A criterion of solvability of the elliptic Cauchy problem in a   multi-dimensional cylindrical domain
  In this paper we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument. 
Tynysbek Sh. Kalmenov, Makhmud A. Sadybekov, Berikbol T. Torebek
math.AP