Exotic parametrization problems (Q1337581)

We study the problem of parametrizing solutions of an overdetermined system of linear partial differential equations \(\{P_ j(X,D)f=0\}\). In the classical parametrization problems, such as the Cauchy and Dirichlet problems, the data are given on a manifold \(M\) whose dimension \(d\) is well-determined by the equations. In exotic problems the data are given on manifolds with dimension \(\neq d\). If the data are given on a manifold \(M\) of dimension \(>d\) then the data must satisfy certain equations. If \(M\) is a hyperspace then these equations are differential equations and there are finitely many of them. In case \(\dim M<d\) infinitely many data are prescribed. Explicit examples are given for the wave equation on a time-like line and on a space-like line. An extension of the ``thin edge-of-the-wedge'' theorem to partial differential equations is given.