The Cauchy problem for Douglis-Nirenberg elliptic systems of partial differential equations

Abstract

Several partial answers are given to the question: Suppose U is a solution of the Douglis-Nirenberg elliptic system L U = F LU = F where F is analytic and L has analytic coefficients. If U = 0 U = 0 in some appropriate sense on a hyperplane (or any analytic hypersurface) must U vanish identically? One answer follows from introducing a so-called formal Cauchy problem for Douglis-Nirenberg elliptic systems and establishing existence and uniqueness theorems. A second Cauchy problem, in some sense a more natural one, is discussed for an important subclass of the Douglis-Nirenberg elliptic systems. The results in this case give a second partial answer to the original question. The methods of proof employed are largely algebraic. The systems are reduced to systems to which the Cauchy-Kowalewski theorem applies.