THE CAUCHY-PROBLEM IN CN FOR LINEAR 2ND-ORDER PARTIAL-DIFFERENTIAL EQUATIONS WITH DATA ON A QUADRIC SURFACE

By means of a method developed essentially by Leray some global existe nce results are obtained for the problem referred to in the title. The partial differential equations are required to have constant principa l part and the initial surface to be irreducible and not everywhere ch aracteristic. The Cauchy data am assumed to be given by entire functio ns. Under these conditions the location of all possible singularities of solutions are determined. The sets of singularities can be divided into two types, K- and L-singularities. K, the set of K-singularities, is the global version of the characteristic tangent defined by Leray. The L-sets are here quadric surfaces which, in contrast to the K-sets , allow unbounded singularities. The L-sets are in turn divided into t hree types: initial, asymptotic, and latent singularities. The initial singularities appear when the characteristic points of the initial su rface are exceptional according to Leray's local theory. These sets of singularity intersect the initial surface at characteristic points. T he asymptotic case, where the set of singularities does not cut the in itial surface, can be viewed as projectively equivalent to the initial case, the intersection taking place at infinite characteristic points . Finally the latent singularities are sets which intersect the initia l surface, but where the solutions do not develop singularities initia lly. In the case of the Laplace equation with data on a real quadric s urface it is shown that the K-singularities and the asymptotic singula rities occur on the classical focal sets defined by Poncelet, Plucker, Darboux et al. There are also latent singularities appearing in coord inate subspaces of R(N). As a corollary a new proof is given of the fa ct that ellipsoids have the Pompeiu property.