Uniqueness in the Cauchy Problem for a Degenerate Elliptic Second Order Equation

There is by now a large literature devoted to the question of local uniqueness in the Cauchy problem for linear partial differential equations (or having linear leading part) — assuming the boundary is non-characteristic. The lecture notes by C. Zuily [4] give an excellent survey and presentation of many of the recent results, as well as very complete references. Various classes of sufficient conditions have been proved and under certain circumstances these are close to necessary. In addition to the references in [4] there is soon to appear [1] by S. Alinhac with further counterexamples and results.