A criterion of solvability of the elliptic Cauchy problem in a multi-dimensional cylindrical domain

TL;DR

This paper establishes a criterion for the solvability of the multidimensional elliptic Cauchy problem in cylindrical domains, linking ill-posedness to spectral properties of associated operators.

Contribution

It introduces a spectral expansion method to determine strong solvability and characterizes ill-posedness via the spectrum of a self-adjoint operator with deviating argument.

Findings

Spectral expansion method for elliptic Cauchy problems

Criterion linking ill-posedness to continuous spectrum

Characterization of solvability in cylindrical domains

Abstract

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.