The Cauchy problem in $\textbf {C}^ N$ for linear second order partial differential equations with data on a quadric surface

Abstract:

By means of a method developed essentially by Leray some global existence results are obtained for the problem referred to in the title. The partial differential equations are required to have constant principal part and the initial surface to be irreducible and not everywhere characteristic. The Cauchy data are assumed to be given by entire functions. 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 Ksets, allow unbounded singularities. The L-sets are in turn divided into three types: initial, asymptotic, and latent singularities. The initial singularities appear when the characteristic points of the initial surface are exceptional according to Leray’s local theory. These sets of singularity intersect the initial surface at characteristic points. The asymptotic case, where the set of singularities does not cut the initial 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 initial surface, but where the solutions do not develop singularities initially. In the case of the Laplace equation with data on a real quadric surface it is shown that the K-singularities and the asymptotic singularities occur on the classical focal sets defined by Poncelet, Plücker, Darboux et al. There are also latent singularities appearing in coordinate subspaces of ${\mathbb {R}^N}$. As a corollary a new proof is given of the fact that ellipsoids have the Pompeiu property.