the x's alone are excluded. If nothing but identities result from the substitution, the N in question has the properties announced at the beginning of this paragraph, and k = r-1; if some relations in u,, u2, . . ., U, - I result, then these new relations treated as (6) were will determine certain of the remaining u's. We substitute them together with the u's previously obtained in FN + 2 = 0. This process must ultimately lead to nothing but identities because there is only a finite number of u's. Conversely, suppose that this necessary condition is fulfilled. By hypothesis, any solution of (7) satisfies XjFO = 0, that is, -

p+ E

aj

-

+ -

ay+

= °

(8)

Differentiating (7) and substituting bf,9/6x and bfObu.,' = 1, 2, . .., kV

in (8) we get

ai_bu -f +~~~~~~~~ ~ -

p a7

=

O,t

k + 1,

r

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669

whence

a= a

+ L

-y =

a,

k + 1, ..., r.

(9)

Suppose the values of Uk + 1' . X* Ur substituted in the coefficients of (3) and seek a function f of x, ul, U2, ... Uk alone satisfying the resulting equations. The equations defining f will be

af

+

ZAPS =0°

(10)

where the A's represent what the a's become after the substitution. The conditions that (10) be jacobian are

+Api+E aux = ?A^ B1 A aApj axj axi

+

i Iz= A au,

(11)

Now we have '

)

axj

p

ba= + PiU aIU,pi~ au, M

bapi bup ai+ , = k + I au, axj axj r

f a'S appi(U k + 1 aUJB aUA

it being understood that Uk + 1, ... ur have been replaced by their values from (7) in the right members. The left member of (11) is therefore

bapi+

axj

,6=1

a

u

+

r

+

X=k+l

bXj

ua.api)

I.= bux buo

By use of (9), this becomes ?)Xja',+

I

apau u

Consequently (11) are simply certain ones of (5) with uk + 1 .. I, ur.eliminated by means of (7). But by hypothesis such elimination leads to identities in x, ui, ..., Uk. Hence, system (10) is jacobian and admits k integrals fi, f2, . ., fk distinct with respect to ul, U2, .. ., Uk. These taken together with (7) will be called a complete set of integrals. It is to be noted that this set reduces to the ordinary complete set if the system (3) is jacobian. We have, therefore, proved THIORZM. A necessary and sufficient condition for the existence of r independent integrals (in the extended sense) of (3) is the existence of an N such that equations (6) are consistent for the determination of some or all of the u's and such that their solution satisfies FN + 1 = 0.

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670

fi- c is, of course,

an

PRoe. N. A. S.

integral of (10), cl being an arbitrary constant.

Put

fi

- cl = 0,

f2

- c2 =

0 .

fk

Ck =O;

(12)

these equations and (7) can be solved to give ul, u2, ..., s, as functions of x and the arbitrary constants c. To apply these results to system (1), suppose u's satisfying (1) are defined by a set of relations

fa(u,X) =0.

a=

1,2,...,r.

(13)

There results

-+ j

a#i, OL,

(14)

that is, the fi, f2, ...,fr constitute a set of integrals of (3) of the type just considered. Consequently, to every system (1) there is an equivalent system (3), and vice versa; and the general solution of (1) is given by (7) and (12). The integrability conditions (2) and (5) are identical. If a system of type (3) is jacobian, the corresponding system (1) is completely integrable. Therefore, if the relations like (6) be employed to eliminate Uk + 1- Ur from those equations of system (1) which contain derivatives of ul, u2, Uk, the resulting system in u1, U2, ..., u is completely integrable, and any solution of it taken with Uk + 1 ***, U, as determined by the relations like (6) satisfies all the equations of (1). If we note that applying the operator Xi to a function of u and x is the same as differentiating that function with respect to xi and eliminating first derivatives by means of (1), we see that the theorem given above is equivalent to a known theorem3 for systems (1). We therefore have developed a theory of systems (3) which corresponds to the theory of systems (1) in the incompletely integrable case. It would seem that no extension of the theory of systems (1) is possible to correspond to the treatment of system (3) by adjoining to it (4) and finally arriving at an equivalent jacobian system: for in the treatment of (1) we are only interested in satisfying (14) by virtue of (13) and not in satisfying (14) identically. 1 The first draft of this paper was written while the author was National Research Fellow at Princeton University. 2 See Goursat, Lepons sur l'integration des equations aux derivkes partielles du premier ordre, Paris, 1921, p. 103. 0O. Veblen and J. M. Thomas, Annals Math., 27, 1926 (288).