This paper represents results obtained under Contract K6ori-201, T. 0. NO. 1, sponsored 'The concept of a "properly solvable Cauchy problem" will he discussed prrsently.hv the Office of Naval Research.
I35 136A. LAX @') the greatest common divisor of p , ( z ) , ap,(z)/az, . . . and ak+, (z)/azk divides #m-k (2) (K = 1, 2, ---, m -1).Incidentally, we observe that, given the principal part of L[u], the set of all lower order terms permitted by part (b) of (A) form a linear space. *Main reszllt for variable coegcientscriteria (A') and (C). Let L[zl] be a linear operator of order m with sufficiently differentiable coefficients. Denote the first order operator a/at-A,(x,t)a/ax by ai. Cauchy's problem for L[%] = 0 is properly solvable if L[u] can be written as (A') L [u] = a? a2 --* 82 [u] + 3P-l a2-l * * a?-' MI [zt; + * -* + 8p-z .. . ap-'M,[zll + . . .where li (x, t ) are real functions and the vi are constant integers2 such that 2 vi = m. In case the coefficients are constant, (A') and (A) are identical. If they are variable, a term of L[%] of order m -I, I < v1 is no longer homogeneous as a polynomial in a/ax, a/at but includes lower order terms. The following equivalent formulation of (A') given by R. Courant, is actually used in the proof: Denote the m-th order operator spa> ---8:. by 17,. From Illrn, form operators 17,/ai of order m -1 by omitting one factor ai (i = 1, 2, . * * , K ) at a time. Similarly, from each Drn/a,, form operators IIrnla, a, of order m -2 and so on, until we arrive at zero order operators. The aggregate of all operators so obtained from Urn span a module S over the ring of smooth functions of x, t. In Part 11, 5 4, we prove: if L [u] has sufficiently differentiable coefficients and can be expressed by (C) then Cauchy's problem for L [21] = 0 is properly solvable. In 3 5, the equivalence of (A') and ( C ) is established.
Remarks about Criterion ( A ) .For equations with constant coefficients and any number of independent variables, L. Giirding (see [4]) has given a necessary and sufficient condition for the proper solvability of Cauchy's problem. It refers to the behavior of the roots of polynomials of degree m (m being the order of L [ u ] ) and is therefore not easy to verify. In the case of two independent variables, the immediately verifiable criterion (A) is equivalent with Girding's condition. Moreover, with the aid of a method developed by R. Courant [u] with N E S , PFor convenience, we order the pI so that v1 2 v2 2 . . . 2 -vk 2 1.
EQUATIONS WITH MULTIPLE CHARACTERISTICS
137(see [3]), criterion (A) can be used in treating more independent variables.The sufficiency of (A) can be proved by Fourier synthesis. However, it can also be proved by the method of characteristics, either via a reduction of our m-th order equation to a first order diagonal systems, or directly, as presented in Part 11, by an iteration process of the Picard type based on an estimate.* The advantages of the latter approach are, that it is valid for arbitrary non-characteristic initial curves, that the domai...